EXTREME VALUE THEORY FOR NON~STATIONARY SEQUENCES
WITH APPLICATIONS TO AIR POLLUTION STANDARDS
by
Gregory Anthony Feeney
Department of Eiostatistics
University of NQrth Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1410
July 1982
EXTREME VALUE THEORY FOR NON-STATIONARY SEQUENCES
WITH APPLICATIONS TO AIR POLLUTION STANDARDS
by
Gregory Anthony Feeney
A Dissertation submitted to the faculty of the
University of North Carolina at Chapel Hill in
partial fulfillment of the requirements for
the degree of Doctor of Philosophy in the
Department of Biostatistics
Chapel Hill, 1982
Reader
Reader
Reader
ii.
ABSTRACT
GREGORY ANTHONY FEENEY.
Extreme Value Theory for Non-Stationary
Sequences with Applications to Air Pollution Standards. (Under the
direction of PRANAB KUMAR SEN and MICHAEL JOSEPH SYMONS.)
The theory on extreme values has been extended to accommodate
certain non-stationary sequences.
Of particular interest are -sequences
of daily maximum ozone concentrations which, while exhibiting obvious
non-stationarity) have previously been treated as either stationary or
independent and identically distributed.
We restrict consideration to those non-stationary sequences which
can be divided into sub-sequences of independent and identically
distributed observations, although we allow for stochastic dependence
between the sub-sequences.
We then extend the classical extreme value
theory to accommodate this dependence structure between the subsequences and determine that, under very reasonable conditions, the
extreme values for each subsequence are asymptotically jointly
independent.
The above result enabled us to obtain expressions for the
limiting distribution of the k-th extreme value for the entire sequence
of observations, for
k fixed.
We extended these results to the
multi-variate situation and considered the case where the sub-sequence
sample size is a random variable.
We also determined the joint limiting
distribution of the sub-sequence maxima.
The above theoretical results were applied to a three-year
sequence of daily maximum ozone concentrations.
The estimation was
conducted under the assumption of two parametric models;
and log-normal;
the Weibull
as well as Cox's proportional hazards model, which
provided a partial non-parametric estimate of the parent distribution.
iii.
These results were compared and the differences between the classical
and the sub-sequence approach was considered.
It was shown that the sub-sequence assumption is justifiable and
the parametric approaches were comparable.
The use of Cox's model has
potential when we are interested in the distribution conditional on
some sets of values of a covariate vector.
i v.
TABLE OF CONTENTS
Acknowledgements
Chapter I:
vi
INTRODUCTION AND LITERATURE REVIEW
1
1.1
Introduction
1
1.2 Historical Remarks
4
1.3 Extreme Value Theory
6
1.3.1
Classical Extreme Value Theory
1.3.2
Extreme Value Theory for Non-identically
Distributed Random Variables
19
Extreme Value Theory with Dependent
Observations
21
1.3.3
1.3.4 Multivariate Extreme Value Theory
Chapter II:
6
31
1.4 Outline of Subsequent Chapters
35
EXTREME VALUE THEORY FOR CERTAIN NON-STATIONARY SEQUENCES
39
2.1
39
Introduction
2.2 The Extreme Value Distribution of the Maximum of
the Sub-sequence Maxima
41
2.3 The Distribution of the k-th Extreme
63
2.4 The Multivariate Extreme Value Distribution
66
2.5 The Joint Extreme Value Distribution
70
2.6 The Effects of Random Sample Sizes on the Extreme
Value Distribution
71
Chapter III: ESTIMATION TECHNIQUES
75
3.1
Introduction
75
3.2
Parametric Models for Air Pollution Data
76
v.
Chapter IV:
Chapter V:
3.3 Parametric Estimation Techniques
78
3.4 Partial Non-Parametric Estimation
89
APPLICATIONS
107
4.1
107
Introduction
4.2 The Parametric Models
111
4.3 The Partial Non-Parametric Models
118
SUMMARY AND RECOMMENDATIONS FOR FUTURE RESEARCH
141
5.1 Summary
141
5.2 Recommendations for Future Research
143
BIBLIOGRAPHY
145
APPENDICES
151
e
vi.
ACKNOWLEDGEMENTS
I wish to thank the co-chairmen, Drs. P.K. Sen and M.J. Symons,
for their invaluable assistance in the course of this dissertation.
In
particular, I appreciate the suggestion of the topic by Dr. M.J. Symons
and the proposal by Dr. P.K. Sen of applying Coxls proportional hazards
model.
I thank the other members of the committee;
Dr. H.E. Jeffries and Dr. W.J. Nelson;
Dr. T.C. Curran,
for their opinions and their
time.
This research has been made possible by the financial assistance
provided by the Australian Government Public Service Board.
I am
grateful for their help and wish to thank Dr. Don MacRae for his
assistance in obtaining the award.
No amount of money or advice can compare with the support that I
have received from my family.
The love and understanding shown by my
wife, Susan, was an invaluable motivation for my work.
My sons, Simon
and Nicholas, made life more satisfying with their tenderness and
humour.
I thank them very much for their patience.
I would also like to thank Dr. Tom Curran for supplying me with
the air pollution data and Dr. Bill Nelson for his assistance in
obtaining the meteorological data.
Mr. Larry Truppi
the meteorological data is gratefully acknowledged.
IS
help in providing
Mr. Yang Yuan
provided much needed assistance in the use of the MAXLIK sub-routine.
I thank him very much.
The typing of this manuscript by Miss Sue Turpie
is also greatly appreciated.
Last, but not least, I would like to thank
all of the friends I made in the Department of Biostatistics.
They made
me feel at home and provided many hours of enjoyable discussion.
I.
1.1
INTRODUCTION AND LITERATURE REVIEW
Introduction
The aim of a statistical theory of extreme va"lues is to analyze
observed values of a variate with a view to characterizing the extreme
values within these observations.
The k-th lower extreme will be
defined to be the k-th smallest observation, while the k-th upper
extreme is the k-th largest observation.
are the minimum and the maximum.
Therefore, the first extremes
The emphasis in the above definition
is that we speak of the k-th extremes in a limiting sense:
in advance and the sample size,
n,
k is fixed
increases indefinitely.
The
extremes are, therefore, not fixed but are new statistical variates
dependent upon the initial distribution and the sample size.
Although
exact results can be obtained, the case with independent observations
is much more tractable than when the observations are dependent.
We
will focus on the asymptotic theory of extreme values, principally
because the results so obtained are, in some respects, distribution
free.
The use of asymptotic theory can produce very simple and compact
characterizations of the distribution of extremes and have practical
applications.
Perhaps uniquely amongst statistical theories the classical
extreme value theory, as described by Gumbel (1958) has been widely
accepted and applied by a broad spectrum of scientists and engineers
(Harter (1978)).
Situations in which a mathematical solution to the
2.
problem involved is expressed in terms of the largest or the smallest
observation abound.
Floods, heavy rains, extreme winds, extreme
atmospheric pressure and other natural phenomena can, if society is not
prepared, cause extensive human and economic losses.
Engineering
decisions, for example, based on a very accurate theory can minimize
the effects of such natural disasters.
We are all aware of the saying
that a chain is only as strong as its weakest link.
Similarly the
breaking strength of materials can be analyzed through the utilization
of extreme value theory.
Recently society, in the industrialized
countries, has been concerned, for health reasons, with keeping the
maximum concentration of a pollutant in the air below some predetermined
level.
Again, we see that consideration of extreme value theory can
assist in the design and implementation of a suitable air pollution
standard.
expensive.
In all of these cases inaccuracies can be disastrously
Other applications are in the failure of equipment, service
time, corrosion, aeronautics, breakdown voltage, geology, naval
engineering, oldest ages and even statistical samples and estimators.
Therefore, any successful theory of extremes unifies a great number of
interesting and socially relevant topics.
One essential condition for the analysis of extremes is that the
distribution from which the extremes have been drawn, and its parameters,
must remain constant in time (or space), or the influence that time (or
space) exercises upon them must be taken into account or eliminated.
For most applications this can be accomplished.
Gumbel (1958) further cited the limitation that the observations
from which the extremes are taken should be independent.
conceded that this assumption is hardly ever realized.
Gumbel further
Brief consider-
3.
tion of the above applications leads us to agree with Gumbel.
Gumbel
then proceeds to justify the independence assumption by proposing the
idea of "distributions based on dependent events should share the
asymptotic properties of distributions based on independent trials
being not teo far fetched.
ll
as
We are not prepared to make this assumption
and will impose some reasonable dependence structure on the observations
and develop the corresponding extreme value theory.
The approach of
Gumbel is widely used by engineers and scientists;
therefore, it is
important to compare his simpl ified approach with any more rigourous
theory.
An application of particular interest in this dissertation is in
the field of air pollution.
The United States Environmental Protection
Agency (EPA) is beginning to express many of the air quality standards
in probabilistic, rather than deterministic, terms.
For example, the
Ozone National Ambient Air Quality Standards (NAAQS) contain the phrase
"expected number of days per calendar year
ll
,
in relation to the
exceedance of 0.125 parts per million (ppm) for a daily
pollution level.
o~one
maximum
It is in the interest of all parties to Aave the most
exact characterization of the extreme value distribution which is also
reasonably robust to the choice of parent distribution.
With this in
mind we will seek the most appropriate dependence structure on the
observations, taking into account the daily variations in pollution
levels due to human activity, such as traffic flow or energy demands on
power plants, as well as changes in the meteorology of the reg ion.
An inherent problem with applications to air pollution, unlike
almost all other applications, is the dynamic nature of the air shed
containing a large urban area.
Because of this, even if independence
4.
could be assumed, it is not possible to utilize the methods developed
by Gumbel (1958) to estimate the parameters of the extreme value
distribution empirically since no long term data set can be said to
apply to one steady state of the air shed.
Typically we may consider
three years of data as being relevant to one steady state of the air
shed, contingent on there being no major changes in public policy which
could drastically change this state.
In this situation we must estimate
the parameters of the parent distribution, assuming a dependence
structure, and thereby directly obtain the extreme value distribution.
The parameters of the parent distribution are estimated using maximum
likelihood techniques, within the assumed dependence framework.
1.2 Historical Remarks
The interest in the distribution of extremes goes back as far as
applications of laws of chance to actuarial and insurance problems.
Gumbel (1958) remarks that the problem did indeed come up in the works
of N. Bernoulli in the eai'ly eighteenth century.
Galambos (1978) states
that accurate and general solutions are impl icitly contained in the
works of Poisson on the theory of sums and the distribution of rare
events.
Most of the very early, pre-1920, studies started from the normal
distribution which hampered the development of extreme value theory,
since none of the fundamental theorems of extreme values are related,
in a simple way, to the normal distribution.
The first study of largest
values for other distributions was made by Dodd (1923).
His work is
based on "asymptotic" values whi ch are s imil ar to the characteri sti c
5.
largest values introduced by von Mises (1923).
The first paper based
on the concept of a type of initial distribution, different from the
normal one is due to Frechet (1927).
He was also the first to obtain
an asymptotic distribution of the largest value.
Most importantly, he
showed that largest values taken from different initial distributions
sharing a common property may have a common asymptotic,distribution.
He introduced the stability postulate according to which the distribution
of the largest value should be equal to the initial one, except for a
linear transformation.
Frechet's paper, published in a remote journal,
like Dodd's work, did not gain the recognition it merited.
The pre-
occupation of statistics at that time with the normal distribution
greatly contributed to the lack of interest in the work of Dodd and
Frechet.
In 1928 Fisher and Tippett published a paper which has become
the basic work on extreme values.
They used the same stability postulate
and found, in addition to Frechet's asymptotic distribution, two others
valid for other initial types.
The authors stressed the slow convergence
of the distribution of the largest normal value toward its asymptote,
and thus explained the relative sterility of all previous endeavours.
Von Mises (1936) classified the initial distributions possessing
asymptotic distributions of the largest value, and gave sufficient
conditions under which the three asymptotic distributions are valid.
Finally, Gnedenko (1943) unified all the previous results into a
complete and rigourous treatment in which he established necessary and
sufficient conditions under which the three asymptotic distributions
exist.
This sound basis for a theory of extreme values led to a
broadening of the scope of future investigations resulting in the
extensive use of extreme values we see today.
6.
1.3
~xtreme
Value Theory
It was not until the early sixties that attention was directed to
extreme value theory in the dependent case.
Prior to this a great deal
of theory was developed for the classical, or independent, situation.
We will consider these two cases separately, using the classical theory
as an introduction to and a building block for, the theory on dependent
observations.
1.3.1 Classical Extreme Value Theory
Classical extreme value theory is concerned principally with the
distributional properties of the maximum and other upper extremes of n
independent and identically distributed random variables, as
n becomes
large.
Let X1, ... ,X n be a sequence of independent and identically
distributed (i.i.d.) random variables. Let Zn denote the maximum of
the n observations, that is,
(1.3.1)
All results obtained for the maxima apply equally to the minima because
of the simple relation,
This dissertation will, therefore, solely discuss the case for the upper
extreme values, the maximum being one of these.
The k-th upper extreme
value will be defined as the {n-k+1)th order statistic, denoted by
7.
Xn-k+l:n ' for fixed k as n tends to -infinity. We
- will see later
that the applications of interest involve the use of. these statistics.
Considering first the maximum, it is the purpose or extreme value theory
to seek conditions on F(x)
sequences of constants
=
Pr{X j
{an} and
~x}
{b n} ,
to guarantee the existence of
an
positive, known as
normalizing or stabilizing constants, such that
H(x) = lim Pr{Z n ~a nx+b}
n
(1.3.2)
exists for all continuity points of H(x) , where H(x)
degenerate distribution function.
is a non,..
Such convergence is called weak
convergence of distribution functions or random variables and we say
that H is max-stable.
distribution,
is,
FED(H)
F,
In such a case it will be said that the parent
belongs to the domain of attraction of H
The exact distribution function of Zn
that
in the i. i . d.
case, is immediate, namely
(1.3.3)
Why, one may reasonably ask, should large sample methods be used?
justification, briefly, is that it can be shown that
The
H can take one
of three possible general forms, regardless of the parent distribution,
F.
Indeed, one of these can take only negative values of x which
precludes its use in all applications of interest in this dissertation.
The choice of the limiting distribution, in fact, depends only on the
behaviour of the upper tail of the parent distribution,
F,
and with
little information on the global properties of F much can be deduced
8.
regarding the asymptotic properties of the maximum.
Unfortunately, the
above argument is not as sound as it first appears.
Although the form
of the limiting distribution depends only on the upper tail of F
the
unknown parameters of the limiting distribution,
an and bn , depend
on the unknown parameters of the parent distribution. Therefore, in the
application of the above theory, the limiting distribution depends on
the global properties of F by way of the maximum likelihood estimates
of the unknown parameters of F.
Notwithstanding the above consider-
ations, it is still desirable, for reasons of the tractability of the
results as well as the general nature of the technique, to investigate
the large sample distributional properties of the maximum.
The theory of extreme values was developed in a series of works by
many authors, referred to in Section 1.2.
This research culminated in
the comprehensive paper by Gnedenko (1943) who formalized the rather
heuristic approach of Frechet· (1927) and Fisher and Tippett (1928).
The
central result of classical extreme value theory is therefore generally
referred to as Gnedenko's Theorem.
Theorem 1.3.1 (Gnedenko's Theorem)
There are only three types of non-degenerate distributions
satsifying (1.3.2).
These are
for
(1.3.4)
¢ex, (x)
where
ex,
x';;; 0
for x > 0
is a positive constant;
If' ex, (x)
=
{ exp(-:-X)C() for x
<
for x
~
0
0
(1.3.5)
e
9.
where a
is a positive constant;
and
(1.3.6)
for all real
x.
These limiting distributions had been discovered by Frechet (1927)
and Fisher and Tippett (1928);
however, Gnedenko (1943) proved the
above theorem in complete generality.
The original proof of Gnedenko·s
Theorem rests on a general theorem of Khintchine (1938) (Theorem 43);
however, de Haan (1970, 1976) gives simpler and free standing proofs.
Independent and identically distributed sequences may be constructed
for which no non-degenerate limiting distribution exists.
Both Gnedenko
(1943) and de Haan (1970) give conditions under which no non-degenerate
limiting distribution exists.
For example, the Poisson distribution
satisfies these conditions.
Gnedenko, in this same paper, was able to provide necessary and
sufficient conditions for the convergence to each of the limiting
distributions.
Theorem 1.3.2
The distribution function,
(i)
¢
a
if and only if,
w(F) =
lim ~ = x-a
t~
--r:ntT
for all positive x,
where
F
00
is in the domain of attraction of
and for some positive constant a,
(1.3.7)
10.
w(F) = sup{x: F(x)
(ii)
~a
< l}
if and only if, w(F)
F* (x)
=
<
00
and
1
F(w(F) - -)
x
for all positive x,
satisfies (1.3.7) for some positive constant a
or
(iii)
A if and only if, for some finite a,
f
w(F)
(1-F(y ) )dy <
00
a
and
1im
t~(F)
for all real
1-F(t+xR(t)) = e -x
1-F(t)
x, where for a(F)
<
t
(1.3.8)
<
w(F)
w(F)
R(t) = (l-F(t))-l J
(l-F(y))dy
(1.3.9)
t
and a(F) = inf{x: F(x) > O} .
Gnedenko was not able to give the characterization of R(t) ,
which is due to de Haan (1970).
Independently of de Haan, similar work
was being undertaken by Marcus and Pinsky (1969) who obtained separate
necessary and sufficient conditions for which a parent distribution of
the form F(x) = 1 - exp(-g(x))
converges weakly to A.
The
11.
conditions of de Haan are more general, not being constrained to a
particular form of F,
and will be preferred.
All of the conditions
of Theorem 1.3.2 are concerned with the behaviour of the upper tail of
the parent distribution, or of a transformation of the parent distribution as in (ii).
In practice the above necessary and sufficient conditions may be
difficult to establish.
In many situations it may be useful to consider
some sufficient conditions derived by von Mises (1936).
Theorem 1.3.3
Suppose that
f.
·e
F is absolutely continuous, with density function
Then,
( i ) F is in the domain of attraction of
for all
x ;;. xl '
I-F x
f!H&
1im
x~
for a.
where xl
=
o
is pas iti ve
(1.3.10)
ex
x in some finite interval
\If
ex
if f(x)
(x1'x O) , f(x)
is pos iti ve
equals zero for
and
1im
xtx o
e
f(x)
> 0
for all
> X
if
ex
is finite and
( i i ) F is in the domain of attraction of
x
<P
o,
(xO-x)f(x)
I-F(x)
=
ex
for ex
>
(iii)
F is in the domain of attraction of A if f
(1.3.11)
and
has a negative
12.
derojvative f'
for all
need not be finite,
x in some interval
f(x)
lim f'(x)
xtx o
(x 1,xO) ' where xo
equals zero for x
1-F(x)
f 2(x)
=
~
xo and
-1
(1.3.12)
It can readily be seen that these conditions are, in most instances,
easily checked.
They may, however, lack specificity.
for convergence to
~
The conditions
are of major concern because, in Theorem 1.3.3,
they are complex and therefore the most difficult to verify.
The
. following necessary and sufficient condition for convergence to
~,
given by Balkema and de Haan (1972) and based on von Mises ' work, will
reduce the complexity of verification.
Theorem 1. 3.4
A distribution function, F, lies in the domain of attraction of
A if and only if there exists a von Mises function, F. (defined below),
such that w(F.)
lim
xtw(F)
Definition 1.3.1:
= w(F) and
I-F(xl
~
=
1
(1.3.13)
A distribution function,
F,
will be called a von
Mises function if it has a density f which is positive and diffentiable
on a left neighbourhood of w(F)
l;m
xtw(F)
d~ ( If
and
H) ) = 0
(1.3.14)
13.
To illustrate the usefulness of the preceeding theory, we will
consider an example in which the parent distribution is of the Weibull
F(x) = 1 - exp(-ax S);
form, that is,
constants a and
S
~
of attraction of
a
for positive
x and positive
The Weibull distribution is not in the domain
because w(F)
=
Consider
00
Thus
1im
t-l>OO
-e
for
x >
o.
~!F({:)) =
Therefore, the Weibull distribution is not in the domain
of attraction of
is
~
-a
0 f x
~
a
The only remaining limiting distribution function
and it can be shown that the conditions of Theorem 1.3.2 are
difficult to establish.
In addition, the conditions of Theorem 1.3.3
do not hold when S is less than one.
However, the conditions of
Theorem 1.3.4 can be verified as follows.
w(F)
=
1im
d
dx
00
and
X-l>OO
lr 1-f~X()) J
14.
{
=
-13
lim 1-13
as x for 13 f 1
x~
a
=1
for 13
= a
F is a von Mises function
Therefore~
the domain of attraction of A.
by Theorem
1.3.4~
F is in
This example shows that the results of
von Mises' work can be very powerful
Having verified the
and~
existence~
~
in a practical sense.
and derived the form, of the
limiting distribution, it is necessary to determine the normalizing
and bn . Using Khintchine's Theorem (Khintchine (1938))~
Gnedenko (1943) determined the extent to which the sequence of normal-
constants an
The theorem is quoted in its extended
izing constants could change.
form as given by Feller (1966).
