I
I
t·
I
I
I
This research was supported in part by the National Institutes of
Health, Grant No. GM-12868 and partially by the Air Force Office
of Scientific Research, Grant No. 68-1615.
I
I
I
I
I
I
I
I
I
I
I
1
,
ON THE ASYMPTOTIC PROPERTIES OF SOME ROBUST
ESTIMATORS IN CERTAIN MULTIVARIATE STATIONARY
AUTOREGRESSIVE PROCESSES
by
Kalyan Dutta
Department of Statistics and Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 762
JULY 1971
,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
ii
TABLE OF CONTENTS
CHAPTER
Page
LIST OF TABLES
ACKNOWLEDGEMENTS
ABBREVIATIONS AND NOTATIONS
ABSTRACT
I
vii
viii
INTRODUCTION AND SUMMARY
1010
1 2
0
II
iv
v
0
1
Introduction and a Review of the Literature
A Summary of the Results in Chapters II-V
1
15
BAHADUR-REPRESENTATION OF SAMPLE QUANTILES IN
SOME STATIONARY MULTIVARIATE AUTOREGRESSIVE
PROCESSES
2 1
2 20
2 030
0
0
0
2 4,
0
III
22
Introduction
Preliminary Notations and Assumptions
Almost Sure (Bahadur-) representation of Z
""'Il
Asymptotic Joint Normality of Z
"'n
22
24
27
44
ASYMPTOTIC DISTRIBUTION OF LINEAR COMPOUND OF SOME
FIXED NUMBER OF QUA.!.'{TILES FOR STATIONARY MULTIVARIATE AUTOREGRESSIVE PROCESSES
46
301 0
302 0
46
Introduction
Extension of Bahadur-representation of Sample
Quantiles over the Entire Real Line
3 3
Asymptotic Normality of a Linear Compound
of Several Qunatiles
304 0 A Useful Application of the Results of
Section 303
3,5
Consistent Estimate of the Dispersion Matrix
of the Vector of Linear Compounds of
Several Quantiles
3 6
Asymptotic Normality of a Linear Compound of
Several Quantiles for Random Sample Size
0
46
0
54
57
0
0
59
0
68
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
iii
CHAPTER
IV
Page
ASYMPTOTIC PROPERTIES OF THE WILCOXON SIGNED RANK
STATISTIC AND RELATED ESTIMATORS FOR MULTIVARIATE
STATIONARY AUTOREGRESSIVE PROCESSES
72
4010
4 2
72
0
0
4030
4030
V
Introduction
Asymptotic Properties of the Wilcoxon Signed
Rank Statistic
Asymptotic Properties of a Class of U-statistics
Asymptotic Distribution of the Median of the
Mid-range Estimator
72
84
94
COMPARISON OF THE PERFORMANCES OF SEVERAL ESTIMATORS
OF LOCATION FOR STATIONARY AUTOREGRESSIVE PROCESSES
99
Solo
5 2
99
0
0
5 030
5040
Introduction
Asymptotic Relative Efficiency of the Median
with Respect to the Mean
Asymptotic Rela~ive Efficiency for Univariate
Stationary Autoregressive Processes
Proposed Topics of Future Research
REFERENCES
100
107
127
129
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
iv
LIST OF TABLES
Page
TABLE
A.R,E, of Sample Median/Sample Mean for First and
Second Order Gaussian Autoregressive
Processes
124
A,R,Eo of 27% Mid-Range/Sample Mean for First and
Second Order Gaussian Autoregressive
Processes
125
AoRoE o of Median of the Mid-Ranges/Sample Mean for
First and Second Order Gaussian Autoregressive Processes
126
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
v
ACKNOWLEDGEMENTS
It is a pleasure and privilege to acknowledge my deep feeling
of indebtedness to my advisor, Professor P. K. Sen, for suggesting
the problems in this dissertation.
His guidance and assistance at
every stage of the work have been illuminating and instructive.
I would also like to thank Professor Wassily Hoeffding, the
chairman of my doctoral committee, for his invaluable suggestions
and comments.
Thanks are also due to Professor N. L. Johnson,
Professor E. J. Wegman and Professor R. L. Davis, the other members
of my doctoral committee, for going through the manuscript.
For financial support during my graduate training, I sincerely
thank the Department of Statistics, the Department of Biostatistics,
and the Air Force Office of Scientific Research and the National
Institutes of Health.
To these institutions I gratefully express
my appreciation.
For her excellent typing of the final manuscript I extend my
thanks to Mrs. Evelyn Nichols.
I wish to express my sincere gratitude to Professor Ro Co Bose
who was chiefly responsible for my decision to come over to Chapel
Hill.
I wish also to thank Professor J. Eo Grizzle of the Depart-
ment of Biostatistics for his encouragement during my graduate
studies and Professor H. Ko Nandi of the University of Calcutta for
stimulating my interest in Mathematical Statistics.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
vi
FinallYJ these acknowledgements would be incomplete without
mention of the active co-operation and enthusiasm of my grandfather.
Mro No Duttapand my parents without whose encouragement I would not
have reached this stage of my academic career o
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
vii
ABBREVIATIONS AND NOTATIONS
Abbreviations
iff
if and only if
r.v.
random variable
distribution function
cumulative distribution function
independent and identically distributed
1. 1. d.
independent and identically distributed random variables
almost sure
p.d.
positive definite
C.L.T.
Central Limit Theorem
A.R.E.
Asymptotic Relative Efficiency
Notations
q variate normal with mean vector
J ,p
....
A
«a 1J.. »
pXq
a
pl .•• a
pq.
~
and dispersion matrix
~
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
viii
ABSTRACT
The following problems are considered in this dissertation.
In Chapters II and III, it is shown that Bahadur's almost sure
asymptotic representation of a sample quantile for independent and
identically distributed random variables holds under certain regularity conditions for a general class of stationary multivariate
autoregressive processes,
This yields the asymptotic (multi-)
normality of the standardized forms of quantiles in autoregressive
processes.
Then the asymptotic (multi-) normality is generalized
for a vector of linear compounds of the standardized forms of quantiles in autoregressive processes and the estimation of its dispersion
matrix is also considered.
Further, the theory of asymptotic (multi-)
normality of a vector of linear compounds of the standardized forms of
quantiles in autoregressive processes is extended for the random
sample sizeo
In Chapter IV, for stationary multivariate autoregressive processes, the asymptotic distribution of Wilcoxon signed rank statistic
is derived and subsequently the theory is generalized for a class of
U-statistics.
Finally, in Chapter V, several rival estimators (median, 27% midrange, median of the mid-ranges and mean) of location parameter for
stationary autoregressive processes are considered and their relative
performances are compared by their asymptotic relative efficiencies,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
CHAPTER I
INTRODUCTION AND SUMMARY
1.1.
Introduction and a Review of the Literature
The central limit theorem in its various forms, the character-
istic functions and the laws of large numbers are the basic tools for
the study of the asymptotic properties of linear estimators (i.e.
linear functions of the unordered sample observations).
However,
in dealing with ordered observations or, in general, a linear function
of ordered observations we require additional tools for the study of
their asymptotic properties.
For example, asymptotic normality of
sample quantiles for i.i.d.r.v. can be deduced by the direct method
given in Cramer (1946, pp. 367-369) or in Mosteller (1946). This
approach becomes increasingly complicated as the number of quantiles
increases.
Also for the study of other large sample properties of
quantiles (e.g., asymptotic normality in the multivariate case,
almost sure convergence, law of iterative logarithm, etc.) this method
becomes too complicated to provide the necessary results.
A recent
asymptotic representation of sample quantiles by Bahadur (1966) not
only simplifies the proof of the asymptotic normality of quantiles
in various univariate as well as multivariate situations but also
provides the access to the study of more refined large sample properties of quantiles.
The other notable non-linear estimators are the
U-statistics, studied by Hoeffding (1948), and rank order statistics,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
2
studied by Chernoff and Savage (1958) and Hajek (1968) among others.
For these statistics either a projection technique is used to decompose
them into two parts where on the first part the classical C.L.T.
and the laws of large numbers, etc., directly apply while the second
part is asymptotically negligible in probability or they are related
as functiona1s of empirical distributions and their large sample
properties are studied with the aid of the large sample properties
of the empirical distributions.
In this dissertation we are nrincinal1y concerned with certain
stochastic difference equations or stationary autoregressive processes
where the successive observ&tions are not independent but are subject
to a chain of dependence.
To be general we start with the following
multivariate model:
k
(1.1.1)
L:
r=O
where ~t
X
r
E
~t
= 0,
(E
(X
1, ••• ,
t,l
t ,
1"'"
k(~l)
, ••• ,
A X
E
t,q
t
~rt-r
X
t ,q
are q
)'
)'
x
0, ±1, ••• ,
is a q x 1 vector,
hr = «a~r))).
,
Js
J , s= 1 , ••• q
q matrices of constants and
(white noise) are i.i.d. q
x
1 random vectors
with an absolutely continuous (q-variate) c.d.f., G(~), ~
E
Rq, the
real q-space.
Meteorological and economic data provide perhaps the most
obvious examples of such a model.
by Whittle (1953).
Its adequacy has been investigated
He has successfully fitted the above model to data
giving the total sunspot area for a series of 120 six-monthly periods,
and for two belts of solar latitude (16°-21° H., and 16°_21° S.).
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
3
For autoregressive errors Hannan (1961) and Eicker (1965)
studied large sample distribution properties of the classical least
square estimators computed under the assumption of the independence
of errors.
Since these estimators are linear functions of the original
observations, they are also linear in terms of the error components.
These errors being independent, they were in a position to apply again
the classical C.L.T. for their purpose.
We briefly present their
principal results below.
Consider a system of regressions
(1.1.2)
where
~t
~t = (X
t
vectors,
Xt =
, 1"'"
+
B*Y
~
~t
.§:.t
.-
£ t = (E t ,l, ••• ,E t ,q)' are q x 1 random
X t,q ) "
(Y t , 1"'"
=
,
Y t, P )
is a p
x
*J
1 vector, ~ * = «Si"'))i=l
, ••• ,q
j=l, ••• ,p
is a (q x p)-matrix of regression coefficients and
(1.1.3)
=
where £j are a sequence of known (q x r)-matrices and
i.i.d. random vectors with covariance matrix
moments of order 2 + 0, 0
> O.
The norm
~t
are r x 1
g and finite absolute
[I£II!/,
of a matrix
£ is
defined by
sup
II~II!/,
II~II JI,=l
where for a vector~,
II ~ II JI,
(1.1.4)
and the components Yt,j of
{ElxiIJl,}l/JI,.
Then it is assumed that
i
~
11£.11 2
Xt
are generated by a process such that,
j=_oo
<00
J
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
4
with probability one, as n increases
lim d 2 ,
n-+oo n,J
(1.1.5)
00
L
y2
= 0,
j = l, ... ,q.
t ,J' ,
t=1
lim y2 ,/d2 .
n,J/ (n,J
n-+oo
(1.1.6)
n
,
= l, .•• ,q
j
(L1. 7)
P jk(h),
I f E(h)
l, ••. , q
j ,k
«P'k(h»).
k-l
then 13.(h) can be written as
J
J, - , ••• ,q
(1. 1. 8)
E(h)
where M(A) is a matrix valued function whose increments are Hermitian
non-negative.
It is assumed that .R(O) is non-singular.
When the errors are independent, the least square estimate
of] * is
XX'Crr')
(1.1.9)
where
~
has
~
-1
t and Y has 1 t as the t-th column, t
N
= l,2, ••• ,n.
(1.1.9) is expressed in the form of a column vector
If
.. in
2* having b *
1J
{q (i-I) + j}-th row, then
Q* = (lq
@
yy,)-1(L ~t
@
It)
t
where
@
denotes the Kronecker product.
by arranging the rows of
We denote by .l2,."
