/
,-
•
EXTREME VALUE THEORY FOR CERTAIN NON-STATIONARY SEQUENCES
by
G.A. Feeney
Australian Bureau of Statistics
Belconnen, Australia
and
P.K. Sen
Department of Biostatistics
University of North Carolina, Chapel Hill, NC
27514
Institute of Statistics Mimeo Series No. 1421
•
December 1982
•
Extreme Value Theory for Certain Non-stationary Sequences
G.A. Feeney
Australian Bureau of Statistics, Belconnen, Australia
P.K. Sen*
University of North Carolina, Chapel Hill, NC, USA
Summary.
-------
Under a Markovian structure on a sequence of random· variables
which can be partitioned into
m(~l)
jointly dependent subsequences
(where within each subsequence the random variables have a common marginal
distribution which may vary between the subsequences), the asymptotic
•
distribution theory of the sample extreme values is developed.
The asymp-
•
*Work supported by the National Heart, Lung and Blood Institute, Contract
totic independence of the subsequence extreme values is also studied.
NIH-NHLBI-71-2243L from the National Institutes of Health.
•
1.
_Introduction.
.... _--------Classical extreme value theory. as developed by Gnedenko (1943). is
based on the assumption that the sequence of observations'
{Xl..• i>l}
-
is
Watson (1954) was able
independent and 'identically distributed (i.i.d.).
to show that m-dependent stationary sequences have the same limiting extreme
value distribution as their associated i.i.d. sequence.
A similar case
holds for other stationary sequences too [See Leadbetter. Lindgren and
Rootzen (1979)].
We will consider the case of some dependent sequence
having a certain non-stationary structure.
{X}
the sequence
In particular. we will consider
as being partitioned into
n
m(~l)
subsequences where within each subsequence the
Xi
jointly dependent
have a common marginal
distribution function, and we allow these to vary between the subsequences.
It should be noted that the construction of the subsequence is quite flexible.'
and. for some relevance to some practical problems. we may refer to the
last section.
Let
x(l) 'denote the j-th observation in the l-th subsequence. for
j
j = 1 •...• n
and
1
l
= l ••••• m.
Since the total sample size is
sample size for the l-th subsequence.
[n/m]
= u.
and
n
1
n.
is approximately equal to
the
nO
=
Also. let
(1.1)
Z(l)
n
1
We refer to
as the l-th subsequence maximum.
l~l~m.
and,we denote
the overall maximum by
z
•
n
= max{z(l).
n
[Actually. the sequence
i
+
(j-l)m+l.
(1.2)
1 '
l~l~m. j~l.l
is reindexed to '
{X~l)}
J
by letting
The main purpose of this investigation is to
~tudy
suitable regularity conditions under which the
ically jointly independent.
zW
n
Prior research on multivariate extreme values
of observations were independent.
In our case, we can see that since the
successive observations are not independent, on letting
X.
-J
j.::.l,
Xi
and
~j+l
x.
(or
-J-
_
X. - (X
-J
(1)
j
(m)
, ... ,X.
J
However, under a Markovian structure
{X., j>l},
we will show that under conditions similar to
those of Sibuya (1960), the
are asymptotically jointly independent.
2.
-J
-
),
1) are not independent vectors, even when the
form an m-dependent sequence.
on the vectors
•
are asymptot-
l
The Extreme Value Distribution of the Maximum of the Subsequence Maxima
--~-------------------------------~------------------- -----------------
Under the usual conditions, given in Gnedenko (1943), de Haan (1970)
and Galambos (1978), it is clear that, whenever it exists
Hi x ) = lim Pr{Z(l)<a(l)x+b(l)}
n l"'"
n l - nl
nl
where
H
l
is a non-degenerate distribution function for
The form of
where
Fl (x)
H
l
= Pr{X. (l)<x}.
J
-
However,
Ho
<-
(l)
n
,
positive, and
.'
l = l, •.• ,m.
is determined by the type of parent distribution,
F ,
l
must be one of the three possible
types of extreme value distributions, see Galambos (1978).
a
(2.1)
are normalizing constants for
In addition
£ =
l, .•. ,m.
l
Therefore
Pr{Z <a x+b }
n-n
n
= Pr{z(l)<a
x+b ;
n -0
n
l
= l, .•. ,m}
(2.2)
where
z
and
positive, and
= max{X.;
n
~
b
n
i
=
l, ... ,n}
are real constants.
We can then re-express
•
(2.2) as
-2-
e·
Pr{Z <a x+b } = Pr
n- n
n
R. = 1, .• . ,m
Z(R.)
nR.
then (2.3) becomes, in the limit;
If it could·be shown that the
are asymptotically jointly independent
lim
m
lim Pr{Z <a x+b }.= II
Pr
0-+00
n- n
n
R.=l n--
since
nR.;
infinity.