Theorem 1.3.5
lim Pr{Z <:;;;a x+S } = H* (x)
n n
n
if and only if, there exist real constants a
Suppose (1.3.2) holds.
*
with non-degenerate H
and b with
We have
a positive such
n~
that~
and
and H* (x)
(1.3.15)
= H(ax+b) for all real x. This result leads to the
following definition.
Definition 1.3.2:
The distribution function
same type as the distribution function
F1 is said to be of the
F2 if there exist two constants
e·
15.
a and b,
with
a positive, such that
(1. 3.16)
for all real
x.
This relation between F1 and F2 is symmetric, reflexive and
transitive. Hence it gives rise to equivalence classes of distribution
functions, called types.
There are many equivalent choices for the normalizing constants.
The non-uniqueness of these constants can be seen from the following
Yn be a sequence of random variables and
Let
simple discussion.
assume that, with some constants
-e
1 im
PdY
n~
an and bn , where an
n ~ a nx + bn} = H(x)
for all continuity points of H(x) , where
distribution function.
Pr{Y
n
~a
is positive,
H(x)
is a non-degenerate
However,
Vb
x + b } = Pr { ..l!. - ..l!. ~ x}
n
n
an an
and so we can see that if an tends to infinity, say, then changing
* ) tending to zero as n tends to
bn to b*n ' which satisfies (bn/a
n
i nfi nity, will have no effect on the 1imit H(x) . Simil arly, we expect
that an can be replaced by an*
as
n tends to infinity.
of the same type and
If HI
satisfying
*
(an/an)
tending to one
H2 are distribution functions
F is in the domain of attraction of HI' then
(1.3.15) is satisfied with a n
and
= an
a '
Q
~n
= bn
+ an
b.'
so that
16.
lim F (a x+13 ) = H2(x)
by Theorem 1.3.5, and hence F is in the
n n
n
domain of attraction of H2 . Thus, if HI and H2 are of the same
type, then D(H 1) = D(H 2 ) .
~,
De Haan (1970) considered sequences {an} and {b n} defined in a
simple way in terms of quantiles of the distribution of Zn' which can
be used when the limiting distribution function,
H,
is continuous on
the whole real line and strictly increasing on {x: 0 <H(x) < l}
.
Theorem 1. 3.6
Let Fn(x) = Pr{Zn"';;;x} and for each y(O<y<l) we define the
y-quantile ~(n) by
y
~(n) = inf{x: F (x) ~y}
n
y
.
(1.3.17)
Let the distribution function H be continuous on the whole real line
and strictly increasing on {x: 0 <H(x) < I} and suppose that (1.3.2)
Let {Yn} and {13 n} be sequences of positive numbers tending
to limits y and 13 respectively as n tends to infinity, where
holds.
o<y
< 13 < 1.
a
n
If we define
= '-:>Q
c(n)
fJ
n
then
lim PdZn",;;;anx+bn}=H(ax+b)
n-xx>
e-
17.
Gnedenko's normalizing constants are a special case of de Haan's, when
Fn == Fn . It is not true, in general, that the relation (1.3.2) holds
with the standard deviation and the mean as normalizing constants;
00
that is, with an = an and bn = ~n' where ~n = J x dFn(x) and
2
2
2
an = J x dFn(x) - ~n' even if ~n and a~ exist for every n.
00
_00
_00
De Haan gives necessary and sufficient conditions under which normalization by moments (mean and variance) is possible.
A connection between
quantiles and centering constants used with the weak law of large numbers
has been given by Geffroy (1958).
We will consider only the simpler normalizing constants of de Haan
(1970) which when applied to the results of Theorem 1.3.2, where Fn
==
Fn
are:
(i)
(1.3.18)
an
= inf{x:
1 - F(x)
.;;;;~}
when F is in the domain of attraction of
(i i )
b
n
e
bn
a
= w(F)
when F is in the domain of attraction of
(i i i)
¢
1
= i nf{x: I-F(x)';;;;-}
n
an = R(b n )
(1.3.19)
If
a
and
(1. 3.20)
18.
where R{.)
is defined by (1.3.9) and F is in the domain of
attraction of A.
As mentioned earlier, another statistic of special interest in
extreme value theory is the k-th upper extreme (or k-th largest value)
for fixed
k.
The k-th lower extreme is related to this statistic in
an analogous manner to which the minimum and maximum are related,
referred to earlier in this section.
Smirnov (1949 and 1952) proved
the following theorem concerning the k-th upper extreme.
Theorem 1.3.7
positive, and {b n} of real numbers, and for
a fixed integer k{>l) , as n tends to infinity Pr{X n- k+1 : n ~a nx+b n}
converges weakly to a non-degenerate distribution function H{k){x) if
For sequences
{an}'
Pr{Zn ~anx +bn} converges weakly to a non-degenerate
distribution function H(x). If H(k)(x) exists, then for
and only if,
a(H) < x < w{H) , where a{H)
w(H)
=
sup{x: H{x) < l}
(k)
H (x)
where H{x)
=
=
inf{x: H{x) >O} and
;
H(x)
k-1
1
t
L IT
[- log H{x)]
is one of three types
~
a
(x) ,
~
(1.3.21)
,
t=O
a
(x) , and A{x) .
The above theorem shows that for i.i.d. random variables any
limiting distribution of Xn- k+1: n has the form (1.3.21) based on the
same distribution function, H, as applied to the maximum, and the
normalizing constants are the same for all
the case where lim (kin)
n~
= A,
k.
Smirnov also considered
0 < A < 1 and Chibisov (1964)
~ .
19.
investigated the case where
zero.
k tends to infinity and
kin tends to
These situations are not within the scope of this dissertation
since they are not relevant to the applications of interest.
This section has reviewed all of the possible non-degenerate
limiting distributions for i.i.d. random variables.
1.3.2
Extreme Value Theory for Non-identically Distributed Random
Variables
Juncosa (1949) showed that much more general classes of limiting
distributions exist if we drop the restriction that the random variables
{X n} are identically distributed.
Let X1, ... ,X n be a sequence of independent but not identically
·e
distributed random variables whose respective distribution functions,
assumed to be non-unitary, are
constants
{an}'
an
F1 (x), ... ,F n (x).
positive, and
If sequences of real
{b n} exist such that
n
1 im
Pr{Z
n
~
a x +b }
n
n
=
lim IT
n-?CO j=l
exists and is of the form V(x) , where
distribution function, then we say that
distribution.
J
V(x)
{Zn}
n
n
(1.3.22)
is a non-unitary
has a limiting
Juncosa (1949) gave some conditions under which such a
limiting distribution exists.
where Fj(X)
F.(a x+b )
Juncosa also considered the special case
= F(ccjx+S j ) . Mejzler
sidered the non-identical case.
(1949,1950 and 1953) also con-
Mejzler1s conditions were more easily
verified than Juncosa1s and they depend on the "Uniformity Assumption
for the Maximum".
20.
-Definition
1.3.3: We say that a sequence
Fl(x)~
... ~F n (x)
of
{an} and {b n} of normalizing
constants satisfy the uniformity assumption for the maximum if
distribution functions and sequences
1i m rna x {F . (a x + b ): 1 ~ j
n~
J n
n
and, for any fixed number 0 < t
[nt]
L
1 im
n-+oo j=1
exists~
t =1
~
{l - F . (a x+b )}
J n
n
which is finite for all
~
where
brackets.
[]
=1
(1.3.23)
= 9( t , x)
(1. 3.24)
~ n}
1 ,
0 < t
~
1 whenever it is finite for
denotes the integer part of the quantity in the
Mejzler then obtained the following result.
Theorem 1.3.8
Under the uniformity assumption for the
distribution
function~
H(x)
~
maximum~
a non-degenerate
is the limiting distribution of
~
for some sequence X1 , ... ,X n of independent random
variables and for some sequences {an}' an positive, and {b n} of
(Zn-bn)/an
normalizing constants if and only if, either
(i)
log H(x)
(ii) w(H) <
00
positive~
(iii)
a(H)
is concave or
and
log H(w(H)-e- x) is concave, where x is
or
is finite and log H(a(H)+e x) , where x is positive,
is concave.
The above approach has been included merely for completeness.
The
relaxation of the assumption of identically distributed random variables
21.
is not necessary in the applications of interest in this dissertation.
1. 3.3
Extreme Value Theory with Dependent Observations
The need for deviating from the assumption of i.i.d. random
variables has been stressed ever since the theory was appl ied to
specific problems.
Indeed, even though the approximation by i.j.d.
random variables may be good, the credibility of the solution should be
challenged if it assumes, in a cavalier fashion, that the variables are
i. i.d. when they are not.
There are numerous and varied ways in which
dependence can be introduced to generalize the classical extreme value
results.
The pioneer in deviating from independence was Berman.
In a series
of papers Berman (1962a, 1962b and 1964) established the basic results
for extremes of segments of infinite sequences of exchangeable variables
and for certain types of stationary sequences.
For the purposes of this
dissertation, the theory on exchangeable variables will be ignored
because, in the applications of interest, exchangeability is obviously
violated.
The first generalization to be considered is that of
stationarity.
The assumption that the Xi
have a common distribution,
F, will, however, be kept.
X1 ,X 2 , ... of random variables which satisfy
Pr{X j ~x} = F(x) , for each j and (ii) for any
Consider sequences
the properties;
(i)
positive integer r,
where
i(k)+r signifies the vector
(i 1+r, ... ,i k+r).
Such a sequence
22.
will be referred to as stationary.
Obviously, exchangeable variables
are stationary, and thus so are i.i.d. variables.
It will then be
assumed that the dependence between Xi
specified way as
li-j!
and Xj falls off in some
The simplest example of this type
increases.
of restriction is that of m-dependence, which requires that
x.J
be actually independent if
li-j! > m.
X.
and
1
Watson (1954) applied this
restriction and developed some extreme value results which, in essence,
said that one could ignore the dependence structure and use the i.i.d.
results.
Gumbel (1958) used this result to justify his application of
the i.i.d. theory in obviously non-independent situations.
The correlation between Xi
of their dependence.
the same type is
as
k
and Xj
is also a (partial) measure
Consequently, another dependence restriction of
ICorr(Xi,X j
tends to infinity.
)!
~
g(li-jl) , where g(k)
tends to zero
Such a restriction may be most useful if the
Xn form a normal or log-normal sequence; however, it could also be
used for some forms of the bivariate distribution. Berman (1962b) used
some simple correlational restrictions to obtain the limiting distribution of A,
for stationary normal sequences.
The m-dependence and
correlational restrictions are too strong and rely too heavily on the
parametric form of the parent distribution.
A more commonly used dependence restriction of this type, for
stationary sequences, is that of strong mixing, introduced by Rosenblatt
(1956).
The sequence
{X n} is said to satisfy the strong mixing
assumption if there is a function
tending to zero as
I P(AnS)
g(k)
the mixing function
II
k tends to infinity, such that
- P(A)P(B) I < g(k)
ll
,
23.
when AE F(X1, ... ,X p) and BE F(Xp+k+1,Xp+k+2"") , for any
and k,
where F(·)
random variables.
p
denotes the a-field generated by the indicated
Thus, when a sequence is mixing, any event A,
"based on the past up to time p ", is "nearly independent" of any event
B,
"based on the future from time p+k+1 onwards
ll
,
when
k is large.
Note that this mixing condition is uniform in the sense that g(k)
not depend on the particular A and B involved.
does
Loynes (1965)
obtained quite a number of extreme value results under the strong mixing
assumption, for stationary sequences.
This strong mixing property seems
rather strong and motivated by the parallel which exists between the
normed maxima and the normed sums of random variables.
Leadbetter (1974) considered a much weaker condition of the "mixing
-e
type" under which the results of Loynes will still be true.
Furthermore,
this condition is satisfied for stationary normal sequences under
Berman's correlation conditions.
In weakening the mixing condition,
Leadbetter notes that the events of interest in extreme value theory,
unlike those for central limit results, are typically the form
or their intersections.
{X1~u"",Xn~u}.
For example, the event
{Z ~u}
n
{X.~u}
1
is just
Hence, Leadbetter (1974) proposed a condition, like
mixing, to hold for events of this type.
Definition 1.3.4:
integers
i1 <
any real
u,
Condition D will be said to hold if, for any
and
(1. 3.25)
24.
where
1im g(i)
=
0 ,
i~
and
Leadbetter showed that Gnedenko's Theorem, as well as a number of other
results, holds under
D.
Leadbetter, Lindgren, and
Rootz~n
(1979)
D(u ) , which imposes mixing restrictions
n
on a certain sequence of values {un} and not necessarily on all values
proposed a further condition,
of u.
e·
Condition D(U n ) will be said to hold if, for any
11 < ... < i p < jl < ... < jq ~ n for which j1 - i p ;;;;. in
Definition 1.3.5:
integers
1~
we have
(1.3.26)
n
= 0 for some sequence £n = o(n) .
n,.lV n
Note that for each n,R. n we may replace a n, .lnV by the maximum of
n
the left hand side of (1.3.26) over all allowed combinations of ils
where
lim a
n~
and j1s
to obtain, possibly, smaller a
which is non-increasing
n, in
n and still satisfies an £ tending to zero
in
R. n for each fixed
as
n tends to infinity.
, n
It is also useful to note that for a i '
n, n
taken non-increasing in an for each fixed n, the condition a £
n, n
e
25.
n = o(n) , may be
n tends to infinity, for each
tending to zero as
n tends to infinity and
.Q.
rewritten as a n,n/\\ tending to zero as
A > O. Leadbetter et al. were then able, using certain results first
established by de Haan (1970), to generalize Gnedenko1s Theorem, as
follows.
Theorem 1.3.9
Let
n} be a stationary sequence and {an}' an positive, and
{b n} sequences of real constants such that Pr{Zn .,;;;anx + bn} converges
to a non-degenerate distribution function H(x) . Suppose that D(u n)
{X
is satisfied for un
= anx + bn
for each real
x . Then H(x)
has one
of the three extreme value forms listed in Theorem 1.3.1.
Having established the possible form of the limiting distribution,
Leadbetter et al. considered the existence of such a 1imit, for
stationary sequences.
Definition 1.3.6:
Condition
D1(u n) win be said to hold if, for the
stationary sequence {X n} and sequence {un} of constants,
lim lim sup n
k-loOO
where
[]
n-loOO
[n/k]
I
j =2
Pr{X.>u ,X.>U }
1
n J
n
=0
p.3.27)
denotes the integer part of the quantity in the brackets.
The above condition bounds the probability of more than one,exceedance
of un among Xl"" ,X [n/k] .
Write Zn
for the maximum of n i.i.d. random variables with the
same marginal distribution function,
F,
as each Xi'
Following
Loynes (1965) this may be called the lIindependent sequence associated
26.
with
{X}"
n
• After establishing some lemmas and theorems, Leadbetter,
lindgren, and Rootzen (1979) were able to prove the following important
theorem.
Theorem 1.3.10
Suppose that D(u n) , D1(u n) are satisfied for the stationary
sequence {X n} , when un = anx + bn for each x and {an} , an
positive, and {b n} are sequences of constants. Then
1im Pr{Z <a x + b }
= H(x) for some non-degenerate H
1im Pr{Z <a x + b }
= H(x) . This theorem shows that, under conditions
D(u n) and D1(un)'
the dependence can be ignored for the purposes of
n~
n
n~
n
n
n
n
n
determining the form of the limiting distribution.
if
and only if
Of course, the
dependence must be considered in the estimation of an and bn .
of this result was proved in Loynes (1965) under conditions which
Part
~
include strong mixing.
While for many applied models the concepts of mixing may be
appropriate, for others it may have serious disadvantages.
the theory considers only stationary sequences.
For instance,
In the applications of
interest in this dissertation, it is reasonable to expect a strong
periodicity of air pollution concentrations corresponding to the day of
the week.
such as
Galambos (1978) cites other situations in which conditions
D(U n ) would not be appropriate.
Consequently, Galambos (1972)
developed a model which describes a situation where asymptotic independence is stressed but with weaker assumptions than
For a given sequence,
and 1978) defined a set,
{X n} ,
*
En'
D(u n) and D1(u n) .
of random variables, Galambos (1972
of so-called exceptional pairs,
with i < j , of the subscripts as follows.
Place
(i,j)
into
(i,j)
En* if
-
27.
it is not reasonable to assume that, as
Pr{Xi~xn ,Xj~xn}
xn tends to max(W(Fi),w(F j
is asymptotically Fi(xn)Fj(x n) . Here, 'Ft(x)
)] ,
denotes the distribution function of Xt . The sequence {X n} is called
*
an En-sequence
if the following three assumptions are satisfied.
contain no pairs from En*
(i)
then the difference
= F'(k)(x , ... ,x ) ,
n
n
k
II
F. (x,)
t=l' t
(1. 3.28)
n
is negligible compared to either of the terms, as
xn ' tends to
supw(F.) .
~l
't
If there is exactly one pair (is,i m) among the components of
*
i(k) = (i1,···,i ) which belongs to En'
then
k
(i i )
k
F. (x )
II
t=l
,t
n
(1. 3.29)
t1m,s
where nk is a constant.
The number of elements in En*
N(n) = o(n 2) .
(iii)
N(n),
is such that
*
Notice that, for an En-sequence,
there is no assumption on the
interdependence of
*
one pair from En'
if (i 1, ... ,i k) contains more than
'1
'k
In other words, consider a subset of the original
(X. , ... ,X.)
set of random variables which contains at least two pairs for which
asymptotic independence cannot be assumed.
This subset can have
arbitrary structure, as is true for the conditions given by Leadbetter,
28.
Lindgren, and Rootzen (1979).
Furthermore, unlike Leadbetter et al.,
was not assumed that the Xj were identically distributed. The
second assumption is far less than asymptotic independence, even in the
it
weak sense of considering each Xj falling below a fixed number, xn '
because the nk can be arbitrarily large. The third assumption is a
natural one, requiring that a positive percentage of all pairs cannot
be exceptional. The above assumptions are, nevertheless, quite complex
and may be difficult to verify in practice.
Galambos (1978) was able to obtain the following result.
Theorem 1. 3.11
*
Let X1"",Xn be an En-sequence.
n
lim
n~
L
j=l
for A positive.
that, for all
1im
[1 -
Let xn be such that,
FJ.(x n)] = A
( 1. 3.30)
Let us assume that there is a constant,
nand
j
L *
,
n[l-F.(x)]
J
Pr{X.
1
n-+oo ( i , j )EE n
~
n
~ K* •
x , X. ~ x }
n
J
n
=0
K* , such
Furthennore, let
(1.3.31)
Then
lim
n-+oo
Pr{Zn~xn}
In particular, if Fj(x}
= exp(-A)
= F(x)
value theory applies to Zn .
for all
(1. 3.32)
j , then classical extreme
29.
In the general situation, where the marginals are not identical,
it can be very complicated to verify (1.3.30).
For example, suppose
X.
a.
J
has a Weibu1l distribution with parameters
J
and S..
J
Then
(1.3.30) becomes
n
13·
exp{-a.(a .x+b .) J} = A
L
1im
J
j =1
n-><x>
where a . and bnj
nJ
nJ
(1. 3.33)
nJ
are both functions of a j
and Sj .
That is
and
13 .
1
b . = inf{x: exp(-a.x J)~_}
nJ
J
n
Numerically, we could estimate aj and I3 j for all
assuming some constraints on the parameters, such as
a.
J
=
and similarly for
a1
for
j=1,. .. ,k 1
a2
for
j=k +1, ... ,k 2
1
a3
for
j=k 2+1, ... ,k
(3 . •
J
n.
by
n
However, it will generally be difficult, if not
impossible, to obtain in a closed form,
the sample size
j=1, ... ,n
a nj
and bnj as functions of
Without this it is not possible to verify (1.3.33).
The only alternative is to use the estimates of a nj and bnj for the
sample size n and try to find a function A of x which gives a good
30.
fit to the part of the left hand side of (1.3.33) inside the limit.
This method is, therefore, data dependent and is not considered
sufficiently general for applications of interest in this dissertation.
The review of univariate extreme value theory, as it relates to the
scope of this dissertation, can be completed by consideration of the
limiting distribution of the k-th largest value (or k-th extreme) in the
dependent case.
Leadbetter, Lindgren, and Rootzen (1979) developed
analogous results to Theorem 1.3.6 in the dependent case.
Galambos
*
(1978) did not develop equivalent results for En-sequences.
Theorem 1.3.12 (Leadbetter et al. (1979))
Let {an}'
an
{b n} be sequences of constants and
positive, and
H a non-degenerate distribution function, and suppose that
D1(u n ) hold for all
lim Pr{Z
n-)OO
then, for each
n
un = anx + bn ,
~a
for all real
n = H(x)
x +b }
n
x.