••
place.
Then Rosenblatt
~
*
If
~
*
is the vector obtained
in the same way, then E(Q *)
=
~
*
the diagonal matrix having d ,in the j-th
n,J
(1956~
has shown that, as n
~
00,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
6
process.
These estimators being non-linear functions of the original
observations are also non-linear in terms of the error components.
Actually weare concerned here with a class of estimators different
from the one considered by Hannan (1961) and Eicker (1965).
Further,
ourproposed class of estimators being non-linear function of the
errors, their technique does not seem to be readily applicable in this
case.
Estimation and the asymptotic distribution of the estimates
of the coefficients of autoregressive processes are available in the
literature (see Grenander and Rosenblatt, 1957).
Also, in the case
of autoregressive schemes, the asymptotic distribution of sample mean
is well known (see Hannan, 1961).
However, for autoregressive
processes, unlike the case of independent observations, we do not
know much about the distribution theories of Hoeffding's U-statistics
or sample quantiles and other related statistics.
To bridge this gap
we want to develop the distribution theory of sample quantiles, any
linear combination of sample quantiles and a class of U-statistics
for stationary autoregressive processes.
First, for a stationary autoregressive process, we consider
the asymptotic distribution theory of a sample quantile.
The chain
of dependence by which the successive observations in the process are
linked makes it difficult to apply the fundamental technique (such
as in Cramer, 1946, PP.367-369) for deriving the asymptotic normality
of a sample quantile.
An alternative approach (see section 2.1) of
deriving the asymptotic distribution of a sample quantile based on
the properties of order statistics seems to be applicable for
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
7
autoregressive processes, but this approach fails to give us more
refined convergence results of sample quanti1es.
These difficulties
have been avoided here by adopting the elegant Bahadur-representation
of a quantile (cf. Bahadur, 1966) and extending it in the context of
autoregressive processes.
With this end in view, we first present
the principal results of Bahadur (1966).
Let w
F(x).
If
s be
= (Xl, X2, ... ) be a sequence of i.i.d.r.v. with d.f.
the population quantile of order p, then it is assumed
that F has at least two derivatives in some neighbourhood of
F" (x) is bounded in the neighbourhood, and F' (0 =
each n
f(~) >
O.
~,
For
= 1, 2, •.• , let Yn (w) be the sample p-quanti1e when the sample
Let Z (w) be the number of observations X. in the
n
1
sample (Xl' ... ' X ) such that Xi
n
(1. loll)
Y (w) =
n
~
+ [(Z (w) n
>~.
Then
nq)/nf(~)]
+ R (w)
n
where
q
1 - p
and
(1.1. 12)
R (w)
n
O(n
-3/4
log n),
as n
~
00,
with probability one.
One important feature of the Bahadur-representation of sample
quanti1es is its ability to yield the asymptotic normality of sample
quanti1es in various non-standard situations where either independence
or homogeneity of the distribution functions or both may be vitiated.
With this motivation Sen (1968 b ) showed that Bahadur's asymptotic
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
8
almost sure representation of the standardized"<form bf a sample
quantile is also valid for any m-dependent process.
The simplified
version of his results for stationary m-dependent processes are given
below.
Let w
=
(Xl' X2,.'.) be a sequence of random variables forming
an m-dependent process, i.e. X. and X. are independent whenever
J
1
Ii - j! > m.
The marginal c.d.f. of Xi is denoted by F(x), and the
joint c.d.f. of (Xi' Xi +h ) by Fh(x, y), for h
p:O < p
(Xl'.'"
(1.1.13)
<
= l, ••. ,m.
For any
1, let Y (w) be the sample p-quanti1e when the sample is
·n
Xn ) and let the empirical c.d.f. Fn (x) be defined by
F (x)
n
n
-1
-co
<
x
<
co
,
where
c(u)
('1. 1. 14)
If
~
1 or 0 according as u~ or. <0 •.
be the population p-quanti1e of F(x) then it is defined by
F(~)
p.
It is assumed that in the neighbourhood of
~,
F(x) is absolutely
continuous and that
a)
f(x)
o
b)
= (d!dx)F(x) is continuous in some neighbourhood of
< f(~)
<
co,
and
F"(x) is bounded in the same neighbourhood of
~.
If v2
is defined by
n,m
m
(1.1.15)
ifn,m
pel - p) + 2
~ (1 - h!n){Fh(~' ~) ~. p2}
h=l
~,
with
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
9
then it is shown that
n Var{Fn(O} = v 2
n,m
Finally, let
~
(1.1.16)
where a
~
n
n
-1/2
log n, as n
+ an }
00.
+
With the above notations we have the following theorem.
Theorem 1.1.2.
If the condition (a) is satisfied, then as n
supn [F(x) - p] + [Fn (0 - Fn (x)] I :x
(1.1.17)
= o(n-3/4
with probability one.
£
+
00,
In}
log n),
If in addition, inf
n v;2
n ,m > 0,
-"JL_ _...:-_..;.;....-=---"..;.;....~'---'-'-
..;.;....-.;:..--.A.~
)(nl/2f(~)[Yn (w) - ~]/v n,m )
(1.1.18)
+
N(O, 1).
Finally, if both the conditions (a) and (b) are satisfied then
[Yn(w) -
(1.1.19)
where as n
+
00, R (w)
n
~]f(~)
+ [Fn (~) - p] = Rn (w)
= O(n-3/4
log n), with probability one.
In the case of a stationary autoregressive process, it is
shown here that for the asymptotic distribution theory of a sample
quantile,we may as well (for large n only) replace the original
process by an mn-dependent process where mn
fixed positive number.
~
K log n, and K is some
Also, it is shown here that Sen's extension
for stationary m-dependent processes of asymptotic almost sure
representation of the standardized form of a sample quantile, as given
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
10
above, can be further extended to the stationary m -dependent pron
cesses.
This enables us to draw conclusions about the asymptotic
normality of sample quantiles for a stationary autoregressive process.
Next, the distribution theory of any linear compound of
several quantiles for stationary autoregressive processes is derived
by extending our previous results for a particular quantile over the
entire real line.
Again, this extension is possible by extensive
use of Sen and Ghosh's (1971) generalization of Bahadur's asymptotic
almost sure representation of the standardized form of a sample
quantile over the entire real line.
Their generalization is briefly
indicated here.
Let the distribution function F(x) be absolutely continuous
with density function f(x) and sup f(x)
x
= fo
<~.
Let the empirical
distribution function be F (x) and
n
gK(n)
K
= n-1/2 (log n),
K ~ 1.
Then Sen and Ghosh proved the following.
Theorem
constant
(1.1. 20)
For every finite s (>0) there exists a positive
and a sample size n , such that for n
s
p{sup
sup
n
x lal<g (n)
K
1 2
/
~
n ,
s
1F (x + a) - F (x) - F(x + a) + F(x)1
n
n
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
11
Hence
(1.1.21)
sup
sup
{n
x la[<g (n)
1 2
/
\F (x + a) - Fn(x) - F(x + a) + F(x)l}
n
K_
O(n
=
with probability one as n
-+
-1/4
K
(log n) -),
00
The above theorem is first extended to the stationary m n
dependent process where m
n
K log n, K being a positive number.
This
extension along with the asymptotic reduction of a stationary autoregressive process to a stationary m -dependent process enables us to
n
prove the same theorem for a class of autoregressive processes.
a~ymptotic
Then
normality of a linear compound of several quantiles for
stationary autoregressive processes follows easily as a corollary
of the theorem.
An application of this result is made by finding the
asymptotic distribution of the 27% mid-range estimate for a stationary
autoregressive process and the result is also extended for the case
of random sample size.
One of the mostly used statistics based on ranks is the socalled Wilcoxon score estimator which is expressed as the median of
the mid-ranges.
This estimator is derived from the Wilcoxon signed
rank statistic (see Hodges and Lehmann, 1963) which can be expressed
as a Hoeffding's (1948) U-statistic as well as a rank order statistic.
The asymptotic distribution of Wilcoxon signed rank statistic for
stationary autoregressive processes is derived by first asymptotically
reducing it to a Wilcoxon signed rank statistic for an m -dependent
n
stationary process and then using Rosenblatt's (1956b) C.L.T. for
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
12
strongly mixing processes.
Since Wilcoxon signed rank statistics can
also be expressed as a U-statistic, attention is concentrated to
generalize this theory to the study of the distribution theory of
U-statistics for stationary autoregressive processes.
under certain regularity conditions.
This is done
These regularity conditions are
somewhat more restrictive than those in Hoeffding (1948) dealing with
independent observations and appear to be necessary in view of infinite
chain of dependence in the series of observations.
It is intended to
follow up the general case in near future.
In this generalization extensive use of Sen's (1963) extension
of asymptotic normality of Hoeffding's (1948) U-statistics is made.
First his extension is given below.
Let~l
' ••. , X be a vector valued sample of size n from an
~n
F(~)
m-dependent stationary process having c.d.f.
~(~a
and
" •• ,
~a
be a statistic symmetric in the arguments
~a
~a
"'"
r
1
; al
)
r
1
< ••• < a
r
•
Let us define the parameter associated with the c.d.f. F as
E{ ~ (X
(1.1.22)
"'al
where a +1 - a i = vi' 0 < Vi
i
~
, ••• , X
m for i = i 1
remaining r - t - 1 values of i, a i + l - a i
For t
= 0,
(1.1.23)
"'a r
;t •• •
>
)}
;tii, while for the,.
m, for t
= 1, ••• , r - 1.
the parameter (1.1.22) is reduced to the following:
g (F)
=
E{~(X , ... , X )Ia
- a 1." > m for i
't' "'a
"'a
i +1
The statistic
~(~a
, ••• ,
1
whenever a + l - a
i
i
>
=
1, ... ,lr - 11.
r
1
m for i
~a
) is termed a non-serial statistic,
r
= 1, ••• , r - 1.
The corresponding
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
13
symmetric estimator
UO~l"'"
!n) based on n observations is then
defined by
Uo ~1 ' •• "
(1.1.24)
X ) =
~n
en - rm + m)_l
r
E
5
0
ep ~0. , ••• ,
where the summation 50 extends over all possible (n 0.
1
"",
a r , satisfying a i +1
-
X
~a
1
)
r
:m + m)
a i > m, i = 1, ••• , r - 1.
sets of
Also let
(1.1.25)
where the summation 5 extends over all 1
assume for all
(0. 1 " ' "
ar):o
<
a i +1
ai
-
~ 0.1
~
< ••• <
ar
~
m + 1, for i
n.
Let us
= 1, ••• ,
r - 1,
(1.1.26)
Let us put
lSa )} - g(F)
(1. 1. 27)
r
for a
= 0, 1, ... , r, where a i +1
-
a
> m for all i
i
= 1, •.. , r - 1,
and let
(1. 1. 28)
(1.1.29)
l;
a' (h 1 ' ••• , h a)
= E{ep a (X~a1 , ••• , lSa a )epa(lS8 1 "", !S8 )}
a
for a = 0, 1, ••• , r - 1; where la i - 8il
i
=
1,2, ... , a, but la i - atl > m,
for all i
(1. 1. 30)
~
t = 1, .•• , m.
1
8
= hi; 0
<
hi ~ m for
i - Btl> m and la i - 8tl
Finally, let
>
m
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
14
Then Sen (1963) proved the following.
~ E{ 1~1 (~i)13} < ~, and (1.1.26) holds, then both
Iheorem 1.1.4.
n
1/2
{U(~l""'~n) - g(F)} and n
1/2
{UO(~l""'~n) - g(F)} have asymp-
totica11y the same normal distribution with zero mean and variance r
We may note that
compound of £t-r' r
~t
= 0,
2
~1'
in (1.1.1) can be expressed as a linear
1, 2, ••• ,~.