R.
= l, •.• ,m;
, (2.4)
tends to infinity if and only if
lim Pr{Z <a x+b }
positive·, and
where
n
tends to
Therefore by equation (2.1) and of Gnedenko (1943)
n-+o>
-e
• (2.3)
n- n
b
n
n
=
(2.5)
are such that
(2.6)
and
(2.7)
for
l
= 1, ... ,m.
The choice of the form of
a
n
and
b
n'
a (R.) . and
nR.
like
depend on the type of marginal distribution functions,
It can be shown that the only sensible choice for
AR.x+BR.
•
a
n
R.
FR.'
and
b (R.)
n '
R.
b
n
will
l, ... ,m.
are such that
equals infinity for all but the dominant sub-sequence(s), for which
it equals
x.
Denote the corresponding normalizing constants by
Then we can see that
-3-
and
a
b
n
••
and
n
(2.8)
where
T
(~m)
t·,
is the multiplicity of
for continuous distributions
however with probability one,
Fl, ... ,F
not agreeing in the tails,
m
T = 1.
From (2.8) we see that the extreme value distribution for the maximum of
the entire sequence reduces to the product of the extreme value distributions
for the maximum of the dominant sub-sequences, if we can assume the asyrnp-
totic joint independence of the sub-sequence maxima.
Some preliminary results need to be established before determining
the conditions under which asymptotic joint independence holds.
Let
m
G(x) = Pr{X.<x}. xEE ,
-
-J~
G[O](x) = pr{x~.t)<x}. l<.t<m and
.(..
pr{xj.t)~x. xj.e.')~y}. l~I.t'~m.
G[.e..e.'](x,y) =
not assumed to be identical, so are the
Pr{X.<x, X
~J-""
J
<v}
-J'+1--'....
Note that the
G[.t]
G[.e..e.']'
Further let
(.t)
~x. X +l ~y}.
j
y ) = pr{x
*
G[.e..e.'](x,
j
and let
-
(.t')
are
Also, let
m
neG)
G(~)/ IT G[.t] (x.t)
(2.9)
.t=l
(2.10)
Our Basic Assumptions are:
(1) For every
.t I .t'
1, ... ,m,
as
Yi' Yt'
+ +00,
(2.11)
and for every
.t • .t' = l •...• m,
-4-'-
•
•
(2.12 )
(II) The vectors
~j
have the Markovian property
prfx- j <Aj I_r
X <Ar •
~j -I}
(2.13)
{Aj , j~l} < {A j • j~l}. j~l.
for all
In defense of this assumption. we may remark that if the
i~l,
form an autoregressive sequence of order
k.
where
k~m.
then we
may write
'.
BXj-l +
X = \.1. + __
_j
where
the
B
is an
mxm
~j'
for all ·j_>l
X. l' . so that assumption (II) holds.
are independent of
~j
matrix of constants and the
Assumption (II) holds also for
-J-
independent (non-stationary) processes.
Choose a sequence
r (n)
y
=
(n)
(n)
(Yl .... 'Ym ).
1 - r
where
-n log G(y_ (n)) (-nrn )
-+-
-r; rdO ."') .
such that
(2.14 )
n
Then. let
(2.15)
Note that
G(l
•
so that
r, <r
n - n'
:e.
(n)
):5..
for all
(n)
G[i) (Yi ) ,
i
= 1 ••.•• m.
-5-
for all
i< [I,m)
(2.16)
•
Lennna 2.1
n
[0(G(Z(n»)] 0
+
1
as
n(-rnnO)+oo
Proof:
m
- nO 1: log(l-r
i=l
(2.17)
)
ni
Now
pr{x(i»y(n)
j
i
i:
for at least one
l<i<m}
+ •.•
Therefore by (2.11),
r
n
=
r* - 0(1)r
n
.
n
so that by (2.17)
m
- nO 1: r
i=l
ni
+
0(1)
0(1)
as
r
*
(2.18)
n
Q.E.D.
-6-
•
•
Lemma 2.2
* *
nO log n (G (r
(n)
• r
(n)
» ..
0
as
n O--
Proof:
Note that
= Pr{at
'1 - G(r(n). r(n»
2r
n
least one of
X
X
<y (n)}
_j' _j+1-
- Pr{X.<y(n), X <v(n)}
-J--j+1-<-
(2.19)
where by (2.12) and the fact that
'.
pr{_Xji1_(n). X '
t l (n)}
_j+1
Pr{at least for one
l~i, i'~m,
we conclude that (2.20) = O(r ).
n
n
xji»yi )
so that (2.19) is
(2.20)
2r +O(r ).
n
n
0(1)
(2.21)
Q.E.D.
Now consider the expression
pr{Z~i)~~i). l~i~m}
i
pr{~j~(n)'.