D(u ) ,
n
If
(1. 3.34)
k=1,2, ...
(1. 3.35)
where H(x) > 0 (zero where H(x) =0).
Conversely, if (1.3.35) holds for some k,
hence (1.3.35) holds for all
Zn
replaces Zn
in (1.3.34).
k
so does (1.3.34) and
Further, the result remains true if
tit·
31.
1.3.4 Multivariate Extreme Value Theory
Bivariate and multivariate extremes are useful in many concrete
problems such as the largest ages of death for men and women each year
for some country, the floods or droughts at two points of the same
river each year, and bivariate extreme meteorological data such as
temperature and pressure.
As part of the theoretical development, we
will need to employ the theory of multivariate extremes.
The bivariate situation has been considered by a few authors.
framework used is as follows:
Let
(Xl'Y1), ... ,(Xn,Y rt )
The
be a sample of
n indeperident random pairs whose distribution function is F(x,y) .
The distribution function of
,
(max Xi' max Vi)
, ";;;y}
Pr{max X.";;;x , max Y.
is
= Fn(x,y)
Sibuya (1960) and Tiago de Oliveira (1975, 1980) have, using slightly
different approaches, obtained useful results in this context.
the approach of Sibuya, we define a dependence function,
~,
Using
as
follows.
Definition 1.3.7:
The dependence function
of the distribution F(x,y)
~(Fl,F2)
, 0 < F1 ,
is the function that satisfies
F2 ";;; 1 ,
(1. 3.36)
where F (x) and F (y) are the marginal distributions of F(x,y).
1
2
For F1 = 0 or F2 = 0, ~(F1,F2) may be defined if the limit
32.
(1.3.37)
or
(1. 3.38)
exists.
Although F1(x):; F~ and F2 (y):; F~, where F~,F~ are
constants, do not determine a point (x,y) uniquely, the value of
F(x,y)
F~}.
is the same for any
Therefore, ~(Fl,F2)
(x,y)
in the set {(x,y): F1(x) :; F~,F2(Y) :;
is a one-valued function defined for all
possible values of F1 and F2 . We can also see that ~(Fl,F2) :; 1 ,
if and only if, X and Yare independent. From the definition
:;
Pr{X~,
Y<y}
Pr{X~}Pr{Y~}
(1. 3.39)
,
Q(F 1,F 2) > 1 «I) corresponds to the positive (negative) association
between the events {X ..,;;x} and
{Y"';;y}.
Theorem 1. 3. 13
If the convergence of P(F 1,F 2) to zero is of such order as
P(l-r,l-r) :; o(r)
where P(F 1(x),F 2(y)):; Pr{X>x,Y>y}, then Z~l) and Z~2)
asymptotically independent, and vice versa, where
( 1. 3.40)
are
e-
33.
Z~I) = max{X;; ;=I, ... ,n} and z~2) = max{Y;; i=1, ... ,n} .
Proof
Let Fn(x,y) = ~n(F~(X),F~(Y))F~(x)F~(Y) . From (1.3.26)
F(x,y) =
~(Fl(x),F2(Y))Fl(x)F2(Y)
and so we obtain
Also, we know that, from (1.3.39)
(1.3.41)
Therefore,
n
n
_ {F1(x)+F2(y)-1+P(F 1 (x) ,F2(y))
F (x)F (y)
1
2
~n(Fl(x),F2(Y)) -
As F (x) and F2(y)
1
then,
tend to one (i .e., for large values of x and y)
lim ~n ( F~ (x) , F~ (y )) = 1im { 1 + P( 1
n~
}h
n~
= exp
{
E
1
-n '
1 ,
1 n
1 im nP( 1 - -E
n~
2
1 - E
n)
r
E2 } ,
n)
34.
E* } ,
. ;;; exp { 1im nP (l - -E* , 1 - -)
n~
n
n
= max(E 1,E 2)
where E
1
and E*
= min(E 1,E 2)
.
Under the conditions of the theorem,
lim ~n(F~(X),F~(Y)) = exp(O) = 1
n-+OO
Therefore,
Z~l) and Z~2) are asymptotically independent.
0
Nair (1976) considered a slightly more general approach in proving
the following theorem.
Theorem 1. 3.14
Let F(x,y)
be a distribution function with non-degenerate
marginals F1(x) and F2 (y) . Then there exist sequences {an}' an
positive, {b n}, {c n}, cn positive, and {d n} such that
lim Fn(a x+b ,c y+d ) = exp{-e-xf(e x-y )}
n
for all
(g,k >0)
n
n
(1.3.42)
n
x and y if and only if there exist functions
g,
hand k
such that
lim 1-F(t+g(t)x,h(t)+k(t)y) = e-xf(e x-y )
t-+oo
1-F 1(t)
( 1. 3.43)
35.
where xo = sup{x: F1(x) < l} and f( . )
is any function.
We will generally be interested in the conditions under which
asymptotic independence can be assumed and therefore results such as
Nair's are of little relevance to the proposed study.
In the extension to multivariate extremes Tiago de Oliviera (1975)
has shown that the multivariate extreme distribution functions split
into the product of the univariate marginals, that is, asymptotic
independence holds, if and only if all the bivariate extreme distribution
marginals split into the product of univariate distributions.
In the
more general situation Galambos (1975) and de Haan and Resnick (1977)
obtain necessary and sufficient conditions for the weak convergence of
the maxima, suitably normed, to a non-degenerate limit distribution
function.
Again, however, in relation to asymptotic independence, they
do not contribute significantly to the work of Tiago de Oliviera (1975).
Note that all current theory on multivariate and bivariate extremes
assumes that the vectors of observations are independent.
As we will
show later in this chapter, a particular dependence case is of relevance
to the applications of interest in this dissertation.
This problem has
not, to date, been addressed in the literature.
1.4 Outline of Subsequent Chapters
Leadbetter, Lindgren, and Rootzen (1979) have extensively covered
the case where {X n} is a stationary sequence and certain reasonable
conditions are satisfied. These conditions D(u n) and D'(u n) seem
eminently acceptable for air poilution applications.
However, itis
quite clear that in such applications, the sequence {X n}, ,cannot be
36.
assumed to be stationary.
For example, air pollution concentrations,
expecially for ozone and carbon monoxide, exhibit a weekly cycle
associated with traffic flow.
We will consider sequences for which certain periodic subsequences are independently distributed:
for example, observations on
the same day of different weeks are independent.
We will, however,
assume that within each sub-sequence the Xn have a common marginal
distribution function and this may vary between sub-sequences. It
should be noted that the construction of the sub-sequences is, thus far,
quite flexible;
the only requirement being that the observations within
a sub-sequence are independent.
Let X3~)
j
= 1, ... ,n~;
denote the j-th observation in the ~-th sub-sequence and
= 1, ... ,s.
~
sample size for the
s
r
~=1
n~
=n
~-th
Since the total sample size is
sub-sequence is
n~(
the
n
::::;n/s), such that
•
Also, 1et
(1.4.1)
for
~
= 1, ... ,s .
Essential to the above arguments is the asymptotic joint
independence of the Z(~).
n~
Investigation of this condition naturally
leads to the topic of mu ltivariate extreme values which was reviewed
ll
II
in section 1.3.
In this section we mentioned that all prior research
had concentrated on the situation where the vectors of observations
were independent.
In the context described earlier in this section, we
can see that since successive observations are not independent, then
e-
37.
X~s) and XJ~i; that is the j-th observation of the s-th sub-sequence
and the (j+1)-th observation of the first sub-sequence;
are not
In the example of air pollution this says that the Sunday
independent.
measurement in the j-th week and the Monday measurement in the following
week (that is, the following day) are not independent.
s dimensional random vectors
Therefore, the
and ~J
X'+ l are also not independent,
as is the case for the usual multivariate extreme value theory to hold,
where X.
=
~J
X.
~J
(X~1), ... ,x~s))1
J
That is, although the corresponding
J
elements of X.
and -J
X'+ 1 are independent, the vectors themselves are
~J
not. In this dissertation we propose to show:that under conditions
similar to those of Sibuya (1960), we may assume that the sub-sequence
maxima are asymptotically jointly independent.
similar results for the k-th upper extremes.
We will also investi9ate
The above results of
classic extreme value theory will be extended to cover the multivariate
extreme value theory.
These considerations will be addressed in
Chapter I I.
In Chapter III we will focus on the problem of estimation of the
normalizing constants a(~)
nQ,
these can be estimated.
and b~~).
There are many ways in which
~
To begin with, we will consider the i.i.d. case
and, using maximum likelihood techniques, estimate the parameters of a
specified parent distribution.
distributions;
We choose to examine two possible parent
the Weibull and the log-normal.
Since we have assumed
that the observations in each sub-sequence are independent, we can apply
the techniques developed for the i.i.d. case within each sub-sequence.
The two methods above do not utilize auxiliary information and are
dependent on the form of the specified parent distribution.
The final
approach is to consider ways in which some covariates can be used in the
38.
estimation so that the results are, in some respects, distribution free.
To do this, we will apply the methods developed by Cox (1972) and
refined by Kalbfleisch and Prentice (1980).
In Chapter IV we will evaluate the various estimates of the extreme
value distribution from a set of ozone concentration data.
The
comparisons between these estimates will indicate the performance of our
methods relative to those when independence is assumed.
Chapter V will contain a summary of the results of this dissertation
and recommendations for future research.
It is the purpose of this dissertation to postulate a model for the
behaviour of air pollution concentrations which is more realistic than
the models being used in the setting of standards.
Once postulated, the
model's parameters are to be estimated in an efficient manner possibly
taking into account auxiliary information.
With this technique we hope
that more accurate characterizations of the extreme value distribution
can be obtained, resulting in better methods for the setting of standards
and for risk assessment.
II.
2.1
EXTREME VALUE THEORY FOR CERTAIN NON-STATIONARY SEQUENCES
Introduction
Leadbetter t Lindgren t and Rootzen (1979) have comprehensively
studied the case where {X n} is a stationary sequence and certain
reasonable conditions are satisfied. These conditions; referred to
earlier as D(u n) and D'(U n); seem justifiable fOT a·i'r pollution
applications. However t it is quite clear that in such ~pplications the
sequence
{X n} cannot be assumed to be stationary.
For example.;
air
pollution concentrations t especially those of ozone and carbonmonoxide t
exhibit a weekly cycle associated with traffic flow over and abov;e the
variation due to meteorological factors.
In this chapter we will t therefore t consider the sequence {X n} as
being partitioned into s jointly dependent sub-sequences of independent
observations;
for example t observations on the same day of the week. over
the different weeks.
sequence the Xi
We will
t
however t assume that within each sub..,
have a common marginal distribution function and allow
these to vary between the sub-sequences.
It should be noted that the
construction of the sub-sequence iS t thus far t quite flexible, the only
requirement being that the observations within a sub-sequence are
independent.
Let xJ!) denote the j-th observation in the !-th sub-sequence t
for
j
=
It •••
,n! and !
=1
t ••• t
s.
Since the total sample size is
the sample size for the !-th sub-sequence t
n!
t
is such that n!
n
40.
s
approximately equals
nls and
I
Q,=1
n~
= n. Also, let
(2.1.1)
for Q,
= 1, ... ,s. We refer to
as the Q,-th sub-sequence maximum.
nQ,
In this chapter we will develop some theoretical results, assuming
Z(Q,)
the above framework, which will enable us to characterize the extreme
value distribution for the complete sequence
We will also
n
consider the k-th upper extreme and extend the previous results to the
multivariate extreme value distribution.
{X }.
Finally, we examine the effects
of treating the sub-sequence sample size as a random variable.
The central result in this chapter is the establishment of
conditions under which the sub-sequence maxima,
ically jointly independent.
z~Q,)
, are asymptotQ,
In Chapter I we saw that all prior research
on multivariate extreme values concentrated on the situation where the
vectors of observations were independent.
In our case we can see that,
since successive observations are not independent, then x~s)
and
X~~ {
that is the j-th observation of the s-th sub-sequence and the (j+1) -th
observation of the first sub-sequence;
are not independent.
In the
example of air pollution this says that the Sunday measurement in the
j-th week and the Monday measurement in the following week;
that is,
the following day;
are not independent.
Therefore, the s dimensional
random vectors X.
~J
and -J
X'+1 are also not independent, as is the case
for the usual multivariate extreme value theory to hold, where
~j
=
(xjl), ... ,xjs)),.
In this chapter we will show that, under
conditions similar to those of Sibuya (1960), we may assume that the
sub-sequence maxima are asymptotically jointly independent.
~ .
41.
2.2 The Extreme Value Distribution of the Maximum of the Sub-seguence
Maxima
Under the appropriate conditions, given in Theorems 1.3.2, 1.3.3 or
1.3.4, it is clear that,
(2.2.1)
where H£ is a non-degenerate distribution function for £ = 1, ... ,s .
The form of H£ is determined by the type of parent distribution,
where F£(x) = pr{xj£) ,.;;;x}.
F£,
However H£ must be one of the three
types given in Theorem 1.3.1.
In addition a{£), positive, and b{£}
n£
n£
are normalizing constants for £ = 1, ... ,5. Therefore,
Pr-{Z ";;;a x+b} = Pr{z{£},.;;;a x+b; ,Q,=l, .. .,s}
n n
n
n£
n
n
{2.2.2}
where
Zn = max{X i ; i=l, ... ,n}
= max{z~,Q,); £=l, ... ,s}
£
and an'
positive,
and bn are real constants.
We can then re-
express {2.2.2} as
;
~=1,
... ,5 }
•
{2.2.3}
42.
If it could be shown that the Z~~)
are asymptotically jointly
~
independent then (2.2.3) becomes, in the limit;
r(
1im Pr{Z n ~a nx +bn}
n-+«>
£ ) -b ( £ )
=
1im
n-+«> Pr
=
IT
~=1
since
n,Q, n~
a(,Q,)
~l;nJJX+
(an ')
n~
n~
(Z{~)-b{~)
s
1im Pr
n-+«>
~ n~
~
n~;
to infinity.
~ =
l
n,Q,
(,Q,)
an~
1, ... ,s;
n
n~
a(~)
')
~= 1, •.. , s ~
n~
b -b(£)
t
J
n n~
~i))
x +
a ( ~) J
a
n,Q,
n~
~(
(2.2.4)
tends to infinity if and only if n tends
Therefore, by Theorem 1.3.5 and equation (2.2.1)
lim Pr{Z
n~
where an'
b -b ( £)
n
~a
s
x+b } = IT Hn{Anx+BJ
n
n
,Q,=1 N N
N
(2.2.5)
positive, and bn are such that
(2.2.6)
and
(2.2.7)
for
,Q, = 1, .. .,s .
The choice of the form of an and b , like a{~)
and b{~)
n
n'
n
~
~
will depend on the type of marginal distribution functions, F~,
~
= 1, ... ,s. Both an and bn will be functions of the sub-sequence
43.
nonnalizing constants a(Q,) and b(Q,), respectively, for Q, = 1, ... ,s
nQ,
nQ,
Having obtained the values of an and bn we can then estimate AQ,
and BQ, and use these in (2.2.5) to obtain the extreme value
distribution for the entire sequence.
To illustrate this technique we will consider the case where all
the marginal distributions are of the Weibull form.
Since the Weibull
distribution
is in the domain of attraction of A,
,
the double
exponential distribution, we have, by (1.3.20)
and
(2.2.10)
where f(·)
is the standard Gamma function.
44.
Let an be equal to one of the sub-sequence values, say a~t) ,
t
thus
a
n
= a(t)
n
for some
t
E
{l, ... ,s}
t
Then,
-
-
log nt liS -1
f
z t exp(-z)dz
o
(2.2.11)
10gnR, 1/S-1
f
z R, exp(-z)dz
o
We also know that nR, is approximately equal to n/s for R,
=
1, ... ,s
Therefore, as nR, tends to infinity we can replace both nR, and nt
by n/s and let n tend to infinity. Thus,
log n/s 1/S-1
-f
z t exp(-z)dz
o
log n/s 1/S-1
-f
z R, exp(-z)dz
o
lim
n-+OO
log n/s 1/S-1
f
z t exp(-z)dz
o
logn/s 1/S-1
-f
z R, exp(-z)dz
o
liS -1
1/SR,
~ 1 (1og nls ) t
f\aR,
ns
1im
= liSt
n-+OO ~ -1 (109 nl s ) liS R, -1
Stat ns
1/SR,
=
SR,aR,
liSt
Stat
lim
n-+OO
(log nl s )
liS -liS
t
R,
~
n
~
n
using L'Hospital's Rule
45.
=
1/(3£
if
(3£ = f\
0
if
B£ < Bt
00
if
B£ > Bt
(a£/a t )
(2.2.12)
.
Now consider
b
n
= b(q)
f or
nq
some
q
}
E {~1
... ~s
.
Therefore~
(2.2.13)
·e
As
£
before~
since n£ is approximately equal to nls
= 1, ... ~s;
infinity.
B~
then~
as
n~
tends to infinity we can let n tend to
Therefore~
= lim
(2.2.14)
n~
~
liB
x 1 im
n-+co
for all
~ (log n/s)
~
log nls 1/B-1
- J
z ~ exp(-z)dz
o
46.
x
s
I/S.Q,
s I l l S -1 s 1
--(logn/s)
+--{logn/s)
q-n2
n S.Q,
ns
1im
n~
(log nl s)
liS -1
.Q,
~~1
n n s
using L'Hospital 's Rule
if
S.Q, = Sq and cx, .Q, = cx,q
if
S.Q, = Sq and cx,.Q, > cx,q
if
S.Q, = Sq and cx,.Q, < cx,q
00
if
S.Q, > Sq
_00
if
S.Q, < Sq
r0
e·
,
00
:::
1-
00
(2.2.15)
We can rewrite (2.2.12) and (2.2.15) by noting that, with probability
one, all of the sub-sequence parameter estimates are distinct.
(
A.Q, =
1
if
S.Q, = St
0
if
S.Q, < St
00
if
S.Q, > St
Thus
(2.2.16)
e
47.
and
B,e
=
0
if
13,e
00
if
13,e > 13q
_00
if
13,e
=
<
13q
(2.2.17)
13q
The only possible combinations of A,e and B,e which would render the
right hand side of {2.2.5} non-zero are;
,e
b
= t = q and A,e = B,e =
00
for ,e
r
t.
A,e = 1, B,e = 0 for
Therefore we choose an
and
n such that
a
=a
nt
(t)
(2.2.18)
b
= b (t)
(2.2.19)
n
and
n
A~
n
t
=
C
(2.2.20)
otherwi se
and
(2.2.21)
otherwise
48.
Since we will be modelling the air pollution data with both the
Weibull and log-normal distributions the corresponding results for the
latter are derived below.
A standard result of extreme value theory, see Galambos (1978), in
the case of the log-normal distribution gives the values of the
normalizing constants as,
and
is such that log X~l)
where the variable
N(~l'O~)
for 1
J
= 1, ... ,5
is distributed as
•
Let an be equal to one of the sub-sequence values, say a(t)
n
t
thus
a
n
= a(t)
n
for some
t
E
{l, ... ,s}
t
Then,
(2.2.22)
e·
49.
crt
log nt r~ 1im
= crQ, 1im ( log
nQ,
n-+<>o
nQ,-+<>O
Q,
crt
= 0Q, 1im
n-+<>o
Q,
b(t)
nt
bUT
nQ,
b(t)
nt
bUT
nQ,
since, as nQ, tends to infinity we can replace both nQ, and nt
n/s and let n tend to infinity. Thus,
by
1im
n-7<Xl
Q,
(
x
1 im
exp
n-+<>o
OQ,(10 g lo g n/s+l0 9 4TI) _ 0t(log log n/s +
{
r exp( fl t -flQ,)
=
Now consider
2(2 log nl s)
~
2
1:94TI) }
2(2 log nl s)
2
= 0Q,
if
crt
00
if
0t > 0Q,
0
if
0t < 0Q,
(2.2.23)
50.
bn
= b(q)
n
'
f or some
I}
q E {, •••
,s
.
q
Therefore,
(2.2.24)
lim
n~
Q,
e·
Now, as before, we can replace both
tend to infinity.
n
q
by
n/5
and let
Thus
(210gn/5)2
{
0Q,(1og10gn/5+10g 47T) _
+
and
!.:
lim
n~
nQ,
1
2(2 log n/s)'2
0
9
(10glo g n/s+l0 9 47T)
!.:
2(2 log n/5) 2
}J}
n
51.
I
0
if
° .R,
= ° q and fl.R, = fl q
00
if
0.R,
= 0q and fl.R,
<
_00
if
° .R,
= ° q and fl.R,
> fl q
_00
if
° .R, > ° q
00
if
0.R, < 0q
I
=
I
~
fl q
(2.2.25)
We can rewrite (2.2.23) and (2.2.25) by noting, as before, that, with
probability one, all the sub-sequence parameter estimates are distinct.