Essentially Hannan (1961) and
Eicker (1965) considered a truncation of this linear compound and
= ~n,t
Y
+ ~n,t
R
where Y
is the truncated part and ~,
R t is the
~,t
wrote ~t
X
residual.
~t'S
Then they approximated the linear function of the
the corresponding function of the Y
~n,t
linear in the independent errors
E
~t
'so
The latter function is again
over a certain number of terms depend-
ing on n, and thereby they were able to use the C.L.T.
~(~l""'~r)
by
But the kernel
of a U-statistic is not, in general, a linear function of
its arguments.
Therefore the above decomposition is not adaptable.
The
non-linear nature of the U-statistic and the chain of dependence of the
successive observations make
it quite complicated to reduce the autore-
gressive process to an m -dependent process in which the above results
n
of Sen (1963) can be incorporated and extended to prove our desired
results.
i)
This, however, is done here through the following conditions:
For every
E >
0, we can find a positive number A such that if we
define the qr-dimensiona1 rectangle
={~
and I
AC
(1.1.31)
, ••• ,~
.1<A,fori=l, ••• , r ; j = l , ••• ,q}
a.,J
l.
qr
is the indicator function of AC = R
- A*, then
al
sup
n
and (li)
ar
:Ix
sup
E[ I c 1HY
, ••• ,Y
)
~n a
~n a
1<a <••• <a <n
A
' 1
' r
- 1
r--
1
3
] <
E
3
sup
E[I~(Y
)1 ] < ~.
~n
a , ••• ,Y
~n a
n 1<a <••• <a <n
' 1
' r
- 1
r-Under the above conditions, Sen's extension of asymptotic
(1.1.32)
sup
normality of U-statistics for m-dependent stationary processes is
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
15
further extended to the mn-dependent stationary process, where
m
n
K log nand K is a positive number.
that the kernel
~(~l' ••• ' ~r)
Again under the assumption
of a U-statistic is either continuous
in its arguments or continuous everywhere except at a finite number
of strips with finite discontinuities, the U-statistic for the
original stationary autoregressive process is asymptotically reduced
to a U-statistic for the above mn -dependent process.
This asymptotic
reduction along with the extension of Sen's (1963) results enable us
to draw cons1usions about the asymptotic normality of aU-statistic
for stationary autoregressive processes.
Finally, several rival estimators (whose asymptotic distributions are derived here) of location parameter for stationary autoregressive processes are considered and their relative performances
are compared with that of process average.
1.2.
A Summary of the Results in Chapters II-V
Let (~l' ~2'·.·' ~n) be a sample of size n from the k- th
order q-variate stationary autoregressive process denoted by (1.1.1).
We assume that in addition to the conditions (cf. Whittle, 1953)
necessary for the validity of the expression for
(but convergent) series in the
E
~t-r
~t
as an infinite
, r = 0, 1, ••• , the following holds:
(1.2.1)
where
II ~t II
,- 1/2
= [~t~t]
• The joint c.d .. £. of ~t and the marginal
c.d.f. of Xt,j are denoted by F(x) .and FU]'(x), and the j--th
variate p (j) -quantile is denoted by
E;~{~;,
where
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
16
j ,•••
=1
F [j] '("C"<., p(j)
(j ) } = p(j) •~O < p(j) < l'.
, 'q.
(1.2.2)
The j-th variate sample p(j)-quanti1e is denoted by
=
z(j)
(1.2.3)
n
xU)
[np (j)] + 1,
r =
n,r'
J' = 1 , ••.
,
q
where X (jl) ::; X (j) ::; •.. s X(j) are the ordered values of Xl ., ••• , X ,"
n .J
n,
n,2
n,n
. ,J
forj=l, ... ,q.
~n
(1.2.4)
Let
~~1) , ••• ,
(.
=
(q»)/
Zn
'
k~
_
-
//1)
(q) '(,
~P(l)"'" ~p(q)J
For the j-th variate, define the empirical c. d. f.
F .(x) = n
(1.2.5)
-I
nJ
n
~
i=l
c(x - X. . ),
where c(u) is defined in (1.1.14).
a
(1.2.6)
where a
n
(1.2.7)
~
n
-1/2
log n as n
~ = «
\i j
_00
1,J
n
+
i )j , j '=
<
x
Also define
I , • '.'. , q ;
and
where f[j](x) = (d/dx)F[j](X);
00
Let
::; x ::;
00.
<
j, j' = 1, •••• q.
1, .•. , q
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
17
Then in connection with quanti1es for stationary autoregressive
processes~
the following two main theorems are proved in chapter II.
(j)
I f in the neighbourhood of ~p (j) ~ f [j] (x) is finite
Theorem 1.2.1.
continuous and
positive~
then as n
+
00,
(1.2.9)
= 0(n- 3 / 4
with probability one.
log n),
Further, if f[j] (x) is bounded in some neigh-
bourhood of
(1.2.10)
=
with probability one, for j
Theorem 1.2.2.
Ii
F(~)
O(n
=
-1/4
log n),
1, ••• , q.
is absolutely continuous at ....p
~
with finite,
-
positive and continuous marginal densities in some neighbourhood of
~.~,
~p.
and i f
»;
is positive definite, then
(1.2.11)
In the first part of chapter III, for stationary autoregressive
processes Bahadur's asymptotic almost sure representation of the
standardized form of a sample quantile is generalized over the entire
real line.
In connection with
Let g(n)
this~the
following theorem is proved.
= n- 1/ 2 log n and let us suppose that F[j](X) be
absolutely continuous and s~p f [j] (x)
=
f oj <
00.·
Then we have
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
18
Theorem 1.2.3.
As n
+ 00,
(1.2.12) sup
sup {n 1/2 IFnj(X + a) - Fn.(x)-F[j](x+a) + F[.](x) I}
J
J
x lal<g(n)
1/4
log n),
= 0(nwith probability one.
With the help of the above
rnultinorrnality,
theote~
and (1.2.10) the asymptotic
derived in dhapter II of a sample quantile (vector)
for a stationary autoregressive process, is generalized here for a
vector of linear combinations of several quantiles.
An application
of this result is indicated and under certain conditions the result
is also extended for the case of random sample size.
Further, the
chapter III contains a method of estimation of variances and
covariances of sample quantiles and hence the variances and
covariances of linear compounds of several quantiles.
In connection
with the estimation problem the following two theorems are proved.
Theorem 1.2.4.
Let ~n (j) = np(j) + o(n l/2 log n), and let Sj be the
~(j)-th smallest observation among (Xl ., ... ,X
n
(1.2.13)
'V"
jj
,
= {F
n
j j , (s . , s . , ) -p
J J
,J
(j) (j')
P
.). If
m n,J
\'n
(j) (j')
}+ t.. { F .., h (s . , s j , ) -p
p
}
h=l nJJ
J
m
+
n
L
h=l
{F j' 'h(s.,s.,) - p
n J
J J
(j) (j')
p
}
where
(1.2.14)
(1. 2 .15)
F .. ,(u,v)
nJJ
=
F .. 'h (u,v)
nJJ
•
n
-1
n
L
i=l
n -1
c(u - Xi .,v - Xi .,)
,J
,J
n
I
i=1
c(u
- Xi ,J.,v - Xi +h ,J.,)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
19
c(u,v)
(1.2.16)
a
{~
for u
~
0, v >
°
otherwise
then
-+-
v jj'
(1.2.17)
p
v jj , ,
Theorem 1.2.5. Let ~(j)
--on,l
C2n(n1/2 log n) where c
(~(j)
- ~(j»
n,2
n ,1
as n
-+-
=.
-+-
~ as n
j, j'
= np(j)
1n
-+-
and c
,J
.).
n,J
- c1n(n1/2 log n)
2n
'
~(j) = np(j) +
n,2
are positive constants such that
~, but c
and c
both converge to zero
- 1n - - 2n ="::"::--==':"="::..s:L::""":::'::""'::':':'::';:'
Also let Sj,i be the
among (Xl ., ... ,X
1, •.. ,q.
=
~~:i-th
smallest observation
Then for each j • 1, •.• ,q, as n
(/j) -
(1.2.18)
n,2
-+-~,
)l(j~)/n(s.
- Sj,l)
n,L
J,2
with probability one.
In chapter IV first, the asymptotic distribution of Wilcoxon
signed rank statistic for stationary autoregressive processes is
deduced and then the theory is generalized for a class of U-statistics.
Let
.... C/.
1
, ... ,X
X
"""'a
~(x
't'
1
, ••• ,X
) be a statistic symmetric in the arguments
....C/.
r
"""'a. •
r'
(1. 2 .19)
Cl.
l
< ••• <
a , and
r
I
s
where the summation S extends over all 1
Let us define
~
't'
(x
, ••• , ~CI.
y
.... a
1
~ C/.
1
<
•••
<
)
r
a
<
r-
n
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
20
~
g(F ) = E{HY
, .. "tY
) la.+ -a. > m for i=l, ••• ,r-1}
n
~n,a1
~n,ar
~ 1
~
n
(1.2.2a)
where ~n,~
Y . and mn are defined in (2.3.5) and (2.3.6) respectively.
Further, let !l""'!r be r.i.i.d.r.v. each having the d.f. F and
lim
n -+-
g(F) = g(F).
Let us put
n
00
(1.2.21)
for a = a,l, ••• ,r, where a + - a
i 1
i
>
m for all i=1,2, ••• ,r-1,
n
and let
i;(n)
(1.2.22)
(1.2.23)
a·a
~(n)(h
a·
1"'"
=
h ) =
a
for a = a,l, ••• ,r; where la 1 - S l • h~; a
i
...
but lai-atl
>
mn , Is.-stl
~
>
<
mn and la.-Stl
~
a.
<
m for i=l, ••• ,a,
~
-
n
>
mn for all i#t=l, ••• ,n.
Finally, let
00
(1.2.24)
~1
I
=
h=l
~(n»
l.h
Then in connection with U-statistics the following theorem
is proved:
Theorem 1.2.6.
section 4.3, n
Under the assumptions (A) and (B), stated in
1/2
{U(~l'."'~n)
- g(F)} is asymptotically normally
distributed with mean a and variance r2~l'
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
21
In the end of the chapter, the asymptotic distribution of the
median of the mid-range estimate (vector) for a stationary autoregressive process is obtained as an application of the earlier
results of the chapter.
In the final chapter several rival estimators (median, 27%
mid-range, median of the mid-ranges and mean) of location parameter
for stationary autoregressive processes are considered and their
relative performances are compared by their asymptotic relative
efficiencies.
Some A.R.E. values are tabulated for univariate
Gaussian autoregressive processes.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
CHAPTER II
BAHADUR-REPRESENTATION OF SAMPLE QUANTILES IN SOME
STATIONARY MULTIVARIATE AUTOREGRESSIVE PROCESSES,
2.1.
Introduction
In this chapter, we are concerned with the limiting properties
of sample quanti1es in a general class of stationary multivariate
autoregressive processes where there is dependence among the successive
observations.
The standard technique of deriving the asymptotic
distribution of a sample quantile for i.i.d.r.v. [cf. Cramer (1946,
pp. 367-369)] usually encounters considerable difficulties in the
multivariate case or in the case of dependent random variables.
alternative simple approach is as follows.
statistic of a sample Xl"."
We let r
F.