V
l~j~nO}
nO
•
= pdx ~y(n)} II Pdx.~(n) Ix <v(n). V s~j-1}
_1 -
j=2
-J -
_S-<O
-7-
Hence
•
no
= Pd~l.::z (n)} .II pdx ..::z (n) 'x. .::z (n)}
(by (2.22»
-J
-J-l
J=2
nO
= pd~l.::z (n)} j~2 pdx ..::z (n) ~j_l~(n)}/pr{~j_l~(~)}
-J
n
n -1
= [G(y(n»] 0[0*(G*(y(n), y (n) »] 0
-
-
(2.23)
Note that by our choice, for every
t
= l, ••• ,m,
(2.24)
whenever
H
exists, where
t
(n)
Y,t
Thus
Ht(x t )
form of
is the limiting distribution function of the standardized
Z(t).
n
=
.:
Combining (2.23) with Lemmas 2.1 and 2.2 we conclude that
t
(2.25)
which proves the desired result; i.e., the sub-sequence maxima are asymp-
totically jointly independent.
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
The maxima
z(l) , •.• ,z(m)
°1
are asymptotically jointly independent if
urn
-8-
•
•
2(1)
each pair of maxima
are asymptotically independent, for
no'
).
i<j = l, ... ,m.
Proof:
(I)
Pr { Z
n
=
0
y~j-1<m)
1
i = 1, .•• ,n '
and
O
Pr { 2
(I)
n
So (2.26) becomes
(m)}
<x , ••• ,2
<x
- l
n - m
°
m
l
~o pdy~l)<Xl"" ,y~m)<x I i~l ~ [y~./'.-(i-j)m)<XoJ}
i=l
1
-
1
-
m j=l ./'.=1
1
-
.(.
no m
h-l
i-I m o o
II { II pr{fh)<x.
n [y~k)<x.] n
n [y~./'.-(l-J)m)<x
1-n
1-1(
1
-./'.
i=l h=l
k=l
j=l ./'.=1
I
°
(2.26)
For notational convenience, let
k = l, .•. ,i-l
=
<x
- m
m
0
XU)
i-k
for
(m)}
no
(1)
(m)
i-I (1)
(m)}
II pdx
<xl" •• ,X
<x I n [XJo ::x l '·.· ,X j ::.xm]
1=1
1
1
m j=l
using Bayes' rule.
:.
1
<xl, .•. ,Z
n
n}
l
no I m
~hi(Xl,···,Xh,Xl,···,xm,···,Xl,···,xm)
II { II
i=l !h=l F(h-l) i (xl' ... , "h-l ' xl' ... ,xm' •.• ,xl' ... , x m) \
where
Fhi (xl .... ,xh,x ' .. :,X ... · .x ' •. .,x )
l
m
m
l
distribution function.
is a
(h+(i-l)m)
(2.28)
variate
Further consideration of (2.28) yields
(1)
(m)}
Pr 2
<Xl' ••. ,2
<x
n n - m
1
m
o
°
{
m
•
o.
n
'II Fh ("h)
h=l
(2.29)
-9-
Let
(2.30)
•
Then.
(1)
Pr{Z
n
1
. (m)
<xl, •.• ,Z
n
"! Fn(x)
I
h h
h1
II
mlln
i=l
<x} ~ II
- m
h~l
m
n
(2.31)
n(h_l)i
Therefore. by definition
n
n
nn (Fl(xl) •••• ,Fm(xm)) ~
m
n
II
II
h=l
i~l
(2.32)
If each pair of maxima are asymptotically independent. then by Lemma 2.1
" .
I 1m
"
n-- 1~
" 1T 1"h"1
for
h
1, •. ,.m.
.:
h
1
Therefore. from (2.32)
lim
n-teo
n
n
nn (Fl(xl)
••••• Fm(xm))
~
(2.33)
1
that is. the maxima
z(l) , •.•• Z(m) are asymptotically jointly independent.
n
n
m
l
Using Theorem 2.2, and a result of Sibuya's. Sibuya (1960), we can
state the following corollary.
Corollary 2.4
The maxima
z(l) •.•. ,z(m)
n
n
l
m
Pij (l-r.l-r)
oCr)
are asymptotically jointly independent if
for all
i<j
1,.~.,m,
where
/
(2.34)
This corollary then gives reasonable sufficient conditions under which
-10-
•
•
•
Let us, briefly, consider the condition, that is,
P(l-r,l-r) = o(r)
(2.35)
This is known as Sibuya's condition, given in Sibuya (1960).
let us suppose that the marginal distribution of
Xl
For example,
is a Weibull (al,Sl)'
Furthermore, assume that the bivariate distribution is derived from the
third Gumbel bivariate exponential distribution referred to in Gumbel (1960).