·e
Thus,
A.R,
1
if
0.R,
0
if
0.R, > °t
00
if
0.R, < 0t
r0
if
° .R,
00
if
0.R, < 0q
_00
if
0Q, > 0q
=
l
= 0t
{2.2.26}
and
B.R,
e
=
= °q
(2.2.27)
The only possible combinations of A.R, and B.R, which would render the
52.
right hand side of (2.2.5) non-zero are;
£
=t
=
q and A£ = B£ =
00
for £ f t .
A£
=
1,
B£
=
0 for
Therefore we choose an and
bn such that
a = a(t)
(2.2.28)
= b(t)
n
(2.2.29)
n
n
t
and
b
n
t
where 0t = max{ok; k=l, ... ,s},
in which case
if
(2.2.30)
otherwi se
and
if
(2.2.31)
otherwise
Consequently, we can find values of an '
£
= 1, ... ,s,
bn , A£ and B£ for
such that
lim Pr{Zn ~a nx+b n }
(2.2.32)
n~
if we can assume that the sub-sequence maxima are asymptotically jointly
independent.
It should be noted that, at least in the cases detailed
e·
53.
above, the parameters A£ and B£;
£
=
1, ... ,s
depend upon the
parameters of the dominant marginal distribution, corresponding to the
sub-sequence most likely to contain the maximum.
lim Pr{Z n ~a nx +b n}
=
We can see that
Ht(x)
n~
that is, the extreme value distribution for the maximum of the entire
sequence reduces to the extreme value distribution for the maximum of
the most extreme sub-sequence.
This result has important implications
in the situation where a substantial proportion of the
missing.
~bservations
are
If either the available data or expert opinion indicate that,
say, Sundays, Mondays and Wednesdays for the particular data set
characteristically produce low air pollution levels, then these days can
be omitted from consideration in determining the extreme value
distribution.
Therefore it is possible to obtain a valid extreme value
distribution from sparse data.
Some preliminary results need to be established before determining
the conditions under which asymptotic joint independence holds.
Let
(Y1""'Yp~
distribution functio'is
be a vector of random variables whose joint
Gi(Y:~~"'YPi)'
i
=
1, ... ,n .
~
"!
Lemma 2.2.1
Let
where Gj
is the marginal distribution function associated with Y.
J
.,
54.
ni is the dependence function. Then,
and
(2.2.34)
for large Y1""'Y
Pi
,
if
(2.2.35)
forall
j<k=l,,,,,P n and large Yj'Y k
bivariate dependence function.
where
nojk is the
Proof
Case I:
Assume that
(Y 1, ... ,Y
) for all
Pi
i = 1, ... ,n are non-
negatively associated such that
(2.2.36)
Now,
Pi
G'(Y1""'Y
1
Pi
) < 1 - p. +
1
Pi
Pi
L G.(y.) + L L Pr{YJ.>YJ.,Yk>y k} ,
j=l J J
j=l k=j+l
(2.2.37)
since
n
n
n
n
Pr( () A.) < 1 - L Pr(A.) + L I Pr(A.A.)
i=l 1
i=l
1
i=l j=i+1
1 J
using simple probability rules.
Thus
55.
(2.2.38)
and so
p.
n
.IT
1=1
S't i
p.
p.
111
(l-P.+ I G·(Y·)+·I
I PdV.>y.,V k >Yk}1
n
1 j=l J J j=l k=j+l
J J
(G 1 (y 1) , ... , Gp . (y .)) ~ .II
1
p.
1=1 {
P1
J
1
I
G. (y .)
j=1 J J
(2.2.39)
Pm
1-P +
Pm
I
I
G.(y.)+
pr{v.>y.,Vk>Yk}}n
J
j=1 k=j+1
J J
m j=1 J
;:;;;
!
L
Pm
m
L
j=1
G. (y.)
J J
Let G.(y.)
=1
= 1, ... ,Pn and let r n tend to zero; that is, consider
Yj
for some m contained in the set {1, ... ,n}.
j
(2.2.40)
P
{
for all
j
= 1, ... ,P n
J
J
- r
n
,
large
. Then,
(2.2.41)
(2.2.42)
(2.2.43)
56.
Since G. {y.}
J
J
tends to one and Pm
~
Pn'
and since, by assumption
then,
1im
n~
if
{2.2.44}
for large Yj'Y k ; which is equivalent to
lim n Pr{Yj>Yj,Yk>Y k}
rr+«'
for j < k
=0
{2.2.45}
= 1, ... ,Pn
By a similar argument this is necessary for
for
j < k
= 1, ... ,Pn.
The required result, for this case, is thus
proven.
Case II:
Assume that
{Y1, ... ,Y p .}
1
associated such that
i = 1, ... ,n
is non-positively
57.
Us i n9 the lower Frechet bound, from Frechet (1940),
p.
G1.(y 1' ... ,y Pi );;..
However, as
Gj(Yj)
1
max
(0 , j=l
I G. (y.)
J J
- p. +
1
tends to infinity, for
1) .
(2.2.46)
j = 1, ... ,Pi '
Pi
G. (Yl ' ... ,Y
1
Pi
);;..
I
j=l
G. (y .) - p. + 1
J
J
•
1
Then,
p.
1
I
n. (G (Y ) , ... , G (y )) ;;..
1 1 1
p. p.
1
1
j=l
G. (y .) - p. + 1
J J
,1
p.
1
L
G. (y .)
j=l
J
J
and
p.
1
n { j-1
I G.J (y J.) - p.1 + 1 }
n
1;;.. lim IT n.(G1(y1), .. .,G
n~ i=l
1
(y
Pi
Pi
)) ~ lim II
n~ i=l
-
.
.
Pi
L G. (y.)
j=l J J
(2.2.47)
Thus,
(2.2.48)
58.
as
G.(y.)
J
J
tends to unity, for j :: 1, ... ,Pn
In this case the lemma
holds without the further conditions.
We now prove that the asymptotic pairwise independence of maxima
implies the asymptotic joint independence of maxima, in the context of
the sub-sequences.
Theorem 2.2.2
The maxima Z (1), ... ,Z(s)
n1
if each pair of maxima
i < j :: 1, .. .,s .
. ns .
are asymptotically jointly independent
Z(l) ,Z(J)
n·1
n·J
are asymptotically independent, for
Proof
n
(
;=1
1
(i-l
( )
I n [X~l)~l""'X.s ~x ]}
1
s j=1 J
J
s
= II pr{x.1)~xl'''''X's)~x
(2.2.49)
using Bayes' rules.
For notational convenience, let
x(j)
i-k
for
k
=
y~j-ks)
1
= 1, ... ,i-1 and i = 1, ... ,n. So (2.2.49) becomes
59.
(2.2.50)
=
n { s
IT
i=l
IT
h=1
(h-l
i-I
pr{y.h)~ I n [y~k)~Xk] n
1
h
k=1
s
n
j=l £=1
[V~£-(i-j)S)~X£]}} ,
using Bayes· rule
(2.2.51)
where Fhi(xl,,,,,xh,xl, ... ,xs, ... ,xl'''''xs)
-4It
distribution function.
is a (h+(i-l)s)
variate
Further consideration of (2.2.51) yields
Let
(2.2.52)
Then,
(2.2.53)
60.
Therefore, by definition
=
S6 .
h1
h=l i=l S6(h_1) i
s
IT
n
IT
(2.2.54)
If each pair of maxima are asymptotically independent, then by Lemma
2.2.1
n
lim
n~
for h
=1
Q
hi
,
= 1, ... ,s; and
lim
n~
for h
IT
i=l
n
IT
i=l
Q
(h-1)i
=1
= 1, ... ,s. Therefore, from (2.2.41),
(2.2.55)
that is, the maxima
independent.
Z~l) , ... ,z~s) are asymptotically jointly
1
s
Theorems 1.3.13 and 2.2.2 can be used, in a straightforward manner,
to prove the following corollary.
Co ro11 ary 2. 2. 3
The maxima Z~l) , ... ,z~s) are asymptotically jointly independent
1
s
61.
if
Pi j (1- r , 1- r ) = 0 ( r)
fo r a11
p .. (F. (x . ) , F . (x .))
lJ 1 1
JJ
i <j
= 1, ... , s
where
= Pr{X.1 >x.1 , X.J >xJ. }
(2.2.56)
Corollary 2.2.3 gives reasonable sufficient conditions under which
lim Pr{Z n ";;;a nx+b n}
Let us, briefly, consider the condition of the above corollary;
P(l-r,l-r) = o(r)
-e
that is,
(2.2.57)
where
This is known as Sibuya1s condition, given in Sibuya (1960).
Suppose that the marginal distribution of Xl is a Weibull
(a 1 ,S1)
and that of X2 is Weibull (a2 ,S2)' Furthermore, assume that the
bivariate distribution is derived from the third Gumbel bivariate
exponential distribution referred to in Gumbel (1960).
That is
where m is a measure of the dependence between Xl and X2 .
we can show that
In fact,
62.
Therefore. the case m = 1 corresponds to the situation of independence.
This bivariate distribution is only defined for m ~ 1, that is for
non-negative association.
x2 are large.
Thus,
Consider the situation where both Xl and
We can, therefore, let Xl
1}
= F1 (l-r
and x2
= F2-1( 1-r)
(2.2.59)
However, since
eand
(2.2.59) becomes
= ~exP{21/mlogr}
1
= r 2 1m - 1
(2.2.60)
63.
Therefore, from (2.2.60),
equal zero;
P(l-r,l-r) = o(r)
as long as m- 1 does not
that is the correlation is not one.
In this
W~ibull
case,
then, Sibuya1s condition is satisfied if the logarithms of the variables
are not perfectly carrel ated.
The condition is, therefore, extremely
general and very reasonable for air pollutant concentrations.
Similarly,
it can be shown that if we assume that the marginals are log-normally
distributed then Sibuya's condition is, again, satisfied if the
logarithms of the variables are not perfectly correlated.
The above discussion, therefore shows that, for the applications of
interest in this dissertation as well as for quite general applications,
the assumption of the joint independence of the sub-sequence maxima is
quite reasonable.
-e
2.3 The Distribution of the k-th Extreme
Under the appropriate conditions, given in Theorems 1.3.2, 1.3.3 or
1.3.4 and by Theorem 1.3.6,
(2.3.1)
for
~
= 1, ..• ,5
k
fixed, and
(2.3.2)
where
H~(x) is as given by
extreme for the
~-th
(2.2.1) and
sub-sequence.
X~~~k+1:n~ is the k-th upper
Consider
64.
1im
n-+oo
~
= H(k}(A
x+B }
~
~
~
(2.3.3)
where
e-
and
Note that (2.3.3) is obtained from Theorem 1.3.7 and a simple
generalization of Theorem 1.3.5 to the situation for the k-th extreme,
made possible
by
the general nature of Khintchine's Theorem (Khintchine
(1938}).
We noted in the previous section that
for
~
A~
and
B~
were such that
=t
(2.3.4)
otherwise
for
~
= 1, ... ,s
where the t-th sub-sequence is the most extreme.
65.
The above results are true for the individual sub-sequences however, we
are only interested in the properties of the k-th extreme of the entire
sequence.
Therefore we consider
where Xn- k+1: n is the k-th upper extreme for the entire sequence of n
observations. Thus
(2.3.5)
The set {X n- k+1:n > anx + bn} can be expressed as the union of a
multitude of sets of the form
(2.3.6)
However, due to (2.3.3) and (2.3.4) the probability of these sets is
zero except for those which involve only the order statistics of the
t-th sub-sequence.
If we assume that each sub-sequence is unique then
there is only one such set, corresponding to the case where all the
k
observations greater than anx + bn belong to the t-th sub-sequence.
Therefore, assuming the asymptotic joint independence of the sub-sequence
k-th extremes, (2.3.5) reduces to
66.
1im
n~
PdX n_k+1: n,;;;; anx + bn}
= 1im [1 - Pr{x{t~k+1'
nt
. nt >a nx+b}]
n
n~
= 1im PdX{t\+1'
nt - .n t ';;;;a nx +b.n}
n~
= H{k){x)
t
(2.3.7")
k-1
I
r=O
/, [-
log Ht{X){
(2.3.8)
We have therefore shown that the distribution of the k-th extreme
can be obtained in an analogous manner in which the extreme value
distribution was derived.
Again we see that the most extreme sub-
sequence is the determining factor for the normalizing constants an
and bn and the distribution function Ht(x) .
2.4 The Multivariate Extreme Value Distribution
Suppose that we are interested in the joint extreme value
distribution for a number of variables.
For example, we may be
interested in obtaining the joint distribution of the largest ozone
concentration, the largest nitrous oxide concentration and the largest
non-methane hydrocarbon concentration.
Let
(2.4.1)
e-
67.
~
for
=
1, ... ,5 and
where xj(~)
i::: 1, ... ,r,
observation of the i-th variable in the
j:::
1, ... ,n~(i),
~:::
1, ... ,s and
size for the i-th variable is
variable in the
n(i)/s
and
I
In addition we shall assume that n~ (i)
i
and
j
, is such that
n~ (i)
~=l
for all
Since the sample
the sample size for the i-th
s
approximately equals
sub-sequence for
i::: 1, .. .,r.
n(i)
sub-sequence,
~-th
~-th
denotes the j-th
n~(i) ::: n( i)
for
n~(i)
i
:::
approximately equals
1, .. .,r
n,Q, (j )
.
Under the appropriate conditions, given in Theorems 1.3.2, 1.3.3
or 1.3.4, it is clear that,
(2.4.2)
where H~
is a non-degenerate distribution function for
~::: 1, ... ,s
The form of Hi is determined by the type of parent distribution,
where Fii)(x)
Fi i )
= Pr{x~(~) <x}. However, H~ must be one of three
types given in Theorem 1.3.1.
In addition
i (~)
an~(i)
,
positive, and
i(~)0) are norma 1 lzlng cons t an t s, f or ,Q, = 1 , ... ,s an d
b nf(,
(1
0
0
1.
= 1 , ... ,r
Therefore
Pr{Z nl
(O)".;;a nl
(o)xo+b
("), ;=1, ... ,1"}
1
nl
= Pr{Z
where
i (~)
(O)<a (o)xo+b (0)' 9;=l, .•. ,s; i=l, ... ,r}
nf(, 1
n 1 1 n 1
(2.4.3)
68.
Zn(i) = max{x}; j=l, .. .,n(i)}
= max{Zi(£)..
0-1 , ... , s}
n£(l)' ~and X~
J
,
denotes the j-th observation on the i-th variable.
We can
express (2.4.3) as
1
j
£= 1, ... , s; i =1, •.. ,r
(2.4.4)
If it could be shown that the z~(~~) are asymptotically jointly
£
independent then (2.4.4) gives, in the limit,
1 im
Pr{ Znl
(.) ~ an
(.)lx·
(.); i =1, ... , r}
l + b nl
n(i)~
i=l, ... ,r
Therefore, by Theorem 1.3.5,
lim
n(i)~
Pr{Zn(i)~n(i)xi+bn(i);
i=I, ... ,r}
i=I, ... ,r
=
s
r
IT
IT
£=1 i=l
(2.4.6)
69.
where an(i)
and bn(i)
A(i) =
Q,
i
=
l, ... ,r , are such that
;rm-
1im
i Q,
nQ, (i)-+ro an Q, (i)
(2.4.7)
and
b
_bi(Q,)
n(i)
nQ,(i)
B(i) = 1im
Q,
i( Q,)
nQ,(i)-+ro
anQ, (i)
(2.4.8)
for Q, = 1, ... , sand i = 1, ... , r .
The rules for choosing an(i) and bn(i) will be the same as
those discussed in Section 2.2.
In addition Corollary 2.2.3 is extended in a straightforward manner
·e
so that the maxima
z~~(~)
;
for Q, = 1, ... ,s and i = l, ... ,r;
asymptotically jointly independent if Pij(l-r,l-r) = o(r)
pairs taken from
.
are
for all
{X~;
j=l, ... ,n(i); i=l, ... ,r}. That is if all such
J
pairs are not pefectly correlated after the logarithm transformation has
been applied.
Again this seems qUite reasonable for the applications to
air pollution that will be considered in this dissertation.
The case for the k-th extreme can also be simply extended to the
multivariate situation to obtain,
1im
n(i)-too
i=l, .. .,r
(2.4.9)
i( Q,)
where Xn(i)-k+l:n(i)
denotes the k-th upper extreme for the i-th
70.
variable of the
~-th
denotes the limiting
sub-sequence and
distribution of the k-th extreme value for the ith-variable.
2.5 The Joint Extreme Value Distribution
A generalization of the theory of Section 2.3, of possible interest
in air pollution applications, leads to the consideration of the joint
extreme value distribution of the sub-sequence maxima.
For example we
may wish to determine the probability that the maximum for each weekday
exceeds a certain value while the maxima for Saturday and Sunday exceed
a different value.
That is we wish to find
; ~=l'."'S}
Under the conditions for which the Z(~)
n~
(2.5.1)
are asymptotically jointly
independent (2.5.1) can be expressed as
(2.5.2)
(2.5.3)
(2.5.4)
e·
71.
This result then allows us to vary the value of xi over the subsequences.
It is easily seen that this result can be extended to the
multivariate situation.
2.6 The Effects of Random Sample Sizes on the Extreme Value
Distribution
In the discussions so far we have considered the sample sizes to be
fixed constants.
variables.
In reality they can be considered to be random
The monitoring equipment could fail so as to end the
sequence of observations or the situation could change so drastically,
through a review of vehicle emission laws for example, that the data set
·e
must be separated from any subsequent data and analyzed accordingly.
such cases the sample sizes,
n£,
are random variables.
In
We would like
to be able to show that, in such cases, the results of sections 2.1 to
2.5 are still valid.
Sen (1972) considers the extremal stochastic processes
{m n(t):
O~t <co},
n;;;' 1, where
t > lin
(2.6.1)
= {
based on the sequence
where each Xi
Suppose
o~
t < l/n
{X 1,X 2, ... } of independent random variables
has a common distribution function F .
{N n; n;;;.l}
random variables such that
is a sequence of non-negative integer valued
72.
n
where
-1
N
n
in probability as
A
-+
n
-+
ro
(2.6.2)
,
A is a positive random variable having an arbitrary distribution.
For every n
~
1 ,
we define a stochastic process
if
(2.6.3)
Nn = a
if
where mk(t)
{m(t}:
is defined by (2.6.1).
a ~t <co}
Pr{m ( t)
Consider then a Markov process
for which for non-negative
~ x} =
t
and
s
(2.6.4)
[H ( x ) ] t
and
a
Pr{m(t+s) <.y
I m(t)
• x}
if
(2.6.5)
= {
[H(y)]s
where
H(x)
y < x
if
y
~
x
is defined by
H(x) = Pr{Z n ~a nx+b n}
(2.6.6)
Sen (1972) shows that:
Under (2.6.2) and (2.6.6),
Skorokhod J 1 - topology on
(3 = a when H is of the type
m
converges in distribution in the
Nn
D[(3,1] to the Markov process m, where
¢
a
,
and
(3 >
a
for the other two
e·
73.
types:
where
m = {m(t), tE [0,1J}
(2.6.7)
and
m
N
n
= {m N (t),
t
E
(2.6.8)
[0,1]}
n
Therefore
lim
N~
Pr{z[Nt]~aNx+bN}
n
n
n
n
= Pr{m( t) ~ x}
= [H(x)]t
for t E [0,1] • Thus
lim
N~
n
PdZ N ~aN x+b N } = H(x)
n
n
n
The above result of Sen (1972) shows that even where the sample
size is a random variable the results obtained for a fixed sample size
are still valid, provided that the random sample size is determined in
some specified way so that (2.6.2) holds.
We note this requires that
the sample size tends to infinity as the number of samples tends to
infinity.
This would preclude the situation of deliberate termination
of monitoring at high concentration times, such as summer.
It would,
74.
however t accommodate the situation where termination of the data set
resulted from a change in the characteristics of the air shed.
In summary then t for most situations arising in the monitoring of
air pollution levels t we lose nothing by considering the sample sizes
as being fixed constants.
III.
3.1
ESTIMATION TECHNIQUES
Introduction
In the previous chapter we showed that the extreme value
distribution depends on the values of the sub-sequence normalizing
constants which, in turn, depend on the parameters of the sub-sequence
parent marginal distributions.
The problem of estimation of these
parameters is rarely addressed in the literature on extreme value theory.
Barlow (1972) obtained bounds on the normalizing constants.
Gumbel
(1958), like others, assumes that a very long time series of data is
available so that, given the form of the extreme value distributions the
normalizing constants can be estimated directly, thus bypassing the
estimation of the parameters of the parent distribution.
of air pollution such data are not available.
In the field
Indeed, even if such data
were available they could not be used because the air shed itself is so
dynamic that one large data set could not justifiably be characterized
by one parametric model.
We must, therefore, be careful not to violate
the premise of extreme value theory and thus consider only sets of data
for which it is reasonable to assume that the air shed has remained
stable over the period of observation.