= np +
Let X
be the r-th order
n,r
X of size n from a distribution F(x).
n
o(~) (0 < p < 1) and denote by sp the p-quanti1e of
Suppose F'(s )
p
= f(s p ) exists and is positive. Then, with c(u)
defined in (1.1.14), we have for every fixed u,
(2.1.1)
= P [at1 east
~
An
= Pn- 1/2
n
~
i=l
r
0
f X
1
, ••• ,
+ n
{c(s
p
-1/2
X
n
_
<
I:"
~
p
+
n- 1 /2 u ]·
u - X.) - F(s
1
P
+ n-
1/2
u)}
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
23
fC(~
n
L:
i=l {F(~
p
P
uf(~p)
2 -
1/2
+
{p(l - p)}
l
0(1)-,
-
and hence, the asymptotic normality can be obtained by using an
appropriate form of a central limit theorem applicable to the double
sequence of random variables {c(~p + n
n
2
1.
-1/2
u - Xi)' i= 1, ... , n},
This approach appears to be applicable for multivariate as
well as dependent random variables.
However, in refined statistical
analysis, we are not merely satisfied with this weak convergence of
sample quantiles.
For i.i.d.r.v. 's, Bahadur (1966) has considered
an elegant asymptotic almost sure representation of a sample
quantile, which is further extended to the case of m-dependent
processes by Sen (1968b). An important by product of this a.s. representation is an alternative proof of the asymptotic normality of the
standardized form of a sample quantile; the cases of several quantiles
or of a linear function of quanti1es also follow more easily from this
representation.
We derive an analogous representation for an m -dependent
n
process, where we let m
n
~
K log n, K being a positive number.
Also,
we show that insofar as the asymptotic behaviour of a sample quantile
is concerned, a stationary autoregressive process may as well be
replaced by a sui table m -dependent process, with m
n
n
~
K log n.
Combining the above, our conclusions about the limiting behaviour of a
sample quantile, including its asymptotic normality, follow readily.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
24
2.2.
Preliminary Notations and Assumptions
Consider a sequence
q(~l)-vectors
in (1.1.1).
t = 0, ±1, ±2, ••• } of stochastic
{~t'
from the multivariate autoregressive process defined
Using the forward operator E as
r
(2.2.1)
E Xt,j = Xt+r,j'
we may rewrite the set of equations in (1.1.1) as
(2.2.2)
q
k
L
L
Q,=l r=o
a~r)Ek-rx
JQ,
t-k,Q,
Again, if we write (when
= E t ,J.,
= 1, ••• , q; for all t.
j
aj~) ~ 0, otherwise take a lower degree
polynomial)
(2.2.3)
where e
a (0)
j Q,
v.
JQ,
(E - e 1(jQ, ») ... \.:IE
_ e kQ,'
(j»)
1, •• "
0
J, Q,
q,
(")
(j)
, ••• , e J are the roots of
kQ,
H
k
L
(2.2.4)
k r
a.(r)E -
r=o
0,
n
j, Q, = 1, ••• , q,
then (1.1.1) may be rewritten as
= £t where '"V =
(2.2.5)
« v.JQ, » J,Q,=
0
Our first assumption is that all the le;i)l, r
j, Q,
, .. . ,q
= 1, ••• , k;
= 1, ••• , q lie in the semi-closed interval [0,1), i.e.,
(2.2.6)
a
<
max
max
. n 1
r= 1 "'" k J , ",= , ••• , q
Ie
(P I = e*
r",
<
1-
Under (2.2.6) and proceeding as in Whittle (1953), we have
(2.2.7)
1:
r=O
B
E:
~r~t-r'
for all t,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
25
where B -~r
('('b ( r
I
~
"-
J'JI.,
)\D J.
.
Jl
0
r
JI., = 1 •••• , q •
b;~)
constant coefficients. where
q).
k'• J'•
roo ts es I V ' s = 1.....
"
t)
-
)\"
-
= 0. 1, ••• are all q
x q matrices of
is a polynomial of degree r in the
q•
1 "'"
If the errors
~t
have
finite second moments. it can be shown (along the lines of Doob (1953.
p. 503»
that
(2.2.8)
where C and g are two positive constants.
We shall see later on that
we do not require the existence of the second moments of
we make (2.2.8) a part of our basic assumptions.
~t'
For clarification
of ideas. we touch briefly the univariate case where q = 1.
g = k - 1.
Thus. when k
2 and the two roots el and
=
1) e r • wh'l
( to e ) • b (r) = (+
r
1 e if el
In general. Ib(r)1 s (k +
insures that for every 0
(2.2.9)
L.
Here
ez are equal
eZ. Ib(r),
.
S ( r + l)(e*)r.
~ - l)(e*)r s crk-1(e*)r. Note that (2.2.8)
0.
>
~ 'b~~)IO
r=O
~
Hence,
• IV
<
00
f or a 11'J.
JI.,
= 1 ••• •• q.
J
Our third assumption is that
(2.2.10)
EI I~tl
1
0
<
00.
for some 0 > 0 (need not be
~l).
In fact. when (2.2.10) holds for 0 = 2. and we denote by
(2.2.11)
then the matrices
E[X
X' h]' h
~t~t~O. ~l"'"
Yule-Walker equations:
~
=
0. 1 •••••
in (1.1.1) satisfy the generalized
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
26
k
(2.2.12)
L~r+h
=.Q
for h = 0, 1, ••• , k,
r=O
~
and the
can then be expressed in terms of the elements
*'
The joint c.d.-f. of ~ is denoted by F(*),
E:
q
R
~,
o•• ,
~0
, the q-dimen-
sional real space and the marginal c.d.£. of Xt ", the j-th component
,J
of
~t'
is denoted by F[j}(X)' j = 1, ••• , q.
By the assumed (absolute)
continuity of G, F and all the F[j} are also (absolutely), continuous.
(q) ) ,
(1)
Thus, there exists a unique vector ~p = ( ~P(I)'OOO' ~p(q)
°
<
p(j)
(2.2.13)
~
-p
<
1 for all j = 1, ••• , q), ;uch that for every
(j)
F[jl ( E;p(j) -
)
E:
>
(where
0,
< p (j)
E:
is termed the vector of population
~o-ordinate wise) p(j)-quantileso
~
For a finite time interval Tn = {t:l
~
t
~
n}, let {XI,o.o,
X
}
~
~
correspond to the chance variables associated with the sample of size n
from the process in (1.1.1) •
The ordered random variables on the j-th
< ••• < x(j); by the continuity of
variate are denoted by X(j)
n,I
n,n
F (and hence of F[j})' ties among the observations can be neglected,
with probability one, for all j = 1, .•• , q.
Z
~
= (Z(I)
,0'"
n
(2.2.14)
-
The sample p-quantile
Z(q»)' is then defined by
n
= x(j) ,
n,llj'
J"
= 1,
... " q
where)l "
J
and [s} is the largest integer contained in s. We are primarily
I 2
interested in the asymptotic behaviour of n / (Z - ~). For this,
....n
we introduce the following notations.
"'P
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
27
For the j-th variate, the empirical c.d.f. is defined by
Fnj(X) = n
(2.2.15)
-1
c (x - X),
i,j
x <
<
-00
1~(1»)
l,Je denote the dispersion matrix of n 1 / 2 1';.
tn1~p(1)
(q) )
... , Fnq ( ~p (q)
- p
shall show that lim
(q51
_
-' by lin-
~
=
~
« njj . » j,j
V
00,
1
1=1
j = 1, ••• , q.
_ p(l) ~
~ . . . ,q'
exists and is finite.
Later, we
Also, we denote by
n~
(2.2.16)
I
and assume that both f[j](x) and f[j] (x) exist for every j = 1, ••• , q.
Other notations will be introduced as and when necessary.
2.3.
Almost Sure
(Bahadu~)
representation of
~
As in Bahadur (1966), we let
I (j) =
(2.3.1)
n
fx . 'op~ (j)(j) - an : ; x::; sp~ (J)
(j ).
+
)
an) ,
j=l, ••• ,q,
where
a
(2.3.2)
n
~
n -1/2 log n
as n
~
00.
Then, the main theorem of this section is the following.
Theorem 2.3.1.
If in the neighbourhood of
E;~{~),
f[j]
(x) is finite
continuous and positive, then under (2.2.6), (2.2.8) and (2.2.10),
as n
~
00,
=
a (n-3/4
log n),
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
28
with probability one.
Further. if f[jl(X) is bounded in some
(j)
~
neighbourhood of sp(j)' then as n
00.
Inl/2{G~j) - g;{~Jf[jJ(i;;{~») + ~nj(i;~{;») - p{j~}1
(2.3.4)
1 4
= o(n- /
with probability one. for j
log n),
1 ••••• q.
=
For proving the theorem. we require certain basic results.
which are considered first.
Let uStdefine
m
(2.3.5)
y
,
L
n
r=O
~n.1
B £i
~r"
-r
,R
00
~n.1
B e:
L
. =
r=m +1
n
where the
~
r
i = 1. 2 ••••• n.
~r-i-r'
,l-"':
are defined by (2.2.7). and where mn is a positive integer
such that
m ~ K log n as n ~ 00; op*K = c > 8,
(2.3.6)
n
K is a positive constant. 0 is defined by (2.2.10) and we write
e*=e""P *
(2.3.7)
where by (2.2.6). p* > O.
(2.3.8)
Let then
y(j»)
n,i •
F*, (x)
nJ
_00
<
x <
j
00
•
be the empirical c.d.f.'s of the q components of the
1. • • •• q.
Xn • i •
The true
c.d.f. of y(j)i is denoted by
n.
(2.3.9)
, n ['lex)
J
= P{ y
(j)'
.s x } •
n.1
_00
< x <
00.
j
The joint c.d.f. of (X, .• X + ,,) is denoted by
i h .J
1.J
1, •• "
q.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
29
. (2.3.10)
F[ . . , ]h(x, y) =p{X . . :;; x, X'+h ., :;; y},
Jd
1.,J
1:,J
(j') ) is denoted by
and the joint c.d.f. of (y(j ) Yn,i+h
n,i'
~
(2.3.11)
F n [j , j
j, j' = 1, ••• , q; h
, ] h (x , y)
p{y(j?
0, 1, ••.•
n,1.
:;;
) :;; y},
x, y(j'
n,i+h
We assume that the joint density
(2.3.12)
also exists for every j, j' = 1, ••• , q; h = 0, 1, ••• and is finite.
Lemma 2.3.1.
Under (2.2.6), (2.2.8) and (2.2.10), forc! (0 <:c{ <,c/ ),
2
there exist two positive numbers
(2.3.13)
I
Cz and c3 such that
(")
-c lo}-(c-c ) g &
I
~ cZn 1
~ c n ( l1o g n)
3
n,1.
p { R J.
j = l , ••• ,q,
where R(ji) is the j-th component of R . and c is defined in (2.3.6).
n,
~n,1.
[Throughout this chapter, we shall exclusively deal with 0:0
with a remark that for 0
>
1, the proofs become comparatively simpler
and we can still work with some 0'
Proof.
~
1.]
By (2.2.8), (2.2.9) and (2.3.5),
(2.3.14)
q
L
s=1
< 0 :;; 1,
r P~o (e'*~ nS
.- }
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
30
=
as m
n
~
K log n, e*
0
~
(log n) og e- c log
= e -p*
and c
= op*K.
,.
Hence, by the Markov
inequality,
I
( )
p{ R j .
n,~
I~
c n
-clio
2
< c -0 n c
2
-
I
( ) 0
p{ R j
0
.1 .: . c 2
n,~
n
-c1
}
1 EIR(j) 10
.
n,~
- (c-c1)
=
=
}
0 (n
0
,I
(log n) ~
Q.E.D.
•
Lemma 2.3.2. Under (2.2.6), (2.2.8) and (2.2.10), for
c
ev~ry,
= Kp*0(>8) ,
(2.3.15)
sup{
IF [j]
(x) - Fn[j] (x)
O(n -d ), j = 1, ... ,q,
I (j) ,y£I (j']}
sup
{sup[ IF[. "]h(x,y) - ''CJ'
F [, j']h(x,y) :x£I
h-1
m
J,J
n
J'
n
n
- , ••• , n
(2.3.16)
=
where 0
I: X£I~J)} =
<
d
<
= 1, •.• ,q,
O(n-d) for all j,j'
c/2 and d can be made greater than 1 by proper choice
of K.