That is
(2.36)
:.
where
m is a measure of the dependence between
Xl
and
X ,
2
In fact,
we can show that
1
m = [l-Corr(logX , 10gx2)l-~
l
Let
and
x2
(l-r).
= F-1
2
Thus,
P(l-r,l-r)
(2.37)
r
Therefore
P(l-r,l-r) =u(r)
is the correlation is not one,
as long as
-1
m
does not equal zero; that
Similarly, it can be shown that if we assume
that the marginals are log-normally distributed than Sibuya's condition
reduces to the correlation not equalling one.
The condition for asymptotic
joint independence is then seen to be very general and quite reasonable.
•
3.
The Distribution of the k-th Extreme
-----------------------------------Let
denote the k-th largest observation in the l-th
n.C k +l : nl
sub-sequence.
Following Gnedenko (1943), de Haan (1970) and Galambos (1978)
x(l)
-11-
we see that, if it exists
(3.1)
I
for
l, ... ,m;
k
fixed, and
k-l
1
r
HI(x) l: -,[-logHe(x)]
(3.2)
r=O r.
where
HI(x)
•
is as given by (2.1).
In an analogous manner to the work
in Section 2 we can show that
= H(k)
I
where
Al and
BI
are as before.
(A x+B )
I
(3.3)
I
We noted earlier that
Al and
BI
were such that
I=t
for
(3.4)
otherwise
for
i.. = 1, .•. ,m; where the
t-th
sub-sequence is the dominant one.
The
above results are true for the individual sub-sequences however, we are
only interested in the properties of the
sequence, denoted by
xn-k+l:
n
.
k-th
extreme of the entire
Therefore
{
= lim{
400 I-Pr x _ + :
n k 1
The set
{x
n-k+l: n
>a x+b }
n
n
n
>a x+b }}
n
n
(3.5)
can be expressed as the union of a multitude
of sets of the form
{X(p)
n -j+1: n
p
p
>a x+b , X(q)
n
n
n -i+l:
q
n
>a x+b , •.• }
q
n
n
"
However, due to (3.3) and (3.4) the probability of these sets is zero
except for those which involve only the order statistics of the t-th
-12-
(3.6)
•
•
sub-sequence.
If we assume that each sub-sequence is unique then there
is only such set, corresponding to the case where all the
greater than
a x+b
n
n
belong to the
t-th
sub-sequence.
the asymptotic joint independence of the sub-sequence
k
observations
Therefore, assuming
b-th
extremes, (3.5)
reduces to
lim
n->oo pdXn _ k+l :
0.7)
We have therefore shown that for a certain
m-dependent
non-stationary
sequence the limiting extreme value distribution can be obtained and is
identical to that for the most extreme or dominant sub-sequence.
:.
4.
The above approach was motivated by the inappropriateness of the
assumption of a stationary sequence adopted by Watson (1954) and more
recently Leadbetter, Lindgren and Rootzen (1979) in relation to air pollution concentrations.
These concentrations are such that consecutive days
are strongly dependent and a weekly cyclical pattern is evident due to
variations in traffic flows.
In addition meteorological factors which
typically have an effect for less than one week destroy the stationarity
of the sequence of concentrations. It did, however, seem reasonable to
assume that observations on different weeks follow a Markovian pattern .
. Of major interest to the U.S. Environmental Protection Agency, for the
setting of standards, is the distribution of the
imum concentration over a long time period.
•
k-th
largest daily max-
This paper attempts to solve
this problem by imposing some reasonable assumptions, detailed above, on
the dependence structure of the observations.
-13-
•
References
1.
Frechet, M.: Les Probabilities Associees A un Systeme D'evenements
Compatibles et Dependants, Paris: Hermann 1940.
2.
Galambos, J.: An Asymptotic Theory of Extreme Order Statistics,
New York: John Wiley 1978.
3.
Gnedenko, B. V. :
Sur la distribution limite du terme maximum d'une
serie aleatoire.
Ann. Math: 44, 423-453 (1943).
4.
Gumbel, E.J.: Bivariate distributions.
698-707 (1960).
5.
Haan, L. de: On Regular Variation and its Application ot the Weak
Convergence of Sample Extremes, Mathematical Centre Tracts 32,
Amsterdam: Mathematical Centre 1970.
6.
Leadbetter, M.R., Lindgren, G. and Rootzen, H.:
Properties
~f
Stationary Processes, Part 1:
Sequences, Sweden:
7.
J. Amer. Statist. Assoc. 55,
Sibuya, M.:
Extremal and Related
Extremes of Stationary
University of Umea 1979.
Bivariate extreme statistics.
Ann. lnst. Stat. Math., 11,
195-210 (1960).
8.
.:
Watson, G.S.: Extreme values in samples from independent stationary
stochastic processes. Ann. Math. Statist. 25, 798-800 (1954).
•