Typically, in line with current
United States Environmental Protection Agency practice, we will consider
no more than three years of observations in any single analysis.
this chapter we will examine methods of directly estimating the
parameters of the sub-sequence parent marginal distributions.
In
76.
We know, from Theorems 1.3.2, 1.3.3 and 1.3.4 that the form of the
extreme value distribution also depends on the type of parent marginal
distribution that is postulated.
It is, therefore, important to choose
this family of distributions with great care, taking into consideration
both statistical and environmental matters.
3.2
Parametric Models for Air Pollution Data
Larsen (1961) stated that carbon monoxide and ozone concentrations
in the Los Angeles area
1I • • •
logarithmically distributed
ll
indicate that the variables tend to be
•
Subsequently Larsen (1964) expressed the
suitability of the log-normal distribution for air quality data in
general terms:
IIAir pollution concentration data usually fit the bell
(Gaussian) shape, if concentration is plotted on a logarithmic scale".
Larsen's results very firmly cemented the appropriateness of the lognormal distribution in the minds of the environmental scientists.
Of
course the ease of computation of the parameters certainly appeals to
the monitoring personnel who wish to get a quick indication of the
effect of various control strategies.
These considerations do not have
a primary place in the detailed statistical analysis of air pollution
data for the purposes of either the setting of standards or the
assessment of performance relative to these standards.
There were many attempts;
Gifford (1974), Mage and Ott (1975),
Benarie (1974), Bencala and Seinfeld (1976);
to justify the choice of
the log-normal distribution from an environmental or meteorological
perspective.
These were either incorrect or based on questionable
assumptions.
For example, it has been proposed, rather circularly, by
77.
Bencala and Seinfeld (1976) that the log-normality of pollutant
concentration frequency distributions can be explained by the near lognormality of wind speed distributions.
We maintain that, since very
little is known, in fine detail, of the formation of pollutants in the
environment, any model postulated for this system does not automatically
endow authenticity to the frequency distribution theoretically derived
from such a model.
The only justifiable test of the adequacy of a
frequency distribution is how well it fits several varied sets of air
pollution concentration data.
It has been suggested that, since we are
usually interested in the larger observations, we should utilize the
family of distributions which gives the best fit in the upper tail of
the distribution.
In the context of the extreme value distribution this
procedure is inappropriate. Although we are interested in the extreme
values the distribution of these depends on the parameters of the entire
parent distribution.
Many other probability distributions have been proposed to
represent air pollution concentrations.
Lynn (l974) considered the
Pearson Types I and IV distributions as well as the Gamma distribution.
Curran and Frank (1975) reasoned that a light-tailed distribution such
as is available within the Weibuil model, should be considered for
fitting light=tailed data sets such as air pollution concentrations.
Johnson (1979) also preferred the Weibull distribution.
Mage (1980) has
postulated the use of the four-parameter log-normal distribution known
as the S8 distribution.
Ott, Mage and Randecker (1979) state that:
"At the present time, there appears to be a growing consensus that the
data analyst should not automatically choose anyone probability model
to the exclusion of others, but, rather, that he should carefully
78.
evaluate the problem he is addressing and the data at hand before
selecting a probability model".
Such a practice is overly data
dependent and could result in a mixture of distributions applicable to
various sites, times and pollutants.
We believe it is desirable to use
one distribution which is sufficiently flexible to fit any set of air
pollution data well.
We maintain that the Weibull distribution is
particularly suited to this task.
For the purposes of comparison, as well as accommodating those who
strongly adhere to the log-normal characterization of air pollution
concentrations, we will consider separately, two possible models.
The
first assumes that the sub-sequence parent marginal distributions are
log-normal while the second assumes the Weibull distribution.
3.3 Parametric Estimation Techniques
Once the distribution family has been specified we will use the
method of maximum likelihood to estimate the parameters.
We must,
therefore, be able to write down the likelihood function associated with
this distribution.
Although we have chosen only two probability
distributions, there are many ways in which the sub-sequences can be
handled, each resulting in a different likelihood function, given the
same probability distribution.
problem has been placed.
Let us recall the context in which our
Classic extreme value theory assumes that all
the observations are independent and identically distributed.
obviously not true for daily air pollution concentrations.
This is
We noted
that it is reasonable to divide the entire sequence into s sub-sequences,
each of which is assumed to contain independent and identically
79.
distributed observations, although the marginal distributions are
allowed to vary over the sub-sequences.
For orientation two likelihoods are presented.
The first is
derived from the classic approach in which the likelihood function,
based on the set of observations
=
where f(.)
n
II
i=l
{x 1, ... ,x n} is given by
f(xi;~)
(3.3.1)
is the probability density function and e is the vector
of pa rameters .
The other extreme could also be postulated in which we assume that
the parameters associated with a sub-sequence are totally independent of
the parameters associated with any other sub-sequence.
Although in
Chapter II we allowed for dependence between the sub-sequences, for the
purposes of estimation, we adopt the strategy which gives the greatest
scope for variation in the sub-sequence parameters.
function, based on the set of observations
Thus the likelihood
{x~~); i=l, ... ,n~; t=l, ... ,s} ,
is given by
(3.3.2)
where
f~(.)
is the probability density function associated with the
i-th sub-sequence and ~(i)
is the vector of parameters of f~;
for
i=l, ... ,s.
In each of the above situations maximum likelihood techniques can
be used to estimate the parameters.
The maximization algorithm which we
will use is called MAXLIK and is described in Kaplan and Elston (1972).
80.
In conjunction with a user supplied subroutine which computes a loglikelihood for a given set of parameter values, this algorithm searches
the likelihood surface to find maximum likelihood estimates of the
parameters and computes an estimate of their variance-covariance matrix.
Recall, from (2.2.1), that
1 im
n-too
(3.3.3)
~
for ~
= 1, ... ,s
; where a{~),
n~
positive, and b{~)
n~
are functions of
the parameters of the parent marginal distributions, which are assumed
to be known.
In the application of this result, however, we use maximum
likelihood estimates of the parameters to estimate the normalizing
constants.
We would like to be able to say that (3.3.3) holds when the
normalizing constants
a{~)
n~
and
b{~)
n~
are replaced by their estimates.
For the purpose of showing this to be the case we will drop the sub~.
sequence indicator,
Equation (3.3.3) is then equivalent to
(3.3.4)
as
n tends to infinity, where the random variable Z has the
distribution function
H and the notation
£
indicates convergence in
di stribution.
Lemma 3.3.1
Let an
respectively.
and
6n
Then,
denote the estimates of an and bn
~
.
81.
H(x)
=
lim Pr{Z n ~a nx +6}
n
,
(3.3.5)
n~
if
(3.3.6)
as n tends to infinity, and
g
0
as n tends to infinity, where the notation
(3.3.7)
p
+
indicates convergence
in probability.
Proof
Since (3.3.6) and (3.3.7) hold then by Slutsky's Theorem,
Therefore, by (3.3.4)
82.
To verify the conditions (3.3.6) and (3.3.7) we will restrict
consideration to the two models proposed in Section
3.2~
namely the
Weibull and the log-normal, however the technique could be applied to
any postulated model.
Firstly let us consider the Weibull model
in which case
~
(3.3.8)
where
vector
(an'~n)
is the maximum likelihood estimate of the parameter
If we employ a first-order Taylor's series expansion
(a~S)
of n(&n~Sn) around the point
(a,S) we obtain
(3.3.9)
for
~** J
[ ;.>
= l(aS J
+ yn
o<y
21 11 'J)
-1: (
< 1
Then,
= _
S~
[10;* n
f
A
/S
*
{a~:a + S~:S
10g[10;* n
J}
(3.3.10)
Since an and Sn are maximum likelihood estimates then !l1 (an-a)
and In(Sn- S)
£0
re-expressed as
~ as n tends to infinity.
£0
Equation (3.3.10) can be
83.
=-
s~ (a.*}-l/S* { a.~ (log n)
1/13*
In
In(a
-a}
n
{3.3.11}
Therefore n(an,Sn) - n(a,l3)
lim (log n) 1/13
n~
rn
g0
as n tends to infinity if
= 0
(3.3.12)
and
·e
lim (log n)1/13
n~
10g(10~ n) = 0
•
(3.3.13)
J
In
It can be shown that this is the case when 13 > 0 and a > O.
have, therefore, shown that for the Weibu11 model
tends to infinity.
1'\
bn - bn
p
~
We
0 as
We also know that
where R(')
is defined by (1.3.9).
Since R{·)
function of
bn , an and Sn and is not directly dependent on n
then a standard result of probability theory gives
p
-+
0
is a continuous
n
84.
as
n tends to infinity.
H(x)
Therefore by Lemma 3.3.1
= lim Pr{Z n .,;;;8: nx+b n}
n~
for the Weibull model.
Let us now consider the log-normal model.
Therefore,
(3.3.14)
where
Tn = (2 log n)~ _ ~(log log n + l~g 4n)
(210gn)'2
and
(3.3.15)
(0 n,8 n) is the maximum likelihood estimate of the parameter vector
(~,a).
Using a first-order Taylor1s series expansion of ~(0n,on)
around the point
(~,a)
we have
A
~n
A
a
for
0<0<1.
Then,
n
= ~*
= 0*
(3.3.16)
e·
85.
= (0 n-~)
exp(~*+o*T ) +
n
(0 n-o)T n exp(~*+o*T)n
(3.3.17)
Therefore ~(Cn,an) - ~(~,o)
.
g0
1::
lim exp(0(210gn)2)
rn
n-+oo
as n tends to infinity if
= 0
(3.3.18)
and
1im
n-+oo
·e
1::
(2 log n)~ exp(0(2 log n) 2)
In
= 0 .
It can be shown that this is the case for all
shown that for the log-normal model
infinity.
~
an
Thus
~
=0
~
(3.3.19)
and o.
6n - bn g 0 as
Thus we have
n tends to
We also know that
A
b
n n
(210gn)
-k
2
(3.3.20)
86.
g
as
n tends to infinity since
"'b n - bn
and
0
£~
0
In(cr n-0) £ 0
Therefore, by Lemma 3.3.1
H(x) = lim
n-+OO
Pr{Z ::;;;a x+b }
n
n
n
for the log-normal model.
We have therefore shown that for the two models to be considered in
this dissertation the extreme value distribution remains unchanged when
we use the estimates, rather than the true values, of the normalizing
constants.
Although we have decided to consider only two families of
distribution functions there are other matters which could affect the
estimates of the parameters.
The first of these is the decision on how
many parameters are needed to adequately fit the data.
The standard
forms of the log-normal and Weibull distribution incorporate two
parameters.
We could add another parameter as a shift operator, if we
believe that there is an actual or theoretical non-zero lower limit to
the variable being considered.
This, we believe, is desirable
considering the nature of the variables we will be analyzing.
Mage
(1980) advocates a further parameter which acts as an upper limit on the
e·
87.
variable of interest.
While we accept that a theoretical upper limit
does exist it will be so large compared to the range of actual
observations as to make estimates of the fourth parameter highly
questionable. We, therefore, advocate the use of a three parameter
version of the distribution function in all cases.
The second
consideration is the degree and level of censoring of the data.
The air
pollution monitoring equipment has a minimum detectable limit (MOL)
under which concentrations cannot be measured.
Therefore, if we note
that r observations are assigned the MOL then all we know is that r
observations were at or below the MOL.
This is known as left censoring.
Where the third parameter, which is a shift parameter, is negative we
have the case of left truncation.
-e
This situation would indicate that
the probability of a value near zero is relatively high.
density f(x;8)
fCT(x;~,8)
The probability
is adjusted for censoring and truncation to obtain
where f CT is given below and 0 is the minimum detectable
1imit.
Algebraically we can express f eT as:
Case I:
where 83 is the shift parameter.
if
x< 0
F(o;8) - F(O,e)
(l-F( 0, e))
if
o< x < 0
f(x;e)
1-F(O;e)
if
x > <5
0
.
fCT(x;~,<5)
e
=
-
-
(3.3.21)
88.
where F{x;e)
is the cumulative distribution function.
Case II:
o
=
F{o;8)
if
f{x;e)
if
{3.3.22}
x> 0
Case III:
{3.3.23}
e-
The likelihood functions can then be modified accordingly.
A consequence of analyzing real data sets is the occurrence of
missing observations.
If it is reasonable to assume that these missing
observations are randomly lost then all of the above results can be
carried through without violation of any statistical precepts.
However
if there is reason to suspect any systematic loss of information, for
example, extremely high pollution days being omitted, then we need to
account for these missing values. The most obvious method would involve
imputing the missing value according to some pre-specified algorithm.
It is not within the scope of this dissertation to either compare the
various methods of imputation or recommend further techniques.
The
e
89.
strategy adopted in the analysis, detailed in the following chapter, is
one of assuming random loss of information.
missing values.
Consequently we ignore
It should be noted, however, that in Section 2.2 we
showed that observations from sub-sequences other than the most extreme
do not contribute to the extreme value distribution.
Therefore those
sub-sequences which are clearly not extreme can be ignored and so missing
observations in these sub-sequences are not a problem.
This result is
particularly relevant for air pollution regions in which the data is
scarce.
Of course as the number of missing observations and their
degree of clustering increases the reliability of the parameter estimates
decrease.
We cannot possibly make reliable inferences from too scant a
data set.
-e
We have now given a description of the parametric techniques which
we will employ to obtain estimates of the normalizing constants. These
will be used in determining the distribution of the maximum and the k-th
extreme for the entire sequence of observations.
3.4 Partial Non-Parametric Estimation
We have seen earlier that there is a degree of controversy over the
selection of a parametric model.
avoid such assumptions.
It would be convenient if we could
This can be done, by assuming some structure
on the relationship between the variable of interest and some covariates
relevant to this variable.
For example, the concentration of ozone
depends on the concentration of both nitrous oxide and non-methane
hydrocarbons, as well as some meteorological variables.
The technique
to be described here was developed by Cox (1972) in the context of life
tables.
90.
Instead of the distribution function we concentrate on the survival
F,
distribution function,
F(x)
given by
= Pr (X > x)
A(X), given by,
and the hazard function,
Pr (x
A(X) = lim
_ f(x)
Consider concentration
~
X < x + lIx
I X;;;. x)
L1x
L1x-+O+
- F(X)
(3.4.1)
(3.4.2)
.
X and suppose a vector z
~
= (zl'"
.,Z )
p
I
of
explanatory variables, or covariates, has been observed.
Note that z
may include both quantitative and qualitative variables.
The principal
problem dealt with in Cox (1972) and Kalbfleisch and Prentice (1980) is
that of modelling and determining the relationship between
However, they also go further and provide ways of estimating
thus
x and z .
A(X) and
F(x) . This technique provides us with a way of estimating the
sub-sequence parent marginal distribution, conditional on certain
covariate values, in a partial non-parametric manner.
Let A(X;Z)
represent the hazard function at concentration
a day with covariate z.
x for
The proportional hazards model proposed by
Cox (1972) specifies that
(3.4.3)
where
AO(X}
is an arbitrary unspecified hazard function for continuous
91.
X and
= (Yl""'Y)
p is a vector of regression parameters. In this
y'
~
model, the covariates act multiplicatively on the hazard function.
This
is the condition which we apply in order to withdraw the assumption of
a particular parametric probability distribution.
The conditional
density function of X given z corresponding to (3.4.3) is
f(x;~)
x
= AO(X) exp(z·y) exp{-exp(z·y) f AO(u)du}
a
(3.4.4)
The survival function is therefore
x
= exp{- J AO(U) exp(z·y)du}
a
F(x;z)
(3.4.5)
·e
where FO(x)
is the survival function associated with AO(X).
The set
of days whose concentrations are greater than or equal to x is called
the risk set at concentration x and denoted by R(x).
For the
particular day at concentration x(i) , conditionally on the risk set
R(x(i)) , the probability that the concentration is on that day, as
observed, is
(3.4.6)
=
for i
= l, ... ,h,
where h is the number of distinct concentrations.
The argument proceeds by noting that, since AO(X)
is completely
unspecified, no additional information about y is obtained from the
92.
observation that no concentration is realized in
(i=l, ... ,h)
(x(i_1),x(i))
This is intuitively the case since we can account for
this observation merely by taking AO(X)
this interval.
to be very close to zero over
If one had additional information on AO(X) ;
example, a parametric form;
for
there would, or course, be contributions to
the inference about y
from the intervals with no concentrations.
partial likelihood for
y
concentrations
L(y) =
The
is now formed by taking the product over all
of (3.4.6) to give
h
II
(3.4.7)
i=l
The partial likelihood, it should be noted, is not a likelihood in the
usual sense in that the general construction does not give a result that
is proportional to the conditional or marginal probability of any
observed event.
Nonetheless, it has been shown, somewhat informally, by
Cox (1972) that the method used to construct this likelihood gives
maximumllpartial ll likelihood estimates that are consistent and asymptotically normally distributed with asymptotic covariance matrix
estimated consistently by the inverse of second partial derivatives of
the log-likelihood function.
It follows that the same asymptotic
results for y hold for estimation from partial likelihood as for the
usual likelihood function.
Unfortunately, it is quite likely that the data will be recorded in
a form involving ties.
If these are small in number a relatively ad hoc
modification of the above procedures will be satisfactory.
To cover the
possibility of an appreciable number of ties, we generalize (3.4.3)
93.
formally to discrete concentrations by;
A(x;z)dx
l-A(x~Z)dx = exp(:l
r)
AO(x)dx
l-Ao(x)dx
.
In the continuous case this reduces to (3.4.3);
(3.4.8)
in the discrete case
A{x;z)dx is a non-zero probability and (3.4.8) is a logistic model.
The typical contribution (3.4.6) to the likelihood now becomes;
(3.4.9)
where
~(i)
is the sum of
~
over the days with concentration x(i)
and the notation in the denominator means that the sum is taken over all
distinct sets of m(i)
days drawn from R{x(i)) , sinc~ m(i)
number of days with concentration equal to x(i)
and
i~l
is the
m(i) = n
This partial likelihood can be maximized to find estimates of y .
Once we have Qbtained the maximum likelihood estimates of y,
we
can consider the estimation of the distribution associated with the
hazard; ei ther for z
- = 0-
or for some othergi ven val ue of z .
-
The contribution to the likelihood of one day with covariates z
and concentration x(i)
is,
(3.4.10)
The likelihood function can then be written
h { ~~D. {
(
) exp(zly)
(
) eXP{zly)}}
(
)
L = i~O
[FO x(i)]
_£- - [FO x(i)+O]
_£,3.4.11
,
94.
where
D.1
X(i)
and
is the set of labels associated with days having concentration
DO
is empty.
It is clear that
L is maximized by taking
for x(i) < x ~ x(i+l)
FO(x) - FO(X(i)+O)
and allowing probability
mass to fall only at the observed concentrations
x(l)"",x(h)'
These
observations lead to the consideration of a discrete model with hazard
contribution 1 - a j
;
at X(j)
for
= 1, ... ,h. Thus we take
j
i-I
=
where
0.
0
1.
=
i = 1, ... ,h
II ex..
j=O J
On substitution in (3.4.5) and rearranging terms, we
obtain
h
11
{ IT
,
exp(z~y)
~J~)
1
(l - a.
JED.
i=1
(3.4.12)
IT
,Q,ER(x(i ) )- Di
as the likelihood function to be maximized.
We can take y
=
y
as estimated from the partial likelihood
function and then maximize (3.4.12) wi th respect to
1 ' ... ,a h Di fferentiating the logarithm of (3.4.12) with respect to a. gives the
0.
,
,
maximum likelihood estimate of a.
exp(z~y)
2
,
exp~;I.y)
JED. 1
- a·
A
,
~J~
as a solution to
2
=
,Q,ER(X( i))
exp (ftY)
If only a single concentration occurs at x(i)'
ai
directly for
a.
,
=
(3.4.13)
(3.4.13) can be solved
to give
1 _ __e_x_p_(~-,(...:..i .l-)2_)__
I
,Q,ER(X(i))
exp(z~y)
~ ~
(3.4.14)
95.
Otherwise an iterative solution is required;
a suitable initial value
for the iteration is a io where
log aio
=
-m( i)
-,
(3.4.15)
which is obtained by substituting
=::
in (3.3.13).
1 + exp(z I. y} log
~J~
a,'
Note that the &i1s can be separately calculated.
The
maximum likelihood estimate of the survival function is then
(3.4.16)
which is a step-function, with discontinuities at each observed
concentration x(i)'
This step-function is not amenable to the
establishment of the conditions of Theorems 1.3.2, 1.3.3 or 1.3.4.
Therefore we cannot ascertain the domain of attraction of this distribution and so we are unable to obtain the extreme value distribution.
Of equal importance is the problem of extrapolation of the survival
function beyond the maximum observed concentration.
Since the step-
function is equal to zero beyond this value the estimated survival
function is of no use in estimating the probability of exceedance of a
concentration greater than the observed maximum.