Proof.
(2.3.17)
By definition in (2.2.7) and (2.3.5), for all XEI(j)
n
Fn[J'] (x)
=
P{y(j) < x}
n,i -
< p{ X, , < x
~,J -
=
p{X
+ R (j?,
n,~
+
<
-
p{X. .
~,J
<
x
+ c2n
x + R(j)}
<
i,j -
n,i
10
-c
R (j i) < c n
2
n,
P{R(j)
-clIo
>
n,i }
+
c n
2
1
}
-c /0
1 }
( )
P{R j
> c n
n,i 2
-c
1
10
}
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
31
10
-c
=
F[j](X) + O(n
-c
) + O(n
1
1) as c
and as f[j](x) is bounded in In (j).
c/2,
<
l
~
By assumption 0
1, and hence,
(2.3.18)
In a similar manner, it follows that as n
+
(2.3.19)
X E: I
00,
(j)
n
Thus, (2.3.15) readily follows from (2.3.18) and (2.3.19), by letting
Similar to (2.3.17), for all x
I
E:
(j)
n
,y
I
E:
(j ')
n
, we have for
large n,
Fn[j ,j , ]h (x,y)
= P{y(j~
n,~
y (j') ~ y}
n,i
< x,
-
, < x
+ R
=
p{ X,
<
P{Xi,j < x +
1.,J -
+
( ,)
P{R J
(J')
n, i
> c n
n,i -
Xi+h,j' -< Y + R(j:)
n,1.+h }
-clio
2
(")
> c n
R J
'n,i+h -
-clio
2
}
-c
~ y
<
( , ')
P{R J,
+ c 2n
>
n,1.+h -
-clIo
=
F[j,j']h(x+c 2n
C
n
2
1
10
-c 1 10
}
}
-Clio
,y+c 2n
)
-(c-c )
+ O(n
1 (log n)go),
by lemma (2.3.1)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
32
=
F[j,j']h(x,y) + O(n
-c 1
/0
) + O(n
-c 1
), as c
1
c/2
<
and as f[j,j']h(x,y) is bounded for x E In(j) ,y E I(j').
n
Again 0
<
1
implies
~
(2.3.20) F [j "]h(x,y) .::. F[. "]h(x,y)+ O(n
n ,J
J,J
-c 1
u
),T x EI
(j)
n
,yE I
(j')
n
Similarly, it also follows that for large n,
~F
n[j,j']h
(x y) > F
(x y)
0 (n'
- [j,j']h , -
c1
)
Vx
<"'
I (j) y
'
'~n
<"'
~
I (j ')
n
Hence the above result and (2.3.20) imply (2.3.16) where d .::. c .
1
Q.E.D.
Let us now define
= «\I*'01(~
nJJ
£ »)j ,j'
=1, .•. , q '
(2.3.21)
where
-Further, if \I
~
')
n covar[F * .(~ (j)
(,»,F * ,,(~ (j(,,»],
j,j'=l, .•• ,q.
nJ P J
nJ
P J
is defined as in section 2.2, then since F . involves
nJ
bounded values random variables and the X. . satisfy the strong mixing
1,J
or ¢-mixing condition (viz, pp. 166-168 of Billingsley, 1968) it is
easy to show that the conditions of his lemma 3 on page 172 hold, which
ensure the existence of lim
n+ oo
where
~.
Then let us define lim \I
n+ oo
"'tl
=~,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
33
I (j) (j ') \
(j) (j')}
+F[jI,j]h\~p(j)'~p(j,»)-2p p
5'
j,j'=l, ... ,q,
(j) \
(J)
and for j = j', h • 0, F[j,j]O ( ~p(j)'~p(j)J is equal to p
(j )
.
Then, we
have the following.
Lemma 2.3.3.
(2.3.24)
lim
n
where
~
Proof.
and
~
~
Under (2.2.6), (2.2.8) and (2.2.10),
-+
v
lim v* (k
=
"'11
n
00
-+
00
"'11
)
£
is finite and
:t
=
jj ' are given by (2.3.23).
V
being a matrix, to prove its finiteness, we have to show
that v .. , is finite for every j,j' = 1, .•• ,q. We first show that v •.
JJ
JJ
is finite for every j = 1, •.. ,q. Then, this will imply finiteness of
V
jj
' for every j,j' = 1, ... ,q.
We note that
v •.
JJ
-
-
p
(j)
(l-p
(j)
)+2
(j)
(j)
L{F[. ']h~(')'~(')
h=l
J,J
P J
P J
00
(
and its first term is bounded by 1/4.
)
To find a bound for the second
term, we write
(2.3.25)
F
/:(j) \
/:(j)
[j,j]h ( ~p(j)'~p(j»)
00
=
p{ \'
I..
I
r=O s=l
b (r\
js
+ ~
l..
=
pix
< /:(j)
i,j -~p(j)'
~(j)
i-r,s - p(J')'
<
~ b(r)
~
r=h s=l
J'S
E i +h -
h-1
\'
< /:(j)
i+h,j - ~p(j)
I
b(-Q
l..
r=O s=l
<
r s -
'
X
~
(j) }
p(j)
E:
}
J'S Hh-r s
'
I
I
I
I
I
I
I
I
I
I
I
I
34
h-1
'i'
L
+ C
<Xl
q
I
L rg(e*)rl€i+h_r
r-h s=1
'
by (2.2.8) .
Let us write
(2.3.26)
and proceeding as in (2.3.14), we have
<
<Xl
,
so that by the Markov inequality,
By (2.3.25), (2.3.26), and (2.3.27), we get
(2.3.28)
~
(j)
.1j)
\
F(j,j]h~ p(j)'~p(j»)
P
I
I
I
I
I
I
I
q
b(r)€
<
(j)
J's i+h-r s - ~p(j)
r=O s=1
'
'i'
L
~
<Xl
L
r-O
q
b(r)
< (j)
h-1
q
I J's€i-rs-~p(j)'L
I
s=1
'
r=O s=1
b(r)€
< (j)
js i+h-rs-~p(j)
'
+
<
-
P
h-1 q
b(r)€
<
(j)
b(r)€
<
L
L
js i-r s - ~ (j)' L L js i+h-r s r=O s=1
'p
r=O s=1
'
~
<Xl
q
C (e*)h
(j)
~p(j)
u)
sl}
,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
35
~
<
q
{I I b(r)E
js i-r,s -
P
<
p(j)
r=O s= 1
<
-
h-l
~(j)
}p{
q
I0 I1b(r)E
js i+h-r,s -
<
r=
~(j) + C(e*)h/2}
p(j)
s=
p{X..
~(j) }p{X
~(j). + 2 C(e*)h/2} + 2P{U
~,J ~ ~p(j)
i+h,j ~ ~p(J)
>
(e*)-h/2}
and using the Taylor theorem, we have from (2.3.28),
Proceeding'in the similar way we can easily show
(,(j)
(2.3.30)
Since
(j)\
2
(j»)
*oh/2
F[j,j]h~~p(j)'~p(j») ~ F[j]~p(j) - O(e )
F[j](~~~~~
= p(j), from (2.3.29) and (2.3.30), we get
(2.3.31)
and by (2.3.31), the second term of V
is also bounded.
jj
By use of the Schwarz inequality and the finiteness of
get
Iv .• ,I
JJ
<
~,
V j ,j'
= 1, .... ,q.
V
jj
' we
This completes the proof of
finiteness of v.
'"
Since
lim ~
n-+""n
to show that
V •• ,
JJ
n
=
lim
~*(~
), for proof of (2.3.24), only we have
n-+~n~
lim V~j"(~)
-+ ~ n J ~
=
v .• "for every j,j' = l, ... ,q.
JJ
being finite, we see that, we must have
(2.3.32)
~ I
/, (j) (j ') \
(j) (j') I
h~l F[j,j']h~~p(j)'~p(j'~- P P
<
~
First
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
36
so that
00
(2.3.33 )
'\ I
;; (j)
(j
'»)
mL+l F[j,j']hl~p(j)'~p(j') - P
(j) (j')
p
I -to
0 as n
-to
00
.
n
Next using the definition (2.3.8), we obtain for every j,j'
(2.3.34)
[*
E Fnj
t (j)
)]
t~p(j)
....
f.
(j)
= Fn[j] \~p(j)
)
=
1, .•. ,q,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
37
so that we can write
(2.3.36)
where
Since
I""
/c{j)
(j/») ""
({j»)""
(;;(j')\1
Fn[j,j']h ep(j)'Sp(j) - Fn[j] ~p(j) Fn[j'] ~Sp(j'~ ~ 1,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
38
m
n
(2.3.37)
IJ 3 1~ I
h/n ~ (K log n)(K log n+l)/2n ~ 0 as n ~ ~
h=l
and similarly
IJsl
(2.3.38)
lim J 1
~ 0 as n ~~.
=
n ~ ~
By (2.3.1S) and (2.3.16), we have
~
("») (') (")
(j)
F [' "] t; (')' t; J(.,) - P J P J
J,J
P J
P J
Also by use of (2.3.33), we get
~
0 as n
~
~,
since by (2.3.1S) and (2.3.16),
= O(n -d log n)
~
0 as n
~ ~,
•
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
39
= O(n
-d
~
log n)
0 as n
~ ~.
Similarly
(2.3.42)
IJ 4
""~
-
L
h=l
{F[., 'Jh
J d
0~ (j)
> (j ') )
- p
(')'" (j')
P J
P
(j) (j')
p
}I
-.- 0 as n -.. "".
The proof of (2.3.24) follows readily from (2.3.36) - (2.3.42).
Lemma 2.3.4.
Let {Z.} be a sequence of m -dependent binomial random
--
n
1.
variables, where EZ
i
= p, i
n~ = [(n + m + 1 - j)/(m
n
J
(2.3.43)
=
c*(n*)
1
1, and
mn~
K log
n~
n
~
"". Let then
+ 1)], j = 1, ... ,m , n* = min
n
j=l, ...
n
Yn
~
-1/2
{(log n*)p(l _ p)}1/2, c *2
l
>
,rnu
(n~), and
J
2.
Then
""\'
(2.3.44)
L.
p{ln -1
i=l
Proof.
n
I
Z
' 1 i
1=
- pi >y n } <
00
Consider the partial sums
=1,2, .. o,m+l.
n
J ,n = Zj + Zj+(m +1) + .,. + Zj+(n~_l) (m +1)' j
n
J
n
S.
Then the lemma follows in the similar way by putting p for p~
J ,n
Lemma 2.1 of Sen (1968b).
The
in
only difference is that instead of
(2.22) of Sen (1968b), we get in this case
(2.3.45)
As m
n
~
n~
J
y 2/2p(1 - p) > (1
n
K log n for large n, n*
large values of n, we have
(2,3.46)
P{n -I
+ 8)10g n*;8
~
>
0, (m + l)n* ~ n.
n
n/K log n so that for adequately
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
40
where c 4 is a finite positive quentity.
large n,
Similarly, for adequately
(2.3.47)
For 0 < 00
we can write
< 0,
c (1 og n ) 1+0/ n 1+0
c [(log n) 1+0 In
4
=
4
°0 lin1+
°)
(0 - 0
= c 4un /n 1+y
where
Now un
y=
~
°-
00 > 0 and un = (log n)l+o /n oO
0 as n
such that un
~ ~
~ E
so that for every fixed
~
for n
nO'
0, there exists a nO
Hence for adequately large n,
c 4 (1 og n ) 1+0/n1+0
(2.3.48)
E >
<
(2.3.46), (2.3.47) and (2.3.48) along with the fact
~
n
-(1+y)
<
~,
n>l
imply (2.3.44).