We will therefore
introduce some further smoothing assumptions which will alleviate both
96.
of the above problems.
We know, from (3.4.2), that
(3.4.17)
We will impose some restrictions on the form of AO(X).
such that the form of AO(X)
hazard function.
For example;
These will be
approximates many possible forms of the
the exponential distribution has a
constant hazard function, AO(X) = k; for the Weibull distribution
AO(X) = _aSx S- 1 which can be approximated by a polynomial; while the
hazard function for the log-normal distribution can be approximated by a
power series using a Taylor1s series expansion.
Therefore, let us assume
that
(3.4.18)
for
PI
negative and
P2
positive, since in our applications the
hazard function is increasing.
We could proceed in either of two ways.
Firstly, from (3.4.18) and (3.4.17) we obtain
(3.4.19)
Also, from (3.4.16), we know that
(3.4.20)
Using
(3.4~19)
and (3.4.20) we could estimate
some criteria of closeness.
PO' PI and P2 based on
97.
Alternatively we could reconsider the likelihood function given by
{3.4.11}.
From {3.4.18} and {3.4.3} we have,
(3.4.21)
and so, by {3.4.17}, we obtain
Therefore,
{3.4.22}
-e
The likelihood function, based on the sample
h
L(x;z}
0:
x
IT
;=1
IT
,
,Q.ED.
ex p{-
I
i=l
(PO
+Plx-(~) +P2 x(i}}
(POX(i) +P1logx(i)
+~P2xZi)} ,Q.ED.
I exp{:~y)}
,
(3.4.23)
where
(3.4.24)
98.
(3.4.25)
and
(3.4.26)
The likelihood, given y,
can be maximized using the MAXLIK algorithm
to obtain estimates of PO' PI and P2.
The underlying hazard function
is, therefore, estimated by
~O(x)
and the distribution function, for a particular value,
~O'
of the
covariate vector, is estimated by,
(3.4.27)
We prefer this second method because of its intuitive appeal and
relative ease of estimation.
Because of the two stage method of estimation of the underlying
parent distribution the maximization routine can only produce an estimate
of variance, given the regression coefficient,
unconditional estimate of the variance.
y.
The likelihood function (3.4.3)
can be re-expressed as
L(x,z)
0:
l( i h=1II
II
Q,ED.
1
P'~(i)
~
We require an
J exp{-p'C(y)}
99.
where
and
Therefore,
h
-e
log L(x,z) =
- -
L
L
log p'~(;) - p·C(Y) + constant
;=1 £ED.
.
1
So,
"'1
L =
o 09
ap
~(1')- I --
h
L
;=1 £ED.
1
pl~(;)
C( y}
and
a2 log L = Ih I
apap'
;=1 £ED.
--
(3.4.28)
1
and
a2 log L
aear'
aC(y)
=
=
ay'
[ aC l (y)
dY'
dC (y)
2
dY'
ac3 (y) ]
dY'
(3.4.29)
100.
where
(3.4.30)
=
ac 2(y)
dY'
h
=
I
log x(i)
i=l
L :£
.Q,ED.
exp(:~y)
(3.4.31)
1
and
aC 3(y)
dY'
h
=
I
h2
i=l
2
x( i)
L
.Q,ED.
:~ exp(:~y)
(3.4.32)
1
From the first stage of maximization we know the estimated variancecovariance matrix of
y.
Denote this by V(y) . Therefore by the
~
~
~
asymptotic normality of the maximum likelihood estimates we know that
2/ 109 L
apap'
a,
(
82 10 9 L -1
apay'
_ a2 109 L I '
8p8y I J
Let the above variance-covariance matrix be denoted by
Then, from standard multivariate normal theory we have,
e-
101.
for large n.
Therefore
(3.4.33)
Thus
(3.4.34)
which can be calculated.
Using the well known conditional result
(3.4.35)
·e
Thus we can estimate the unconditional variance by
A(A)
Ve
since
= El2
-1
E22 ~
(A)(
r
-1)1
E22
I
(AlA)
~12 + V P r
(3.4.36)
V(ply) is known from the second stage of maximization.
This method then produces estimates of variance of the estimates of
the parameters of the initial hazard function.
The distribution function for a particular value, :0' of the
covariate vector is thus estimated as
(3.4.37)
To obtain the exteme value distribution we first determine to which
domain of attraction this distribution belongs.
102.
Using Theorem 1.3.2 we know that the distribution is not in the
domain of attraction of
~
ex
since w(F)
=
00.
Now,
A
I-F(tx;~o)
1 im
A
I-F(t;~o)
t~
=
lim exp{-exp(~Oi){Potx+Pllog(tx) +~P2t2x2_pot-P1l09t-~P2t2}}
t~
(3.4.38)
A
However
PI
is negative and so according to Theorem 1.3.2,
F(x;~O)
is
not in the domain of attraction of ¢ex
Now since
and
then,
A
I-F(x;:o)
A
f(x;~o)
Thus,
=
(3.4.39)
103.
and
A
11"m L
d
X~ x
[ l=F(X;~o}] = 0
.
(3.4.40)
n
f(x;~O}
A
According to Definition 1.3.1 the distribution function
von Mises function.
F(x;~O}
is a
Therefore, by Theorem 1.3.4, the distribution is in
the domain of attraction of A.
Let us examine this approach more fully.
Having obtained the
·e
A
b
A
n
= infO - F(x;~o) ~ lin}
(3.4.41)
and
(3.4.42)
Thus,
A
F(bn;~O}
=1 -
lin
,
104.
since F(')
is a continuous function.
So
(3.4.43)
-exp(zo·y){pOb +P1l0gb +P2(b )2} :: -log n
- n
n
n
"'-
which can be solved iteratively for b
Equation (3.4.42) can then be
n
solved using numerical intergration. Analogous to the work in Chapter
II on the parametric model, we can consider only that sub-sequence which
is most extreme for calculating the extreme value distribution.
The
criterion for determining the most extreme sub-sequence is not as clear
as was the case in the parametric model.
However, the appropriate sub-
sequence could be ascertained by an examination of the plots of the
estimated distribution functions.
As we showed in the parametric case we would also like to be able
to replace the normalizing constants
an and bn by their estimates
and still obtain the same extreme value distribution.
Since
then
(3.4.43)
:: log n
After a little manipulation this equation reduces to
~
.
105.
= -po(b" n-b n)
- P1log(b" n/b n ) -
"2 2
n-b n)
~P2(b
We know that the maximum likelihood estimates,
PO'
(3.4.44)
13
1 and 13 2 , are
such that
A
p.
1
as
P
..,.
p.
i
1
n tends to infinity.
probability, to zero as
= 0,1,2
Thus the left hand side of (3.4.44) tends, in
n tends to infinity.
Therefore the right hand
side of (3.4.44) must also tend, in probability, to zero; which is true
if and only if
as n tends to infinity.
Also since
is a continuous function of bn ,
also true that
where R(')
PO' 131 and 1)2 then it is
o ,
as n tends to infinity.
Therefore by Lemma 3.3.1
106.
H(x)
=
lim Pr{Z n ";;a nx+b}
n
n-+<JO
that is, we can replace the normalizing constants by their estimates
without invalidating the limiting result of extreme value theory.
In this section we have detailed the method by which a partial nonparametric approach can be used in extreme value theory, using auxiliary
information.
It will be instructive to compare the results of the two
parametric models with those from this method.
following chapter.
This will be done in the
IV.
4.1
APPLICATIONS TO AIR POLLUTION DATA
Introduction
In the preceeding chapters we have developed a theory which enables
us to obtain a limiting distribution of the k-th extreme; where
k
= 1,2, ... , is fixed; for certain non-stationary sequences of
observations we have decided to restrict discussion to two parametric
families for the parent marginal distributions and one partial nonparametric approach which utilize's the proportional hazards model of
Cox (1972).
The estimation techniques chosen are based on a maximum
likelihood algorithm called MAXLIK and allowance is made
~r
left
censoring of the observations.
To illustrate this theory we have chosen a data set of air
pollution observations. The main variable of interest will be daily
maximum ozone concentration.
This was selected because the ozone
standard was the first to be expressed in probabilistic terms by the
United States Environmental Protection Agency.
Auxiliary variables to be
considered in the application of Cox's model are daily maximum carbon
monoxide concentrations (mg/m 3), daily maximum temperature (OF) and
daily average wind speed (mph).
To elicit the most relevant comparisons
between the various theories we believed it would be best to consider an
Air Quality Region for which few daily maximum ozone concentrations were
above the standard.
One such borderline region is Portland, Oregon.
Between 21 June 1978 and 20 June 1981 only two days exceeded the standard
108.
of 0.125 parts per million (ppm).
The data set corresponding to this
region and time period is summarized in Table 4.1.
The auxiliary
variables referred to were obtained from different sites.
Because of
the way in which the monitoring network is organized in the United States
Environmental Protection Agency it is neither possible nor meaningful to
use ozone and carbon monoxide measurements from the same site.
The
carbon monoxide concentrations were measured at a sampling site in downtown Portland, Oregon at 718 W. Burnside.
The site chosen for the ozone
monitoring is in suburban Portland, downwind from the downtown site.
Under this arrangement it is reasonable to associate the carbon monoxide
emitted in the downtown area with the ozone measured in the suburban
downwind areas.
The meterorological variables were measured at the
International Airport at Portland, Oregon and, therefore, are considered
to represent the meteorological conditions of the area.
In this chapter we will present and comment on the results from
both the parametric and the partial non-parametric models.
We will then
compare the resulting extreme value distributions and discuss the
consequences of any differences.
This comparison will concentrate on
three summary statistics of the extreme value distributions;
the mean and the probability of exceeding the standard.
the median,
Recall from
(1.3.21) that
k-1
= H(x) I /' [r=O
where k = 1,2, ...
is fixed.
.
109 H(x) {
(4.1.1)
109.
For the models we will be considering in this analysis H(x)
has
the form,
H{x}
= exp{-exp(-x})
.
(4.1.2)
From {4.1.1} we then have
Median(X n_k+1: n)
where X~-k+l:n
= an Median(X n* _k+1: n)
+ bn
'
(4.1.3)
has the distribution function H(k) (x) .
Similarly,
(4.1.-4)
Now, let mk
= Median(X *n_k+1: n)
k-1
I
r=O
, then
r\ [- log H(m k){
=
0.5 .
(4.1.5)
This can be solved quite simply, using a programmable hand calculator,
and from (4.1.2) and (4.1.3) the medians are presented in Table 4.2.
To obtain the means we will employ a little calculus.
be the moment generating function of X*n- k+1: n '
Gk(t)
= /00
exp(xt)h(k)(x)dx
_00
From (4.1.1) we can also show that
Let Gk(t)
Then,
( 4.1.6)
llO.
= h(x) [-logH(x)]k-l/(k-l)!
h(k)(x)
(4.1.7)
where
h(x) = exp( - x - exp( -x))
(4.1.8)
.
Therefore,
=
1
(k-1)!
00
f
exp[-(k-t-1)x-exp(-x)]exp(-x)dx
.
(4.1.9)
_00
Using the transformation
Gk (t)
z = exp(-x)
we obtain
= Ii1=tl
Tk=nT'
=
(4.1.10)
{k~l(k-~-tJ}
v=l
r(l-t)
k v
.
(4.1.11)
Thus,
(4.1.12)
Differentiating with respect to
1-t
IfR:H1
r 1-t
Therefore,
_
t
we get,
k-1
I
v=l
-1
(k-v-t)
.
(4.1.13)
111.
k-1
Gk(t)!
t=O
= ['(1-t)
It=O - v=1L
(k_v)-1
,
(4.1.14)
since Gk(O) = r(1) = 1 .
However,
00
r'.(1-t)\
=-1 exp(-u)logudu
t=O
0
= K = 0.5772156645 ...
K
is known as Euler's number.
*
-e
E Xn- k+1: n
=K
k-1
-
I
v=1
(4.1.15 )
So, from (4.4.14),
(k-v)-
1
(4.1.16 )
.
Therefore, using (4.1.4) and (4.1.16) the means are presented in Table
4.3.
Tables 4.2 and 4.3 will be used to obtain the means and medians of
the extreme value distributions, for comparative purposes.
4.2 The Parametric Models
It was decided earlier to restrict consideration to two parametric
families of parent marginal distributions;
namely, the Weibull and the
log-normal families.
For the Weibull model, we let
(4.2.1)
112.
for
x~
a
and
8 =
(a,S,T) where a,S>
a.
Let
S
= {i:
x.>e}
1
and
s = {i:
x.
1
=e}
That is, S is the set of days for which the daily maximum ozone
concentration was less than or equal to the minimum detectable limit,
e. Therefore, compressing the results of Section 3.3, we express the
likelihood function as,
II
iES
where
n(T)
=
L:
if
T ~
a
if
T
<
a
since, for our data set, the minimum detectable limit is attained and
thus Case III is impossible.
by
Then, the log-likelihood function is given
113.
(4.2.2)
This form of the log-likelihood function is the most suitable for input
into the maximization algorithm, MAXLIK.
See Appendix 1 for a listing
of the computer program.
For the log-normal model we let,
(4.2.3)
for x > 0 and
e = (~,cr,T) where cr
>
o. The resulting log-likeli-
hood function is then,
- L
iES
_
'2·
...
log(l-n*{T»
'"ES'
I..
(10
9
L log{cr{x.-T)/2TI)
iES'
-
(Xcri -T ) -~J
2
~ L
-
iES
log(l-n*(T»
where
n*(T}
if
T
if
T ~
< 0
=
o
;
0
(4.2.4)
114.
and ¢(.)
is the standard normal distribution function.
Again this form of the log-likelihood function is the most suitable
for input into the maximization algorithm, MAXLIK.
See Appendix 2 for
the listing of the computer program.
The initial values for the parameters of the two models were
obtained by the method of moments.
value for
T
as zero.
We firstly considered the initial
The method of moments in the log-normal case is
obvious, leading to initial values
~O
and
°0
such that,
n
~O
(4.2.5)
= l/n L log Xi
i=l
and
°0 = {l/ (n- 1)
n
.L
1=1
( log xi -
2
k
~0)}2
(4.2.6)
.
The information required ;s contained in Table 4.1.
In the Weibull case
we note that
EX
= T + a-liS r(l+l/(3)
(4.2.7)
and
Var(X)
where
r(·)
= a-
2/ S {r( 1 + 2/(3) - r 2(l + 1/(3)}
is the standard Gamma function.
r(1+2/(3)
r 2 (1+1/S)
=
1 + Var(X)
(EX)2
(4.2.8)
Thus
(4.2.9)
115.
for
T::
O.
Since the right hand side of (4.2.9) can be calculated
from Table 4.1 we obtain an estimate of S,
function.
using a table of the Gamma
Equation (4.2.7) is then used to obtain an initial estimate
of a..
Once the estimates of the parameters of the parent distributions
have been obtained we then assess the goodness-of-fit to the empirical
distribution function.
This is best tested using the Kolmogorov-Smirnov
one-sample statistic On'
°n : supx
defined by
(4.2.1O)
IFn(x) - F(x} I
where Fn (') is the empirical distribution function and F('} is the
postulated parent marginal distribution function. Graphical plots will
also be useful in examining the goodness-of-fit.
These results will be
obtained using SASGRAPH. See Appendix 3 for the program listing.
Finally the normalizing constants are calculated, using numerical
techniques and, following the rules given in Section 2.2, the values of
A~
and
B~
are obtained for each sub-sequence.
See Appendix 4 for the
listing of the FORTRAN programs used in this step.
The results of the above analyses are given in Tables 4.4 and 4.5.
The graphical plots of the estimated distribution functions against the
empirical distribution functions are presented in Figures 4.1 to 4.8.
Both the Weibull and log-normal distributions fit the sub-sequence
data extremely well, although the log-normal distribution gives a
slightly better fit in all cases.
It is, however, interesting to note
the large differences in the values of the threshold parameter,
The Weibull distribution assumption generates estimates of
T
T.
which are
116.
very small, positive and, with one exception, not significantly
different from zero.
This result seems intuitively appealing.
On the
other hand, the log-normal distribution assumption generates estimates
of
1
which are very large, relative to the scale of observations,
negative and significantly different from zero.
This situation implies
a much more even distribution of the censored observations and permits
the realization of a zero concentration.
The controversy surrounding
the choice of model is therefore not resolved by this analysis.
We can,
however, maintain that both are eminently suitable for modelling air
poll ution data.
The Kolmogorov-Smirnov one sample statistics presented in Tabes 4.4
and 4.5 show that we would accept all the sub-sequence models, for both
the Weibull and log-normal distributions, at the one per cent level, yet
reject the single sequence models.
The apparent excellent fit shown in
Figure 4.8 is therefore negated by the large number of observations.
We
believe that this result is consistent with the hypothesis that the
observations are not independent and identically distributed, as is
usually assumed.
This confirms the choice of the sub-sequence
corresponding to the days of the week.
Tables 4.4 and 4.5 both show that the sub-sequence corresponding to
Tuesdays is the most extreme, resulting in the largest estimates for
both a(Q,)
and b(Q,) .
nQ,
nQ,
Using the sub-sequence model we can then state
that, under the assumption of a Weibull distribution;
1im
n-+oo
Pr{Xn_k+l: n ~ O. 0191x + O.1133}
k-l
= exp(-exp(-x)) L
Jr exp(-rx)
r=O r.
(4.2.11)
117.
for k = 1s2 s' ••
fixed.
Thus
lim Pr{Xn_k+l:n~x}
n-+<Xl
= ex p (-ex p(0.1133-XJJ kIl -1, ex p (t(0.1133-X)J
0.0191
for k = 1s2 s.•.
fixed.
r=O r.
0.0191
{4.2.12}
While under the assumption of a log-normal
di s tri bution s
= ex p [_ex p (0.l091-X)J kIl ~ ex p (t(O.1091-X))
0.0237
r=O r!
0.0237
-e
{4.2.13}
for k = 1s2s•.. ; fi xed.
These distributions were used to produce Table 4.6 using Tables 4.2
and 4.3.
The three summary statistics, the means median and probability
of exceeding the standards show that the log-normal sub-sequence model
is slightly more conservative than the Weibull sub-sequence model.
The
single sequence model s in both cases s is more conservative than the
seven sub-sequence model for the maximums however, this situation is
reversed s for the mean and medians as we consider less extreme values.
An extreme value of possible interest to the United States Environmental
Protection Agency is the fifth extreme over a three year period.
For
the data set analyzed above the log-normal sub-sequence model is
substantially more conservative than the other three models with regard
to the mean and median.
The most startling difference between the
models occurs when we consider the probabil i ty of exceed; ng the ozone
118.
standard of 0.125 ppm.
The single sequence model is much more sensitive
to the choice of distribution than the sub-sequence model.
This is due
to the large differences in the estimates of a(£) . Consequently we
n£
would recommend that decisions based on these probabilities be viewed
with great caution.
In summary, the results indicate that, for the data set considered,
the sub-sequence model is more appropriate than the single sequence
model.
The choice of distribution function between the Weibull and the
log-normal has not been resolved.
On a more practical note we can say
that the Portland Air Quality Region is well in compliance with the
present standards.
4.3
The Partial Non-Parametric Models
As mentioned in Section 3.4 the first stage of the partial non-
parametric technique involves the maximization, for each sub-sequence,
of
L(y)
=
~
i=l
eXP(~(i)Y)
I
£ER ( x ( i)
where
~(i)
(4.3.1)
exp(r~y)
, m( i ) )
~ ~
is the sum of z over the days with concentrations
x(i)
and the notation in the denominator means that the sum is taken over all
distinct sets of m(i)
days drawn from R(X(i)) .
This maximization
was accomplished using the MAXLIK algorithm, after the data was
manipulated into a suitable form using SAS.
in Appendix 5.
The MAXLIK program is listed
119.
The covariates chosen for the analysis were daily maximum carbon
monoxide concentration (mg/m 3), z1; daily maximum temperature (OF),
z2;
and daily average wind speed (m.p.h.),
maximization are presented in Table 4.7.
z3'
The results of the
Because of the demonstrated
applicability of the sub-sequence model we did not consider the single
sequence model in the use of Cox's model.
We can see that the regression
coefficients for average wind speed are, with the possible exception of
the Saturday sub-sequence, not significantly different from zero.
The
signs of these coefficients are, with one exception, negative indicating
that as the average wind speed is increased the expected ozone
concentrations also increase.
mechanism for ozone formation.
This seems to be counter to the accepted
However, one explanation may be that
when wind speed is low the pollutants emitted by motor vehicles in the
downtown area are not transported into the suburban areas where the
ozone monitor is located.
In this situation we would expect the ozone
concentrations to be low.
The regression coefficients for daily maximum
temperature are, counter to experimental information, positive and
generally significantly different from zero.
It is difficult to
rationalize this effect, however one explanation could be that wind
speed and temperature are negatively correlated.
These unexpected
relationships may provide some scope for discussions with Environmental
Engineers. The regression coefficients for the major covariate, daily
maximum carbon monoxide concentration, is negative and significantly
different from zero.
formation of ozone.