Lemma 2.3.5.
Let ~nj
>
as n
~
= np(j) +
0, X(j)
n, ].JIll
E
0(n 1 / 2 log n), 0 < p(j) < 1. Then,
I(j), j • 1, ... ,q, with probability one,
n
~.
Using Lemma 2.3.4, the proof follows along the same line as in
Lemma 2.4 of Sen (1968b), and hence, is omitted.
Proof of theorem 2.3.1.
take d > 1.
n
~~,
Thus, to prove (2.3.3), it suffices to prove that as
for every j
(2.3.49)
In Lemma 2.3.2, by proper choice ofK, we
= 1, ••• ,q,
sup{ I[F~j (x) - F: j
•
(~~1~»]
0 (n- 3 / 4 log n),
-
[Fn[j l (x) - Fn[j l (s~1~ ))l I :xdn (j)}
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
41
with probability one, and
(2.3.50)
with probability one.
To prove (2.3.49), we consider a sequence {b } of positive integers
n
such that b
~ n 1 / 4 as n ~~.
n
Then the proof follows by the same
technique as in Lemma 1 of Bahdur (1966), provided his (13) also holds
for the m -dependent case where m
n
n
~
K log n for large n.
By his
G (n ,w) (see (5) and (13) of Bahadur) is equal to
definition,
n
r
'"
(j)
- [Fn[j]!;p(j) + rn
-3/4
1
.\
'"
'(j)\
log nj - Fn[j ~p(j1]J
~
where r (
b ) is a positive integer. The above can also be expressed
n
1
as n~ (2 (j) - p (j», where 2 (j) 's are m-dependent binomial valiab1es
i~l
and p
(j)
i
i
'"
(j)
-3/4
f[j]
(~p~1~} ~
• Fn[j] !;p(j) +rn
follows from 0
<
o < p(j)~cln-1/2
n
L(j»)
)
~
log n - Fn[j] ~p(j)'
It therefore
that for n sufficiently large,
log n for all r ~ bn'
3 4
select Yn - c! n- / log n, with
Hence by (2.3.44) we can
c!2/2c~ sufficiently large, and this
along with (2.3.45) and the Bonferroni inequality gives us
(2.3.51)
where
(2.3.52)
1
2 1 2
vn • - 7; log n + ci n / (10g n)2/2[~n1/2 log n +
-1/4 + c*2/2c*
1
2[1
• log n
+ 0 (1)]
<1n1/410gn]
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
42
If -1/4 + C~2/2c*2 is chosen greater than 1, then use of (9) of
Bahadur (1966) completes the proof of (2.3.49).
To prove (2.3.50), we consider the same sequence {b } as defined above.
n
Let us then write n(j) - .;(J('j»+ ra /b (where a is defined by (2.3.2»,
r,n
p
n n
n
(')
(')
(')
J 1 ),
for r - 0,+
J be the interval (n r,n
J ,nr+
- 1, ... ,+
- bn , and let I r,n
,n
Since both Fnj and F~j are non-decreasing, we have
r - -b , .•. ,b -1.
n
n
for all x e: I (j)
r,n'
F (n(j»- F* (n(j) ) < F ,(x) - F*,(x) <F (n(j) )-F*,(n(j»
nj r+1,n
nJ r,n
nj r,n
nj r+1,n - nJ
nJ
and consequently, we get
(2.3,53)
IFnj(X) - F* (x)l<
max IF (n(j»
nj
- s-r, r+1 nj s,n
- F* (n(j»1
nj s,n
+ IF* (n (j) ) - F* (n(j»1
nj r+1,n
nj r,n
This implies
(2.3.54)
sup
I Fnj (x) - F~j (x) I
xe:I (j)
n
+
max
IF* (n (j) ) - F* (n (j ) ) I
-b <r<b -1 nj r+1,n
nj r,n
n-- n
Let us now
~enote
by Gn[j] (x) -
(2.3.55) ElF ,(n(j»
nJ r,n
P{IR~:il 2
x},
°< x
<~.
- F*.(n(j»1
nJ r,n
I
< n- 1
Elc(n(j) - y(j)
i-1
r,n
n,i
<
y (j) < n (j)
n,i -
r,n
or n(j) < y(j) < n(j) + x}dG
(x)
r,n - n,i - r,n
n[j]
Then,
I
I
I
I
I·
I
I
I
I
I
I
I
I
I
I
I
I
I
I
43
I n,i I ->
+
-c/o
where 0 <
O(n
° < 1.
p{ R (j )
-c
) + O(n
1
c n
2
-c/o
1
}
), by Lemma 2.3.1,
We now choose a 0': 1/3 < 0' < 3/4 and K such that
c - 0P*K > 8, so that in Lemma
2.3.1 we may take c > 4. This
1
-c o'T 1/4
1
implies c 0' > 4/3 ~ ~ n
<
Also, c (1 - 0') > 1. Thus,
1
1
n>l
by using the Chebychev inequality and (2.3.55), we obtain that as
=.
n -+-
Qi),
-c 0'
(2.3.56)
for all r • 0,
n -+-
1
n
± 1, ••• ,± bn .
Hence, by the Bonferroni inequality, as
=,
(2.3.57)
P{ max IF (n(j»
Irl<b
nj r,n
n
°'
- F*,(n(j»1 > Kl - o' n- l }.::. K ;l-e:,e: >0.
2
nJ r,n
- 2
The last equation implies that as n -+- =,
(2.3.58)
with probability one.
Now, for every j (- 1, ..• ,q),
process.
(2.3.59)
{y~~l,
i=1,2, •.. } forms an mn-dependent
Hence by (2.3.49), it follows that as n -+- =,
max
IF* (n(j) ) - F* (n(j»1 • o (n- 3 / 4 log n),
nj r+l n
nj r,n
-b n<r<b
-1
'
-- n
with probability one.
and (2.3.59).
Then (2,3.50) follows from (2.3.54), (2.3.58)
This completes the proof of (2.3.3).
Finally, if f[j](X) is bounded in some neighbourhood of
~;1~),
(2.3.4) follows directly from (2.3.3), as, under this condition,
Lemma 3 of Bahadur (1966) extends directly to our case.
This
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
44
completes the proof when Z (j) is defined by a single order statistic.
n
If Z(j) is defined as an average of two successive order statistics,
n
say, the
~ n -th
and
(~ n+l)-th
ones, (where
~ n-<
np(j)
(~+1»'
n
<
-
upon
noting that the difference of empirical c.d.f's at these two points
is equal to lin, it follows from (2.3.3) and Lemma 2.3.5 that the
difference between the values of F[j](x) at these two points is also
o(n- 3 / 4 log n), with probability one, as n
~
=.
Consequently (2.3.4)
holds for the general case when z(j) is any inner point of two sucn
Hence the proof of Theorem 2.3.1 is
cessive order statistics.
completed.
2.4.
Asymptotic
Join~
Normality of
1n
1£ we let
(2.4.1)
T ••
JJ
I
-
j,jl- l, ... ,q,
where \)jjl and f[j](E;;1}») are defined in section 2.2, then we have
the following.
Theorem 2.4.1.
If
F(~
is absolutely continuous at
~
.....
with finite,
positive and continuous marginal densities in some neighbourhood of
~,
and if
~
is positive definite, then under (2.2.6)- (2.2.10),
'"
(2.4.2)
Proof. By virtue of (2.3.3), Lemma
as n
~
=,
2.3.5,
(2.3.49) and (2.3.50),
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
45
[n l / 2{F*.
(~(j)
nJ '\ p (j)
) - p(j)} , j
l, ... ,q] have the same limiting distri-
III
bution, if they have one at all.
Now, by (2.3.8), F*. involves an
nJ
average over zero-one valued m -dependent random variables, on which
n
a direct multivariate extension of the Hoeffding-Robbins (1948)
C.L.T. for m-dependent processes (with a straightforward extension
for an m -dependent process with m
n
n
~K
log n) or a multivariate
extension of the C.L.T. for strongly mixing processes by Rosenblatt
(l956b)yields that
[nl/2{F~j(~~{~»)_ p(j)},
j ... l, ... ,q] has
asymptotically, a multinormal distribution with mean
matrix ~(~), defined by (2.3.21) and (2.3.22).
Q and
dispersion
The rest of the proof
follows trivally by a use of Lemma 2.3.3 along with (2.4.3) and
(2.4.1) •
Q.E.D.
Note:
In the above theorem if the condition that
definite is not satisfied and
f«
q) then we choose f
~,
~
is positive
is positive semidefinite with rank
linearly independent variables and prove
asymptotic normality for this subset.
Since the other variables will
be linearly dependent on them, in this case, the asymptotic distribution for the whole set of variables will be a singular normal distribution with rank f.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
CHAPTER III
ASYHPTOTIC DISTRIBUTION OF LINEAR COMPOUND OF
SOME FIXED NUMBER OF QUANTILES FOR STATIONARY
MULTIVARIATE AUTOREGRESSIVE PROCESSES
3.1
Introduction
In the first part of the chapter, for stationary autoregressive
processes Bahadur's asymptotic a.s. representation of the standardized
form of a sample quantile is extended over the entire real line.
With
this extension, the asymptotic multinorma1ity, derived in Chapter II
of a sample quantile (vector) for a stationary autoregressive process,
is generalized in section 3.3 for a vector of linear combinations of
several quanti1es and an application of this result is considered in
the next section.
The section 3.5 contains a method of estimation of
variances and covariances of sample quanti1es and hence the variances
and covariances of the linear compounds of several quanti1es.
The
chapter is concluded with an extension of the above asymptotic mu1tinormality of the vector of linear compounds of several quanti1es for
random sample sizes.
3.2.
Extension of Bahadur-representation of Sample Quanti1es over
the &lUre Real Line.
For a fixed j, if p(j) is varied in (2.2.13) over the open
interval
~,1)
then we get a quantile process (see Kiefer 1967, 1970).
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
47
In this section the result (2.3.3) (extension of Bahadur-representation
of sample quantiles to the autoregressive case) is first extended for a
quantile process.
Then asymptotic normality for linear function of
several quantiles follows immediately as a corollary of this result.
Let us suppose that F[j](X) is absolutely continuous and
sup f[.](x)
x
J
=f
.
OJ
<
00.
Similar to (2.3.15), we have
~
Fn[j](X) ~ F[j](X) + O(n
-d
)
and
uniformly in x, where we take d
>
1.
Hence it follows that
=
(3.2.1)
o(n
-d ),
d > 1.
~
Then absolute continuity of Fn[j](x) is obtained by (3.2.1) and the
absolute continuity of F[j](X)'
.....
Hence the random variables
z~:i =
( .)
F [.](y J ), i • l, ... ,n are distributed uniformly over (0,1).
n J
n, i
the empirical process [V*.(t):
nJ
(3.2.2)
°< t
< 1] by
--
1 2
V*.(t) - n / [G*.(t) -t];
nJ
nJ
G*.(t)
nJ
= n- 1
°
Let g(n) • n- 1 / 2 log n.
(3.2.3)
(3.2.4)
Define
< t
< 1 •
Define
K .(t) = sup{lv:j(t) - V~j(t + a)!: lal ~ g(n)}, 0 < t < 1;
nJ
a
K* = sup {K .(t)} •
nj
nJ
O<t<l
Then we have the following theorem.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
48
Theorem 3.2.1.
For every j
s~p lal~~~n)
(3.2.5)
= 1, •.• ,q,
one~
+ ~,
1 2
n / !Fnj (X + a) - Fnj(X) - F[j](X + a) + F[j](x)1
O(n- 1 / 4 log n),
=
with probability
as n
provided (2 02 06). (20208) and (202.10) hold.
The proof of the above theorem depends on the following lemmas.
Lemma 3.2.1.
For every finite s(> 0) and j (= 1, ... ,q), there exist
a positive constant C (1) and a sample size n j' such that for
s
s,
n -> n s, j'
P{K~j ~ Cs (1) n- 1 / 4 log n} ~ 4n- s
(3.2.6)
Hence as n
+~, K*. = O(n- 1 / 4 log n), with probability one.
nJ
Let ~n
= t[n 1 / 2 ]-1
N,n
Proof.
(0 ~ t ~[n1/2]), [x] being the largest
integer less than or equal to x.
function being equal to 1 (0
~
t
For uniform distribution the density
~
1), by the same technique as in
(2.3.51) and using (9) of Bahadur (1966) we get for large n,
P{K (!;;i ,n)
nj
(3.2.7)
>
c
* -1/4
n
log n}
1
~
4e
-vn
,i
= 1,2, ••• [n1/2 ]
where c * > 0
1
and
2
* -1/4-1
v . - 1/4 log n + ct /2 log n (1 + c1 n
).
n
The rest of the proof of (3.2.6) is exactly the same as in Theorem 4.1
of Sen and Ghosh (1971).
The second part of the lemma follows by taking s > 1 in
(3.2.6).
Note:
Actually the above lemma is the extension of Theorem 4.1
(extension of Bahadur's (1966) result to the entire real line) of Sen
and Ghosh (1971) to the mn -dependent process where mn
for large n.
~
K log n
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
49
n
Lemma 3.2.2.
1
c(t -U j).
i - 1,2, •••• n, G .(t)-u
i '
nJ
i=l
1, ••• ,g, as n + 00.
Let Ui,j
Then for every j
=
l:
= F(Xi,j)'
1 2
sup n / 1G jet) - Gn*j(t)I
O<t<l
n
(3.2.8)
= O(n- 1 / 4
log n),
with probability one.
Proof.
As in Lemma 3.2.1, let us take the grid points ~n
°
1 2
[n / ].
x-,n
< Q, <
for t
E:
Now Gnj(t) and
G~j(t)
= Q,[n 1 / 2 ]-1,
being non-decreasing in t,
[~Q, ,n'~Q,+l.n]' we have
so that
2
n 1 / [Gnj
(~ Q"n ) - G*nj (~ Q,+l,n )]
_<
n 1 / 2 [G (t) - G*nj ( t )]
nj
< n 1 / 2 [G (~
) -G* (~
)]
nj ~Q,+l,n
nj ~Q"n
This implies
(3.2.9)
<
+
R,=
By Lemma 3.2.1, as n
with probability one.
as n
+
00,
+
max
/2
O,1, ... [n 1 ]
n1/2IG~j(~Q,+1,n) - G~j(~Q"n)
00, the second term is of order n- 1 / 4 log n,
Hence for (3.2.9) it is sufficient to prove that
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
50
1 2
max
n / 1G .(~
) n=0 " 1 • • • , [ n 1/2 ]
nJ.Q, , n
N
(3.2.10)
G*'(~n
nJ
N
,
n
)1
= O(n- 1 / 4 ),
with probability one.
Similar to (2.3.55), here we get
1
E[ n 21 GnJ. (~nN,n ) - G*j
n
(3.2.11)
(~nN,n ) I]
We now choose a °1: 3 / 7 < 01 < 13/14 and as before K such that c >
1
-{(c - 1)01- I}
1
1
n
2
2 < and also
This implies (c 1 - 2)01
> 3/2 ~
n>l
2
1
(c 1 - 2)(1 - 01)
>
1
4'
(3.2.11), we get as n
as n
~
00
Thus, by using the Chebychev inequality and
~
00,
p~}1 GnJ«~ ,n) - G~J«~ ,n) Ill') - C1Y-'1}~ (K3J- c1)'I
(3.2.12)
for all
4.
Q,
= O,l, ... ,[n 1/2 ].
Hence, by the Bonferroni inequality,
00,
(3.2.13)
(3.2.10) is implied by the Bore1-Cante11i Lemma and (3.2.13). Q.E.D.
Lemma 3.2.3.
~
n
~
With the notations used above, for every j = 1, ... ,q,
00,
(3.2.14)
sup
t
sup n1/2 1G j(t + a) lal<g(n)
n
with probability one.
G~j(t
+ a) I = O(n-1/4 log n),
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
51
Proof.
We have
(3.2.15)
1 2
sup
n / 1G .(t + a) - G*.(t + a)!
lal<g(n)
nJ
nJ
sup
O<t<l
< sup
- O<t<l
1 2
sup
n / 1G j(t + a) - Gnj(t)I
lal<g(n)
n
+ sup
O<t<l
+ sup
O<t<l
l 2
sup
n / /G*.(t + a) - G*.(t)
la!<g(n)
nJ
nJ
I
By Lemma 3.2.2 and Lemma 3.2.1 it follows that the second term and
the third term of (3.2.15) are of order n- 1 / 4 log n, with probability
one, as n
Hence for proof of Lemma 3.2.3, it is sufficient to
~~.
show that, as n
(3 . 2 . 16)
sup
O<t<l
~
00,
2
sup
n 1 / JG .(t + a) - G .(t)!=O(n- 1 / 4 log n),
!aJ<g(n)
nJ
nJ
with probability one.
I f we denote by
H • (t)
=
H*.
nJ
=
nJ
1 2
sup
n / 1G .(t + a) - G .(t)1
lal<g(n)
nJ
nJ
and by
sup H • (t)
O<t<l nJ
and consider the grid points
~~,n = ~[n1/2]-1, ~ = 1, •.. [n 1 / 2 ], then
as in (4.8) of Sen and Ghosh (1971), we have
(3.2.17)
H*j
n
<
-
3[
max
/2 H '(~n )]
nJ N ,n
l.::.~.::.[nl]
Hence for (3.2.16), it is enough to show that, as n
~
00,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
52
max
H
1.:sB- .:sJ n 1/2] nj
(3.2.18)
(~
= D(n -1/4
)
Q.
,n
log n),
with probability one.
Again
(3.2.19)
=
max /
l~ . ::.rn 1 2 ]
<
max /
- l.::.t.:sJn 1 2 ]
I
sup n 1/21 G. (~n
+ a) - G . (~n
)
lal<g(n)
nJ N,n
nJ N,n
1 2
sup n / , G j
1 al <g(n)
n
(~n
N
,n
+ a) - G*.
nJ
(~n
+ a)
"- ,n
1
*
max /
sup n 1/21 G '(~n
+a) - G*'(~n
)1
l<.Q,.::.[n 1 2 ] lal<g(n)
nJ "-,n
nJ "-,n
+
max /
+
1 ]
2
1.:sB- .::.[ n
n 1/21 G. (~n
) - G*. (~n n) 1
nJ
n
nJ N'
To,
,
By Lemma 3.2.1 and Lemma 3.2.2, the second term and the third term
in (3.2.19) are of order n- 1 / 4 log n, with probability one, as n
~
00
and by the same technique as in (2.3.54) and (2.3.59), we obtain, as
n
~
co,
(3.2.20)
max 1/2
l~.Q,.:Jn
<
sup
n
] lal<g(n)
1/2
max 1/2
max n
l<.Q,<[n
] Irl<b
1
G. (~n
+ a) - G*. (~
+ a)
nJ To"n
nJ
,n
1/21 G j (~Q.
n
-n
-1/4
+ D(n
log n),
I
(~n ,n+ ran /b)h I
+ ran /b n ) - G*j
n
,n
To,
with probability one.
Therefore for proof of (3.2.18), it is only required to show that,
as n
~
00,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
53
1 2
max 1 2
max n / 1G j(~n + ra /b ) 1<R. .:~J n / ] 1r l2.bn
n
N , n
n n
(3.2.21)
G*j(~n
n
N
,
+ ra /b )1
n
n n
with probability one.
We choose a 02: 1/2 < 02 < 13/14 and as before K such that c > 4.
1
-{(c - 1/2)02 - 3/4}
1
This gives (c - 1/2)02 >7/4 ~ 1: n
< 00 and also
1
n>l
Hence, similar
tJ
(3.2.12), we have by the Chebychev inequality
1/2
1/2-c 1 1-0~
(3.2.22) P nl.G j(i;n + ra /b ) -G*'(~n + ra /b )1> (K n
)
n
lv , n
n n
nJ lv , n
n n 4
f
< (K n
4
for a11.Q.= 1, ... ,[n
inequality, as n
(3.2.23) P
+
1/2
]; r = O,+l,
•.. ,+b.
- n
1/2 - c 1 02
)
Hence, by the Bonferroni
00,
l
max 1/2
max n 1/2, G j(~n + ra / b )-G *j(~n + ra /b ) I
12..Q...::Jn
] Ir l2.-b n
n
lv , n
n n
n
lv , n
n n
>
and by use of the Bore1-Cante11i lemma, (3.2.23) implies (3.2.21).
Proof of theorem 3.2.1.
-
00
< x <
(3.2.24)
00.
Now since
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
/1
I
54
< sup sup
n 1/21 [F *j (x+a) - ~F ['] (x+a)] - [F* . (x) - ~F ['] (x)]
- x la!<g(n)
n
n J
nJ
n J
+ sup
x
I
l 2
sup n / 1[F ,(x+a)-F[,](x+a)] -[F*,(x+a)
lal<g(n)
nJ
J
nJ
- Fn[j] (x+a)]
I,
by use of Lemma 3.2.1, Lemma 3.2.2 and Lemma 3.2.3, the theorem
3.2.1 easily follows.
3.3 Asymptotic Normality of a Linear Compound of Several Quantiles
Let us suppose that Z (j) (j) denotes the sample quantile of
,
n,Pi
J
be the corresponding
pi ) for the j-th variate and ~(j)
order
population quantile, j = 1, ... ,q.
p (j)
i
We want to determine the asymp-
to tic distribution of the vector of linear compounds
(1/2
~
I
i=l
a
(1)
(1)
Pi
(., (1)
~
. (1) -
n,Pi
~
(1) '\
(l»)"
.. ,
n
1/2
Pi
~
l
I
a
(q)
(q)
(q) Z
i=l Pi
n,Pi
(q) -~.
(q)
~
(qY
Pi
'-I , .. , t', J'-I
were
- •... q. are kno wn c ons tants
th e a (j)( , ) 's ,1.-,
h
Pi J
Let us assume that
f[,](~(j), \>0.
J \' Pi (J»)
j=l, ... ,q, and let us denote
by
(3.3.1)
I
(j)
n.Pi
(j)
where as before an
a
< x <
n -
'V
n -1/2 log n as n
en
~ (j )( .)+
J
a },
n
j = l , ... ,q.
Pi
i
get Z (j} (j} e: I(j}
n,Pi
n,Pi
as n
= {x: ,(j)
p (j)
-+
00.
By use oj; Lemma 2 .3. .5, we
for i:::;l, •• "t; j:::;l, ••• ,<t, witlL probability one,
00
and under the condition that
r]
sup sup f
(x) is finite,
l~J.~.' t x
J
Lemma 3 of Bahadur (1966) extends directly to our case.
-+-
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
55
Similar to (2.3.4), in this case, by (3.2.5) and Lemma 3 of
Bahadur (1966) we get simultaneously for i = 1, •.. ,t, j = 1, ... q, as
n -+
00,
(3.3.2)
=
with probability one.
an n
>
O(n
-1/4
This implies that for every
0 such that, as n
(3 • 3 • 3) P{n 1/2! [ZO) (J')
n,Pi
-+
00,
-
<,
log n),
E
> 0 there exists
~(j)(j ) ]f [j] (~(j»)
+ [Fnj \f:o
I~(j)(j ~\ - Pi(j)]
<,
(j )
Pi
Pi
Pi
I~
for i=l, ... t, j=l, ..• ,q} .=:.l-n
From Lemma 3.2.2, it follows that
1 2
sup n / jF .(x) - F*j(x)
l.::J~q x
nJ
n
(3.3.4)
sup
I -+
0 as n -+ 00,
P
and this implies
.l(n1/2[Fnj(~(j~j»)-p?)], i
(3.3.5)
= 1, ... ,t, j= 1, ... ,q)
Pi
~ ( n 1 / 2 [ Fnj\~
* II
'U~
0»)
(j)
.