This is totally consistent with the theory on the
Carbon monoxide concentrations should be highly
correl ated with non-methane hydrocarbon concentrations as both t"'esult
from motor vehicle emissions.
Hydrocarbons, together with nitrous
120.
oxides are precursors to the formation of ozone.
Therefore we would
expect an increase in carbon monoxide concentrations to be associated
with an increase in ozone concentrations.
This is the case.
Having obtained the regression coefficients we then undertake the
second stage of maximization, using MAXLIK.
The likelihood function to
be maximized is given by,
(4.3.2)
where C1 (Y) , C2 (Y) and C3 (Y) are given by (3.4.24), (3.4.25) and
(3.4.26) respectively. These three statistics, being functions of x(i) ,
:(i) and
y,
will be calculated prior to maximization, using SAS, to
reduce computation in MAXLIK.
6.
The MAXLIK program is listed in Appendix
In addition, the estimates of the unconditional variances of the
estimates
PO' PI and P2 were derived using the equation (3.4.36) and
implemented in SAS.
The computer programs used are listed in Appendix 7.
The result of this stage of the analysis are presented in Table 4.8.
It is apparent from these results that the estimate of the hazard
function varies drastically over the sub-sequences.
The variances
associated with the first and last parameters are very large, due mainly
to the contribution of the term V (E(p!1)).
A
y
--
since the hazard function is highly sensitive.
This is to be expected
Different estimates of
the regression coefficients will result in large differences in the
hazard function.
It should also be pointed out that, in all cases, the
estimates of PI were all zero.
Therefore the hazard function is
e-
121.
estimated to be a straight line with positive slope.
We then have that
(4.3.3)
It could be argued, by opponents of the present method of standard
setting, that meteorological conditions should be taken into account so
as to reduce the impact of freak weather conditions on the attainment
of the standard.
Environmental Engineers could also be interested in
assessing the change in the probability distribution of ozone
concentrations under the hypothetical situation in which motor vehicle
emissions of hydrocarbons were reduced by, say, forty per cent.
of these situations we wish to estimate
where
~O
A{X;~O)'
and thus
In both
F{x;~O)
,
is the vector of covariate values considered suitable for the
particular situation. Thus
(4.3.4)
Two scenarios will be considered, each of which specify some
reasonable values for the meteorological covariates while the value of
the carbon monoxide covariate will vary from low to medium.
Specifically,
the alternate values of :0 are
:~1) = (6, 85, 5)
that is;
I
daily maximum carbon monoxide concentration of 6 mg/m 3, daily
maximum temperature of 85°F and daily average wind speed of 5 m.p.h.;
122.
wh i 1e
~62) = (15, 85, 5)
The estimates for the normalizing constants, obtained using
numerical techniques are given in Table 4.9.
listed in Appendix 8.
The computer program is
As in the parametric approach it is obvious that
the sub-sequence corresponding to Tuesdays is the most extreme.
There-
fore, using (2.3.8) and the fact, established in Section 3.4, that
F(x;:O)
is in the domain of attraction of A,
we have;
lim Pr{X n-k+l'. n <0.0210x +0.1144 I zo(l)}
-
n~
=
for
k = 1,2, ...
lim
n-+<xJ
exp(-exp(-x)) k 1 ~ exp(-rx)
r=O r.
I
(4.3.5)
e-
fixed and
Pr{X n-k+l'.n <0.0488x+0.2979 I -zO(2)}
k-1
= exp(-exp(-x))
for k
=
1,2, ...
L ~ exp(-rx)
r=O r.
(4.3.6)
fixed.
Using these distributions the means, medians and the probability of
exceeding the standard for the k-th extreme values were derived and
presented in Table 4.10.
z(l)
-0
We can see that the results for the covariate
are similar to those for the parametric models while those for the
2 ) are, as expected, much less conservative. Therefore we
covariate
:6
_
~
123.
can assert that, if the carbon monoxide concentrations were consistently
in the vicinity of 15 mg/m 3 we would expect the Portland Air Quality
Region to be in violation of the standards.
This approach may be very useful in the prediction of the behaviour
of ozone concentrations under various scenarios, described by covariates
such as nitrous oxide and non-methane hydrocarbons.
The preceeding analyses show that if we are merely interested in
obtaining the extreme value distribution for a set of air pollution data
then the parametric approach, using either the Weibull or log-normal
distributions is sufficient.
If, however, we are more interested in
examining the extreme value distribution under a particular set of
auxiliary conditions then the partial non-parametric approach has
potential.
TABLE 4.1:
Region: Portland, Oregon
Pollutant: Ozone
Measurement: Daily I-hour maximum
Summary of Ozone Data
Site: Milwaukie High School
Collection Method: Instrumental
Maximum Detectable Limit: 0.003 ppm
Maximum
Mean
Concentration Concentration
(ppm)
(ppm)
Day of
the week
No. of
No. of
Missing
RMeasurements
Days
Sunday
5
144
12
0.102
Monday
6
144
12
Tuesday
7
137
Wednesday
1
Thursday
Time Period: 21 June 1978 20 June 1981
Analysis Method: Chemiluminescence
Standard
Devi ati on
Log-Transformed Concentration
(ppm)
Mean
Standard
Deviation
0.029
0.017
-3.731
0.682
0.115
0.026
0.015
-3.864
0.737
19
0.192
0.029
0.024
-3.818
0.806
139
18
0.087
0.026
0.015
-3.872
0.720
2
143
14
0.087
0.027
0.016
-3.847
0.798
Friday
3
146
11
0.102
0.028
0.018
-3.841
0.839
SatuY'day
4
145
12
0.120
0.027
0.017
-3.807
0.719
Over all
days
-
998
98
0.192
0.027
0.018
-3.826
0.758
~
e
e
e
N
..".
.
125.
TABLE 4.2:
Medians of the k-th Extreme Value
k
Median
1
0.3665 an + .bn
-0.5178 an + bn
-0.9836 an + bn
2
3
4
5
6
-1. 3008 an + bn
-1. 5414 an + bn
-1. 7352 an + bn
·e
TABLE 4.3:
Means of the k-th Extreme Value
k
Mean
1
0.5772 an + bn
-0.4228 an + bn
2
3
4
5
6
-0.9228 an + bn
-1.2561 an + bn
-1.5061 an + bn
-1. 7061 an + bn
TABLE 4.4:
Day of
the week
~
A
Estimates for the Weibull Model
A
A
a~
S~
(s. e.)
(s. e.)
(s.e.)
oCt)
T~
n~
p-value
(a)
A(Q.)
b
n~
A(Q.)
a
n~
B~
A~
Sunday
5
283.430
(115.716)
1.617
(0.133)
0.00175
(0.00093)
0.0899
0.098
0.0838
0.0096
00
00
Monday
6
378.143
(181.121)
1.658
(0.156)
0.00088
(0.00116 )
0.0873
0.111
0.0742
0.0083
00
00
Tuesday
7
61.928
(20.276)
1.150
(0.100 )
0.00273
(0.00035)
0.1123
0.032
0.1133
0.0191
0
1
Wednesday
1
329.006
(158.154)
1.607
(0.153)
0.00128
(0.00103)
0.0765
0.197
0.0745
0.0087
00
00
Thursday
2
372.579
(207.004)
1.691
(0.192)
0.00012
(0.00158)
0.0811
0.152
0.0779
0.0087
00
00
Friday
3
164.609
(82.257)
1.448
(0.162)
0.00124
(0.00117)
0.1037
0.043
0.0905
0.0117
00
00
Saturday
4
274.221
(118.960)
1.582
(0.140)
0.00149
(0.00097)
0.0635
0.311
0.0808
0.0095
00
00
Over all
days
-
189.241
(29.504)
1.473
(0.050)
0.00164
(0.00033)
0.0756
0.00001
0.1073
0.0100
a.
Obtained from the large sample result;
p
= eXP(-2nD~) . Owens (1962).
I-'
e
N
e
.
(J")
e
e
e
e
TABLE 4.5:
Estimates for the Log-normal Model
,,( R,)
nR,
A(R,)
a
nR,
BR,
AR,
0.193
0.0862
0.0131
00
00
0.0788
0.167
0.0732
0.0096
00
00
-0.010
(0.001)
0.0779
0.190
0.1091
0.0237
0
1
0.329
(0.013)
-0.020
(0.002)
0.0582
0.390
0.0766
0.0116
00
00
0.182
(0.010)
-0.068
0.0696
0.250
0.0764
0.0090
00
00
(b)
0.262
(0.006)
-0.045
(b)
0.0858
0.117
0.0872
0.0122
00
00
4
-3.078
(b)
0.330
(b)
-0.022
(b)
0.0517
0.461
0.0824
0.0124
00
00
-
-3.054
(0.007)
0.337
(0.002)
-0.023
(0.0006)
0.0533
0.003
0.1109
0.0136
(s.e.)
A
OR,
(s.e.)
A
TR,
(s.e.)
D( R,)
nR,
p-value
(a)
5
-3.092
(0.075)
0.342
(0.007)
-0.019
(0.003)
0.0756
Monday
6
-2.872
(0.060)
0.256
(0.028)
-0.033
(0.004)
Tuesday
7
-3.384
(0.027)
0.514
(0.006)
Wednesday
1
-3.141
(0.041)
Thursday
2
-2.385
Friday
3
-2.675
Saturday
Over all
days
Day of
the week
R,
Sunday
a.
b.
A
lJR,
(b)
(b)
Obtained from the large sample result;
Estimated variance was negative.
p = exp(-2nD n2) .
b
Owens (1962).
~
N
'-J
.
128.
TABLE 4.6:
Summary Statistics for the Parametric Models
Weibull
one sequence
k
seven sub-sequences
Mean
l"'1edi an
Pr(X n_k+1: n>0.125)
Mean
Median
Pr(X n_k+1 : n>0.125)
1 0.113
0.111
0.157
0.124
0.120
0.413
2 0.103
0.102
0.013
0.105
0.103
0.101
3 0.098
0.097
< 0.001
0.096
0.095
0.017
4 0.095
0.094
<
0.001
0.089
0.088
0.002
5 0.092
0.092
< 0.001
0.085
0.084
< 0.001
6 0.090
0.090
<
0.001
0.081
0.080
<
0.001
e·
Log-normal
one sequence
seven sub-sequences
Mean
Median
Pr(X n_k+1: n>0.125)
Mean
Median
Pr(X n_k+1 : n>0.125)
1 0.119
0.116
0.299
0.123
0.118
0.400
2 0.105
0.104
0.050
0.099
0.097
0.094
3 0.098
0.098
0.006
0.087
0.086
0.015
4 0.094
0.093
<
0.001
0.079
0.078
0.002
0.090
0.090
<
0.001
0.073
0.073
<
6 0.088
0.087
<
0.001
0.069
0.068
< 0.001
k
5
0.001
-
e
TABLE 4.7:
Day of
Jl,
the week
A
Y1
•
e
Estimates of Regression Coefficients of Cox·s Model
s.e. (Yl)
(C.O. )
A
Y2
(Temp. )
s.e. (Y2)
A
Y3
s.e. (Y3)
(Wind Speed)
Sunday
5
-0.1974
0.0329
0.0154
0.0040
-0.0220
0.0229
Monday
6
-0.0600
0.0146
0.0046
0.0036
0.0350
0.0221
Tuesday
7
-0.1197
0.0236
0.0128
0.0057
-0.0020
0.0436
Wednesday
1
-0.0884
0.0191
0.0088
0.0047
-0.0029
0.0273
Thursday
2
-0.0538
0.0145
0.0097
0.0033
-0.0212
0.0243
Friday
3
-0.0825
0.0160
0.0161
0.0040
-0.0388
0.0263
Saturday
4
-0.1276
0.0252
0.0157
0.0038
-0.0487
0.0231
......
N
~
TABLE 4.8:
Day of
the week
Estimates of Coefficients of Hazard Function
A
t
Po
s.e. (PO)
Sunday
5
9.141
Monday
6
Tuesday
A
A
PI
s.e. (PI)
(a)
P2
s.e. (P2)
18.115
0.0
0.083
748.937
400.498
6.661
21.686
0.0
0.070
1300.611
608.213
7
27.633
18.085
0.0
0.127
41. 723
222.810
Wednesday
1
11.453
28.659
0.0
0.103
1274.184
754.548
Thursday
2
10.703
29.053
0.0
0.097
1089.190
721.871
Friday
3
14.577
16.391
0.0
0.062
544.700
363.934
Saturday
4
13.246
20.605
0.0
0.087
821. 226
489.138
a.
The parameter
PI
was constrained to be negative and so the variance of
zero when the estimate was zero.
PI
given
y
was set to
This standard error is then only derived from the first term in
(3.4.36)
I--'
W
o
tit
tit
~
e
131.
TABLE 4.9:
Estimates of Normalizing Constants for Cox's Model
Covariate ~al)(a)
Day of
the week
e
b(.Q,)
n.Q,
Covariate ~a2)(b)
A(.Q,)
n.Q,
b( .Q,)
n.Q,
A(.Q,)
an
.Q,
a
Sunday
5
0.1027
0.0105
0.2658
0.0257
Monday
6
0.0738
0.0073
0.0981
0.0095
Tuesday
7
0.1144
0.0210
0.2979
0.0488
Wednesday
1
0.0709
0.0074
0.1095
0.0110
Thursday
2
0.0692
0.0072
0.0906
0.0092
Friday
3
0.0732
0.0086
0.1154
0.0127
Saturday
4
0.0786
0.0085
0.1505
0.0153
a• ~a 1 ) = (6, 85, 5)
b.
I
:~2) = (15, 85, 5)
I
(1) _
a.
:0
b.
:~2) = (15, 85, 5)
- (6, 85, 5)
t-'
e
e
W
N
.t
e
133.
1.0
t.,
1.1
D."
D
I
8
T 8.6
It
I
•
U
T
·e
I
o
.
0.5
F
U
II
C
I.t
T
I
.o
•.
a~
"
J
,A
I
~
I
I
8.2
KEY:
log-normal D.F.
Weibull D.F.
0.1
* Empirical D.F .
•.• Jr.,,,...,..........,........,
''''i'''''''''.........,...,s,. ..,...,
PO,
• . . ,... ., . .. . . .,,...,. .
, .., . .,. . ., .. .
, .T"'j,. ...
, . .,...,.,,..,.
.
.,...,.
.
0.00
0.02
0.04
CONCENTRATION
,...,u,....,...,...,
••..,....,..-,..........
,
T"'i.........
0.06
0.118
0.10
(PPM)
Figure 4.1. ESTIMATED DISTRIBUTION FUNCTIONS FOR WEIBULl
AND lOG-NORMAL r~ODElS COMPARED WITH Et·1PIRICAL DISTRIBUTION
FUNCTION FOR SUB-SEQUENCES OF SUNDAYS.
134.
,A
,
I
I
~
,I
I,
I
,,J
1.31
t
I
0.21
1
KEY: --- Log-normal D.F.
--- Weibull D.F.
* Empirical D.F.
~
C.l
i
~
0.0 J'rl....
1 .r"¥I....
I "'"'I1....1....
1.,..11""111....
1.,..1'MIIMI....
I .....""'11'r"TI.,..I",",I1....1....
1 ....
1 .....
1 Ir-T'.,..I.....
' IMI....
I ....
I "'"'I1....
1""1"
• .,..1.-.,I....
I"\""'"'"'I1....
1....
1 T"IMI"'I.,...I.,.,11....
1....
',.,-1
1
0.00
0.02
0.04
0.06
0.08
0.10
0.12
CONCENTRATION (PPM)
Figure 4.2. ESTIMATED DISTRIBUTION FUNCTIONS FOR WEIBULL
AND LOG-NORMAL MODELS COMPARED WITH EMPIRICAL DISTRIBUTION
FUNCTION FOR SUB-SEQUENCES OF MONDAYS.
135.
o. 0 'r-P"""'~"""''''''''''''''''''''''''--''''--''''''-''''''''-''''''''''''''''''''''''''''''''''''I''''"T'.........--.~r--.,...,....,..
0.00
o.n
0.06
0.09
0.12
0.15
0.18
CONCENTRATION
(PPH)
Figure 4.3. ESTIMATED DISTRIBUTION FUNCTIONS FOR WEIBULL
AND LOG-NORMAL MODELS COMPARED WITH EMPIRICAL DISTRIBUTION
FUNCTION FOR SUB-SEQUENCES OF TUESDAYS.
136.
1'·1
·'·1
"I
~
"!
0.8i
o.7J
D
I
s
T
R
I
8
3
. .i
U
T
I
0
If
0.5
e·
F
U
If
C
0.4
T
I
0
Iii
I
0.3
I
0.2
KEY:
0.1
--- Log-normal D.F.
--- Weibull D.F.
* Empirical D.F.
0.0
0.00
0.01
0.02
0.03
.0.04
0.05
0.06
0.07
0.08
CONCENTRATION (PPM)
Figure 4.4. ESTIMATED DISTRIBUTION FUNCTIONS FOR WEIBULL
AND LOG-NORMAL MODELS COMPARED WITH EMPIRICAL DISTRIBUTION
FUNCTION FOR SUB-SEQUENCES OF WEDNESDAYS.
0.09
137.
0.7
D
I
8
T 0.6
R
I
B
U
T
·e
I
0
0.5
F
U
N
C
O.t
.
T
I
0
.
0.1
0.2i
--- Log-normal D.F.
--- t~e i bu11 D. F.
* Empi rica1 D.F.
0.1
o. (I "l""""""'~"""'~PY'""""""''"''''''''''''-''''''''''''''''''''''''''''''I''"''""""".......-r .........'''''''''Tn''I""""'nTI",,,,,,"''''''''1T'
0.00
0.01
0.02
0.03
0.0'
CONCENTRATION
0.05
0.06
0.07
0.08
(PflH)
Figure 4.5. ESTIMATED DISTRIBUTION FUNCTIONS FOR WEIBULL
AND LOG-NORMAL MODELS COMPARED WITH EMPIRICAL DISTRIBUTION
FUNCTION FOR SUB-SEQUENCES OF THURSDAYS.
0.09
138.
1.Gl
G.91
,
.~
,
ji
=II
O.S..:l
0·,1
i
il
I
8
T
R
I
0.6
a
u
T
I
0
0.5
e·
N
F
U
N
C
O.4~
j
T
I
(I
N
~
0.3
0.2
KEY:
Log-normal D.F.
We i bull D. F.
0.1
* Empirical D.F.
o. 0 ~....-r~..........................r-r-r~I"T'T""""""""'''''''''''''''''''''''''''''''''''''''''''''''...-r'''''''''''''''''''''''''''''''''''''''''''''
0.00
0.02
0.0"
0.06
lUiS
0.10
CONCENTRATION (PPM)
Figure 4.6. ESTIMATED DISTRIBUTION FUNCTIONS FOR WEIBULL
AND LOG-NORMAL MODELS COMPARED WITH EMPIRICAL DISTRIBUTION
FUNCTION FOR SUB-SEQUENCES OF FRIDAYS.
139.
1.0 -4
. .1
~
···i
1.7
D
I
8
T
R
I
t.6
T
I
0
I.S
8
U
·e
II
F
U
II
C
T
I
0.'
0
Ii
0.3
0.2
KEY:
--- Log-normal D.F.
-- - We i bu11 D. F•
* Empirical D.F.
0.1
I. 0 '-...,.....""""""~~........~...,.,...,......~ ........'f"T't".........._ _.........,....................,.....I""I"T'.................,.......-r'"T"
0.10
0.12
0.0'
0.06
0.08
0.00
0.02
CONCENTRATION (PPM)
Figure 4.7. ESTIMATED DISTRIBUTION FUNCTIONS FOR WEIBULL
AND LOG-NORMAL MODELS COMPARED WITH EMPIRICAL DISTRIBUTION
FUNCTION FOR SUB-SEQUENCES OF SATURDAYS.
140.
• iII-
i.U....!
"Ii
cia
III
uosJ
~
j
-!j
~
0.81
~
,
0
D
I
0
'1
8
T
R
G.61
U
~
I
II
T
I
0
N
uoS1
e·
F
.u
N
c
T
I
0
N
u..l
003
1
0.2
KEY:
0.1
~
0.0 jL,.:1:,., ,..",..." , 1 ,
0.00
O.~
,..-,....,...,
0.06
, , ,-. ,
0.09
, , ..",..." , T'i ,
0.12
Log-normal D.F.
Weibull D.F.
* Empirical D.F.
, ..,.., ,
0.15
.."..-, , T"i , ,.."...,..,
0.18
CONCENTRATION (PPHl
Figure 4.8. ESTIMATED DISTRIBUTION FUNCTIONS FOR WEIBULL
AND LOG-NORMAL MODELS COMPARED WITH EMPIRICAL DISTRIBUTION
FUNCTION FOR SINGLE SEQUENCE MODEL.
'I
"
V:
SUMMARY AND RECOMMENDATIONS FOR FUTURE RESEARCH
5.1 Summary
This dissertation has been concerned with extending the theory on
extreme values to accommodate certain non-stationary sequences.
Of
particular interest are sequences of daily maximum ozone concentrations
which, while exhibiting obvious non-stationarity, have previously been
treated as either stationary or independent and identically distributed.
We firstly restricted our consideration to those non-stationary
sequences which could be divided into s sub-sequences of independent
and identically distributed observations, although we allowed for
stochastic dependence between the sub-sequences.
For example, the days
of the week can be considered to constitute seven natural dependent subsequences of independent and identically distributed observations.
We
then extended the classical extreme value theory to accommodate this
dependence structure between the sUb-sequences and determined that,
under very reasonable conditions, the extreme values for each subsequence are asymptotically jointly independent.
This result enabled us
to obtain expressions for the limiting distribution of the k-th extreme
value for the entire sequence of observations, for k fixed.
This
expression was found to depend entirely on the extreme value distribution
of the most extreme sub-sequence.
We extended these results to the
multivariate situation and considered the case where the sub-sequence
sample size is a random variable.
We also determined the joint limiting
142.
distribution of the sub-sequence maxima.
The above theoretical results depended on the efficient estimation
of the sub-sequences normalizing constants.
We initially considered a
parametric approach to this estimation problem.
Two distributions, the
Weibull and the log-normal, were proposed as suitable models for
characterizing the distribution of air pollutant concentrations.
Estimation techniques for the parameters of these models were detailed
and consequently estimators of the normalizing constants were obtained.
We then considered a partial non-parametric approach based on Coxls
proportional hazards model which utilizes auxiliary information.
This
technique, together with assumptions which produced smooth hazard
functions, rather than the usual discrete representation, allowed us to
obtain estimates of the sub-sequence normaliZing constants in addition
to the domain of attraction of the underlying distribution.
The estimation techniques mentioned above were applied to a
sequence of three years of daily maximum ozone concentrations for
Portland, Oregon.
Auxiliary variables, or covariates, considered in the
partial non-parametric approach were daily maximum carbon monoxide
concentrations, daily maximum temperature and daily average wind speed.
The results of the parametric analyses determined that both the lognormal and Weibull distributions are equally appropriate for the
modelling of ozone concentrations.
The existence of seven dependent
sub-sequences of independent and identically distributed observations,
corresponding to the days of the week, was confirmed.
Significant
differences in the limiting extreme value distributions were found
between the sub-sequence and the single sequence models.
The partial
non-parametric approach concentrated on showing how, depending on the
143.
value of the covariate, the limiting conditional extreme value
distribution varies greatly.
This approach is considered useful in
determining the effect on the distribution of air pollutant
concentrations under various combinations of covariate values.
The remaining section of this chapter discusses additional problems
that require further investigation.
5.2 Recommendations for Future Research
The major assumption of this work which could, justifiably, be
challenged is that the observations on the same day of different weeks
are independent and identically distributed.
tend to refute this assumption.
·e
Seasonal variation would
One method of addressing this problem
is by forming twenty-eight sub-sequences corresponding to the season by
day of week combinations and expanding the number of observations fourfold to maintain the same large sub-sequence sample size.
It would be
useful if an alternate strategy could be developed which did not require
such large total sample sizes.
The multivariate theory could be developed and applied to a
situation in which the distribution had a more concrete application.
The air pollution field will provide an avenue for the application of
this theory only if the standards are set in a multivariate context.
For example, if ozone, hydrocarbons and nitrous oxides are controlled
simultaneously.
We consider it important for more data sets to be analyzed, in a
similar way, to determine if the results can be translated and applied
to other air pollutants and other types of air sheds.
144.
The partial non-parametric approach should also be applied to
other primary air pollutants as well as to a more comprehensive range of
covariates.
The application of this technique to the contour plotting
of ozone concentrations versus hydrocarbons and nitrous oxide
concentrations, being undertaken by some environmental engineers, needs
to be developed fully.
There are other areas in which research could be profitably
directed.
(1)
These include the following:
The extension of the partial non-parametric method to the
multivariate situation.
(2)
Consideration of other flexible parametric models, such as the
Burr distribution.
(3)
The application of these techniques to data on water pollution.
(4)
The application of the non-parametric approach to develop a
distributional analogue of intervention analysis.
...
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the Domain of Attraction of exp(-e- X )II, Annals of r4athematical
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_________ (1964), "L imi t Theorems for the Maximum in Stationary
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CHISIBOV, n.M. (1964), liOn Limit Distributions for Order Statistics
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Theory of Probability and its Applications, 9, 142-148.
COX, n.R. (1972), IIRegression Models and Life Tables", Journal of the
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,
Paper No. 75-51.3, 68th APCA Annual Meeting, Boston, Massachusetts.
DODD, E.L. (1923), liThe Greatest and the Least Variate under General Laws
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FRECHET, M. (1927), IIS ur la Loi de Probabil ite de Llecart Maximum
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------
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~1u1tivariate
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----- (1960),
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,
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---------
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ll
,
Annals of Mathematical Statistics,
,
e
•
.
e
APPENDIX 1: LISTING OF SUBROUTINE FOR LOG-LIKELIHOOD
UNDER THE WEIBULL MODEL
C
C
C
C
40
C
THE DATA ARE DENOTED BY D(I),I=1,1096. THIS SUBROUTINE CALCULATES
THE LOG-LIKELIHOOD FUNCTION WITH PARAMETERS B(1),B(2),B{3) FOR A
SELECTED DAY OF THE WEEK. THE LOG-LIKELIHOOD IS DENOTED BY RLK.
SUBROUTINE LGLIK(RLK,B)
IMPLICIT REAL*8{A-H,0-Z)
COMMON/DATA/D(1096),V(1096)
COMMON/MLECAM/
2XL(75),XU(75),FOPT(10),FEST(75),Z(75),XTK,XTK2,
3IFR(75),LIRX(75),IOPT(40),IDIST(20),LABEL{75,7),KTRAN,NT,NR,NS
DIMENSION B(3)
LENTRY= IOPT (14·)
PLACE CONSTRAINTS ON THE PARAMETERS B(l), B(2) AND B(3)
DO 40 1=1,2
IF(B(I).LE.O.) GO TO 400
CONTINUE
IF(B(3).GE.0.003) GO TO 400
DEFINE THE MINIMUM DETECTABLE LIMIT
YMDL=0.003
RLK=O
AL=B(l)
BE=B(2)
e
C
C
C
200
250
300
400
T=B(3)
SELECT THE DAY OF THE WEEK. IN THIS CASE IT IS 2
CORRESPONDING TO THURSDAY
DO 300 1=2,1096,7
CALCULATE RLK DELETING MISSING OBSERVATIONS FIRST
IF(D(I).EQ.O.) GO TO 300
V(I)=D(I)-T
IF(D(I).EQ.0.003) GO TO 200
RLK=RLK+DLOG(AL*BE)
RLK=RLK+(BE-1)*DLOG(V(I))-AL*(V(I)**BE)
IF(T.GE.O.O) GO TO 300
RLK=RLK+AL*((-T)**BE)
GO TO 300
IF(T.GE.O.O) GO TO 250
A=AL*((-T)**BE)-((YMDL-T)**BE))
RLK=RLK+DLOG(l-DEXP(A))
GO TO 300
RLK=RLK+DLOG(l-DEXP(-AL*(YMDL-T)**BE))
CONTI NUE
IOPT(36)=IOPT(36)+1
RETURN
IOPT(ll)=l
RETURN
END
I-'
e
e.
U1
N
,
e
e
e
APPENDIX 2:
C
C
C
C
C
C
e
LISTING OF SUBROUTINE FOR LOG-LIKELIHOOD
UNDER THE LOG-NORMAL MODEL
THE DATA ARE DENOTED BY D(I),1=1,1096. THIS SUBROUTINE CALCULATES
THE LOG-LIKELIHOOD fUNCTION WITH PARAMETERS 8(1),B(2),B(3) FOR A
SELECTED DAY OF THE WEEK. THE LOG-LIKELIHOOD IS DENOTED BY RLK
SUBROUTINE LGLIK(RLK,B)
IMPLICIT REAL*8(A-H,0-Z)
COMMON/DATA/D(1096),V(1096)
COMMON/MLECAM/
2XL(75),XU(75),FOPT(10),FEST(75),Z(75),XTK,XTK2,
3IFR(75),LIRX(75),IOPT(40),IDIST(20),LABEL(75,7),KTRAN,NT,NR,NS
DIMENSION B(3)
LENTRY=IOPT(14)
PLACE CONSTRAINTS ON THE PARAMETERS B(2) AND B(3)
IF(B(2).LE.0.) GO TO 400
IF(B(3).GE.0.003) GO TO 400
DEFINE THE MINIMUM DETECTABLE LIMIT
YMDL=0.003
SET THE VALUE OF PI
PI=3.14159
RLK=O.
DMU=B(l)
S=B(2)
.-.
Ul
W
C
C
C
90
95
200
250
300
400
e
T=B(3)
SELECT THE DAY OF THE WEEK. IN THIS CASE IT IS 2
CORRESPONDING TO THURSDAY
DO 300 1=2,1096,7
CALCULATE RLK DELETING THE MISSING OBSERVATIONS FIRST
IF(D(I).EQ.O.) GO TO 300
V(I)=O(I)-T
IF(T.GE.O.O) GO TO 90
G=(OLOG(-T)-DMU)/S
CALL MDNOR(G,H)
E=H
GO TO 95
E=O.O
IF(O(I).EQ.0.003) GO TO 200
RLK=RLK-DLOG(V(I)*S*DSQRT(2*PI)
RLK=RLK-0.5*((DLOG(V(I))-DMU)/S)**2)-DLOG(1-E)
GO TO 300
P=(OLOG(YMDL-T)-DMU)/S-E
CALL MDNOR(P,R)
RLK=RLK-DLOG(R)-DLOG(l-E)
CONTINUE
IOPT(36)=IOPT(36)+1
RETURN
IOPT( 11 )=1
RETURN
END
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APPENDIX 3: LISTING OF THE SASGRAPH PROGRAM
TO PLOT DISTRIBUTION FUNCTIONS
COMMENT THIS JOB PLOTS THE EMPIRICAL DISTRIBUTION FUNCTION AND THE
ESTIMATED WEIBULL AND LOG-NORMAL CDF'S AGAINST CONCENTRATION. IT
ALSO CALCULATES THE KOLMOGOROV-SMIRNOV ONE SAMPLE STATISTICS
GOPTIONS HSIZf=7 VSIZE=10 COLORS=(BLACK);
DATA A;
INPUT X @@;
IF X=O THEN DELETE;
X=X/lOOO;
Sl=INT((_N_-1)/7)-((_N_-l)/7);
S2=INT((_N_-2)/7)-((_N_-2)/7);
S3=INT ((_N_-3 )/7) - ({_N_-3) /7) ;
S4=INT ((_N_-4 )/7) -{ {_N_-4 )/7) ;
S5=INT{ (_N_-5)/7)-{ {_N_-5)/7);
S6=INT{(_N_-6)/7)-({_N_-6)/7);
S7=INT{ (_N_-7)/7)-( {_N_-7)/7);
CARDS;
DATA B1;
SET A;
IF 51=0;
PROC SORT; BY X;
I--'
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DATA C1;
SET B1;
SIZE=139;
MU=-3.14112;SIGMA=O.32946;TAUL=-O.02031;
ALPHA=329.00599;BETA=1.606762;TAUW=-O.00128;
FL=PROBNORM((LOG(X-TAUL)-MU)/SIGMA);
FT=PROBNORM((LOG(-TAUL)-MU)/SIGMA);
FL=FL/(l-T)-T/(l-T);
FW=l-EXP(-ALPHA*((X-TAUW)**BETA));
E=(_N_) / SI ZE;
DL=ABS(E-FL); DW=ABS(E-FW);
PROC GPLOT;
LABEL FL=DISTRIBUTION FUNCTION;
LABEL X=CONCENTRATION (PPM);
PLOT FL*X=l FW*X=2 E*X=3/0VERLAY VZERO HZERO;
SYMBOL1 I=L3 L=l;
SYMBOL2 I=L3 L=3;
SYMBOL3 V=STAR;
PROC MEANS MAX MAXDEC=4; VAR DL DW;
etc. for other days of the week
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APPENDIX 4, PART 1: LISTING OF SUBROUTINE FOR ESTIMATION
OF NORMALIZING CONSTANTS UNDER THE WEIBULL MODEL
C
C
400
500
THIS SUBROUTINE CALCULATES THE NORMALIZING CONSTANTS FOR THE
WEI BULL MODEL USING THE COMPLETE AND INCOMPLETE GAMMA FUNCTIONS
IMPLICIT REAL*8{A-H,0-Z)
S=139
A=329.00599
B=1.60672
T=0.00128
SIZE=S
ALPHA=A
BETA=B
TAU=T
BNL=«DLOG(SIZE))/ALPHA)**(l/BETA)+TAU
X=DLOG(SIZE)
P=l/BETA
CALL MDGAM(X,P,GI,IER)
GC=DGAMMA(P)
ANL=(SIZE/(BETA*(ALPHA**(l/BETA))))
ANL=ANL*GC*(l-GI)
WRITE(~,400) BNL,ANL
FORMAT(2X,FIO.4)
CONTINUE
END
I-'
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APPENDIX 4, PART 2: LISTING OF SUBROUTINE FOR ESTIMATION
OF NORMALIZING CONSTANTS UNDER THE LOG-NORMAL MODEL
C
C
C
THIS SUBROUTINE CALCULATES THE NORMALIZING CONSTANTS FOR THE
LOG-NORMAL MODEL USING THE NORMAL AND INVERSE NORMAL
PROBABILITY INTEGRAL WITH NUMERICAL INTEGRATION
IMPLICIT REAL*8(A-H,0-Z)
S=139
A=-3.14112
V=0.32946
T=-0.02031
SIZE=S
DMU=A
SIGMA=V
TAU=T
W=(OLOG(-TAU)-DMU)/SIGMA
CALL MDNOR(W.B)
C=(1-1/SIZE)*(1-B)+B
CALL r~DNRIS(C,D)
BNL=TAU+DEXP(DMU+SIGMA*O)
Q=O.O
E=OoOOl
DO 200 1=1000,10000,100
P=O.O
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200
300
400
500
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DO 100 J=l, I
Z=BNL+(J-O.5)*E
U=(DLOG(Z-TAU)-DMU)/SIGMA
CALL MDNOR(U,F)
U=l-(F-B)/ (I-B)
U=U*E
P=P+U
CONTINUE
D=DABS(P-Q)
IF(D.LT.O.OOOOOl) GO TO 300
Q=P
CONTINUE
Q=P
ANL=Q*SIZE
WRITE(3,400) BNL,ANl,P,Q,D,I,J
FORMAT(2X,5FIO.4,2IIO)
CONTINUE
END
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APPENDIX 5:
LITING OF SUBROUTINE FOR COX'S MODEL (FIRST STAGE)
C
C
THE DATA ARE DENOTED BY X(I),OZONE,Cl(I),CO,C2(I),TEMPERATURE,AND
C3(I),WIND,FORI=1 TO THE NUMBER OF DISCRETE VALUES OF X FOR THE
PARTICULAR DAY OF THE WEEK. THIS SUBROUTINE CALCULATES THE PARTIAL
LOG-LIKELIHOOD WITH PARAMETERS B(1),B(2),8(3). THE LOG-LIKELIHOOD
IS DENOTED BY RLK
SUBROUTINE PALIK(RLK,B)
IMPLICIT REAL*8(A-H,0-Z)
COMMON/DATA/X(50),Cl(50),C2(50),C3(50)
COMMON/MLECAMj
2XL(75),XU(75),FOPT(10),FEST(75),Z(75),XTK,XTK2
3IFR(75),L1RX(75),IOPT(40),IDIST(20),LABEl.(75,7),KTRAN,NT,NR,NS
DIMENSION B(3),S(50)
LENTRY=10PT(14)
S(1)=DEXP(B(1)*Cl(1)+B(2)*C2(1)+B(3)*C3(1))
C
CALCULATE THE SUMS OVER THE RISK SET
DO 1=1,48
S(I+l)=S(I)+DEXP(B(1)*Cl(I)+B(2)*C2(I)+B(3)*C3(1))
100 CONTINUE
RLK=O.O
DO 200 1=1,49
RLK=RLK+B(1)*Cl(1)+B(2)*C2(I)+B(3)*C3(I)
RLK=RLK-DLOG(S(I))
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APPENDIX 6:
C
C
C
C
C
LISTING OF SUBROUTINE FOR COX'S MODEL (SECOND STAGE)
THE DATA ARE DENOTED BY X(I),I=I,49 CORRESPONDING TO THE DISCRETE
VALUES OF X. AND D(I), THE NUMBER OF OBSERVATIONS WITH THAT VALUE
THIS SUBROUTINE CALCULATES THE LOG-LIKELIHOOD WITH PARAMETERS
B(1),B(2) AND B(3) FOR A PARTICULAR DAY OF THE WEEK
THE LIKELIHOOD IS DENOTED BY RLK
SUBROUTINE PALIK(RLK,B)
IMPLICIT REAL*8(A-H,O-Z)
COMMON/DATA/X(50),D(50)
COMMON/MLECAM/
2XL(75),XU(75),FOPT(10),FEST(75),Z(75),XTK,XTK2,
3IFR (75) ,LI RX (75) , IOPT (40) ,101ST (20) ,LABEL (75,7) ,KTRAN, NT ,NR, NS
DIMENSION B(3),S(50)
LENTRY= IOPT (14)
Cl=3.826149
C2=-477.509837
C3=0.074699
A=-B(1)*CI-B(2)*C2-B(3)*C3
RLK=O.O
DO 200 1=1,49
T=B(I)+B(2)/X(I)+B(3)*X(I)
IF(T.LE.O.O) GO TO 400
T=DLOG(T)
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APPENDIX 7:
LISTING OF SAS PROGRAM FOR UNCONDITIONAL VARIANCES
COMMENT THIS SAS PROGRAM CALCULATES THE ESTIMATE OF THE
VARIANCE (OVER GAMMA) OF THE EXPECTED VALUE OF P GIVEN GAMt1A;
DATA A;
INPUT X Cl C2 C3 DAY NO;
X=X/lOOO;
X1=X**{-1);
X2=1
All=NO;
A12=Xl*NO;
A13=X*NO;
A22=Xl**2*NO;
A33={X**2)*NO;
CARDS;
PROC MATRIX;
FETCH K
DATA=A{KEEP=X2 Xl X All A12 A13 A22 A33);
B=11.45271//0.0//1274.183;
P1=K( ,1) ;P2=K{ ,2) ;P3=K( ,3);
P=P111 P211 P3;
T=((B'*P ' )##2) I;
TS=TIITIITIITIIT;
Kl=K(l:46,4,56 7 8);
.......
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K2=Kl#/TS;
K3=J (l ,49, I) ;
K4=K3*K2;
Bll=K4(l ,I} ;
B12=K4{l,2} ;
B13=K4(l ,3);
B22=K4{l,4} ;
B33=K4(l,5);
F1=BllIIB121IB13;
F2=B1211 B2211 B11;
F3=B13I IB111 IB33;
D2LPP=F11IF21/F3;
COMMENT ENTER THE VAR-COV MATRIX FOR GAMMA AND THE SECOND
PARTIALS FROM PREVIOUS RUN;
D2LPH=19.097733 274.867581 32.2403851
-3006.360415 -31405.593 -4026.9738081
0.329867 5.673509 0.613090;
D2LPG=D2LPH
V=0.000364 -0. 0000 3£~ -0.0000061
-0.000034 0.000022 -0.000112/
-0.000006 -0.000112 0.000744;
Z=(D2LPPI ID2LPG ' )//(D2LPGI IV);
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W=INV(Z);
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APPENDIX 8: LISTING OF FORTRAN PROGRAM FOR ESTlt4ATION OF
NORMALIZING CONSTANTS UNDER COX'S MODEL
C
C
400
THIS PROG~Wj CALCULATES THE NORMALIZING CONSTANTS UNDER THE
COX MODEL USING THE NORMAL PROBABILITY INTEGRAL
IMPLICIT REAL*8(A-H,O-Z)
81=11.45271
82=0.0
83=1274.1836
S=139
CA=0.5551
PI=3.14159
SIZE=S
A1=81*CA
A2=B2*CA
A3=B3*CA
BNL=-Al+DSQRT(A1**2+A3*DLOG(SIZE))
BNL=BNL/A3
X=(8NL+Al/A3)*DSQRT(A3)
CALL MDNORD(X,Y)
ANl=(1-Y)*(A3**(-O.5))*DEXP«Al**2/2)/A3)
ANL=SIZE*ANL*DSQRT(2*PI)
WRITE(3,400) BNL,ANL
FOR~lAT (2X, 2FlO. 4)
END
!
......
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