- Pi ( j ) ]
' .1=1, ... ,t, J=l,
... ,q ) .
Pi
Now, by (2.3.8), F*. involves an average over zero-one valued
nJ
m -dependent random variables, on which a direct multivariate extension
n
of the aoeffding-Robbins (1948) central limit theorem for m-dependent
processes (with a straightforward extension for an m -dependent process
n
with m
n
'U
K log n) or a multivariate extension of the central limit
theorem for strongly mixing processes by Rosenblatt (l956b) yields that
or' n 1/2{F*
(r:(j»)
nj\("
(j)
i lca
ly a
- Pi (j)}.
,1= 1 , ... ,t j =1 , ... ,q] h
as asymptot
Pi
mu1tinorma1 distribution with mean
Q and
dispersion matrix
e:
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
56
matrix
Ilv * (jj '~\
(3.3.6)
J) i,i'=l, ••. ,t
U'nii'
j,j'=l,oo.,q
where
v* (jj , )
(3.3.7)
nii'
Hence by (3.3.5), Lemma 2.3.3
[nl/2{Fnj~(j~j))- Pi(j)}
and above, it follows that
, i=l,."t;j=l, .•• ,q] has asymptotically a
Pi
mu1tinorma1 distribution with mean
Q and
dispersion matrix
(Iv (jj '),\
~\ii' )}i,i'=l, ... ,t
(3.3.8)
j,j'=l,oo.,q
where
,,(jj')
vii'
(3.3.9)
=
lim v* (jj , )
n+
OO
nii'
V
i,i'=l, ••. ,t;j,j'=l, .•• ,q
and its exact expression is given by (2.3.23) with p(j) and p(j')
replaced by Pi (j) and Pi' (j') respectively.
((v ii~' )))
We assume the matrix
to be positive definite.
Finally, the asymptotic distribution of [n 1 / 2 {F .e(j).'·\-p (j)}
nJ
(J ~
i
'
Pi
i=l,oo.,t, j=l,oo.,q] being multivariate normal, [-n1/2{Fnj0~~~jy-Pi(j)},
~
~~ (j)(j)j\
(j )
_
._
i=l,oo.,t, j=l,oo.,q] and [n
{F
Pi
},i-1,oo.,t,J-1,oo.,q]
nj
p.
have the same asymptotic distribution.~
1/2
Therefore, by (3.3.3) and the above arguments, it follows that
as n
+00
(3.3.10)
,
,I
cL / n l / 2 [Z(j)
n
,Pi
() j
~(j)(J')]'
i=1, .• ,t;j=1, ..• ,q)+N
,.J
Pi
where
(3.3.11)
~
=
(O,H)
t , q '"
((niif)))i, i' =1, •.. ,
t
j,j'=l,oo.,q
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
57
and
(j j ') =
nii'
(3.3.12)
(j j ,
\i ii'
Hence for any vector
1
Ii
f [j ] ,,~
2
(j»)
11 (j , »
)
,(j) f [j , ]
(j ') , i, i -1, ... , t
Pi
Pi'
. . '-1
J,J
- , .•• ,q.
0
,
= (d , ••• ,d q ) , the asymptotic distribution
1
of the linear combination
n
1/2
is univariate normal with mean zero and variance
(3.3.13)
and this implies that the asymptotic distribution of the vector
(
n
1/2
fL
~
(1) (z(l)
_!" (1)
\
• 1/2
(q)
(1)
(1) '0 (1»)" .• ,n
L a
()
i-1 Pi
n,P i
Pi
i-1 Pi q
a
is multivariate normal with mean vector
(3.3.14)
where
(3.3.15)
3.4.
T*
~
~
(z
(q)
_!" (q)
( ) '0
n,Pi q
Pi(q)
)~
and dispersion matrix
tT~.,)\
=
~
=
JJ
V j,j'-l, .•. ,q
I f
a(j) aU')
n(jj'), j,j'=l, ... ,q.
i=l i =1 P (j) P (j') ii'
i
it
A Useful Application of the Results of Section 3.3.
Mosteller (1946) considered the case of more than one quanti1es
with the idea of studying the properties of estimates based on a subset
of quanti1es.
He termed such estimates as inefficient statistics.
In
the book by Sarhan and Greenberg (1962) for small sample sizes and
various known parent distributions optimal or best linear ordered
estimators of location and scale parameters are considered.
These
estimators demand the complete knowledge of the covariance matrix of the
sample order statistics up to a unknown multiplicative constant.
In
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
58
large samples it becomes extremely laborious to compute the above covariance matrix.
There are two different approaches to the large sample
problem. (i) To consider a subset of fixed number of quanti1es and to
base our estimate as a linear function of the corresponding sample
quanti1es.
These include the so-called mid-ranges which are often more
efficient than the sample median. (ii)
To consider a subset of quanti1es
of size depending on the sample size and to base our estimate as a linear
function of the corresponding sample quanti1es.
These include the so-
called Trimmed mean and Winsorized mean which sometimes provide useful
estimates.
Often, for symmetrical density functions, we employ the sample
median as an estimate of the location parameter.
It is known that for
the class of regular density functions, we can always find more efficient
estimates from the class of mid-ranges, provided that the parametric form
of the distribution is known.
ally
distribute~~it
1961).
If the variables are independently norm-
is known that the 27% mid-range is optimum (See Sen.
In our case, for each j = l, .•. ,q, the 27% mid-range estimator
/zU) . + ZU) . \/2. Here
\n p (J)
n p (J»)
, 1
' 2
_
(1)
(q) I
U - (U
, .•. ,U
) being a vector of linear combination of two
for the location parameter is
"'l1
n
quantiles,
UU)
n
=
n
IOu
--n
is asymptotically multinormally distributed with mean
£1/2 (population median vector) and dispersion matrix
given by (3.3.14) and (3.3.15) with t = 2, P1
(j)
a (j)
a (.) = p (j) = 1/2 for j ,j , =-l, .•. ,q.
P1 J
2
Note:
(j)
!*,
where
!*
is
=.27, P2 (j) =.73 and
By the theory, developed in 3.3 for linear combination of fixed
number of sample quanti1es, the asymptotic distribution of Trimmed mean
and Winsorized mean cannot be obtained for stationary autoregressive
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
59
processes.
hold.
For such statistics our strong convergence results do not
However, the distribution of such statistics for stationary auto-
regressive processes may be derived by some weak convergence results.
Such studies are proposed for future work.
3.5.
Consistent Estimate of the Dispersion Hatrix of the Vector of
Linear Compounds of Several Quanti1eso
The following theorems enable us to estimate the variances and
covariances of sample quanti1es and hence the dispersion matrix of the
vector of their linear compounds.
Theorem 3.5.1. Let ~(j)= np(j) + o(n 1 / 2 log n), and let s. be the
---
n
J
smallest observation among (Xl ., .•.• X j)' If
,J
n,
m
(3.5.1)
"
(.)
(")
v .. , = {F .. , (s .• s . , ) -p J p J
JJ
nJJ
J J
} +
n
(.) ( " )
Ih=l
{F .. , h (s . ,s . , ) -p J P J
}
nJJ
J J
where
(3.5,2)
(3.5.3)
(3.5.4)
F .. ,(u,v)
nJJ
=
Fnjj'h(u,v)
=
c(u.v)
=
n
-1
n
\'
L
c(u - Xi ., v-X . . ,)
,J
1..J
i=l
n
-1
n
I
i=l
t
.,)
c(u - Xi " v - X
Hh ,J
,J
for u .:. 0, v
otherwise
then
(3.5.5)
p
Remark~
In practice, of course, we take
~(j)
n
> 0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
60
Proof .. By lemma 2.3.5, as n
lity one.
+
00,
Sj
£
I~j) and Sj'
£
I(j') with probabi-
Clearly~
(j ') )
v .. , - v .. , = F .. ,(s.,s.,) - F[j "] (:t,; (j)
(.),t,; (")
JJ
JJ
nJ J
J J
'J
P J
P J
(3.5.6)
m
+
~(')
(")~
I {Fn'j'h(s.,soI)-F[.
"] t,; 1')'t,; ~oI)}
h=l J
J J
J,J
P J
P J
n
(j) (j')J
~ (j) (j , )
¥[J' "]h t,; (j)'t,; (JoI)) - p
p
h=m +!
,J
P
P
\'
co'
-
L
n
m
~F ., 'h(s.s.,)
- F[" ,](Sj's,,)}
h=l ~ nJ J
J J
J ,J
J
+ In
- h=mI +ll.(F [j
00
n
By
(2,3.33)~
as n
+
I
j ]h
~ (j)' t,; p (j' ~~ l'p
(j)
(j , )
p
3
(j) (j')
p
the third term and the fifth term in (3.5.6) tend to zero
Also the second term and the fourth term in (3.5.6) are of
00.
similar nature.
Hence to prove (3.5.5), it is sufficient to show that
m
n
J}I
(j , )
sup
I L {F ." (s , ,s . , ) - F [j '1] h ~ (j)
(j)' t,; (j')
(j) h=O nJJ
J J
,J
P
CP
s.£I
005.7)
J
+
pO,
n
Sj,dn(j')
as n
+
Actually the left hand side of (3.5.7) is the combination
00.
of the first and the second term in (3.5.6).
Again to prove (3.5.7) we see that
m
n
(3.5.8)
sup
II {F "'h(s"soI)
(j) h=O nJ J
J J
s,£I
J
s
n
01£1
J
m
(j ')
n
<
sup
(j)
s.£I
J
n
solEI
J
(")
n
J
0
n
( (j)
(j')
I IF "'h(s.,s,,)-F[, "]h ~ (''It,; (,,)1
h=O nJJ
J J
J ,J
P J P J
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
61
(3.5.9)
with probability one as n+
as n -+-
00
and use of (3.5.9) in (3.5.8) yields,
00,
m
(3.5.10)
'\n
~ (j)
(j ') )
sup (') I L {F oo'h(Sj'Sol) - F[j,Jd]h ~ (')'~ ("') }
h=O
nJJ
J
p
J
P
J
soe:r J
J n
s e:r(J')
m
j'
n
I
n
Ih=O IFnJJoo'h(Sj's.,)
J
- F[, "]h(s.,s.,)1
J,J
J J
-1/2
2
+ O{n
(log n) } ,
with probability one.
(3.5.11)
sup (')
Hence, it is sufficient to show that)
m
n
I
IF 'j'h(S.,Sol) - F[, j'Jh(s. s.,)1 -+-p 0,
El J
h=O nJ
J J
J~
J, J
n
s El (j ')
j' n
s
j
as n -+- "".
To prove
C3. 5 .11}
considex' a pa'rti.cu1ar n and as in (2.3.50)
(j)
(j)
take n
= ~ (.) + r a Ib , r = O,+l, ... ,+b.
r ~n
n n
- n
P J
Since F "'h(s.,s.,)
nJJ
J J
and F[. ']h(s.,s.,) are non-decreasing in s. and
Jtj
J J
J
S."
J
for s. e: r(j) and
J
r,n
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
62
S .,
J
E: I (j ')
r,n '
and
Therefore,
<
F .. ,(s.,s.,) - F[. "]h(s"s,,)
nJJ
J J
J~J
J J
which gives us
(3.5.13)
IF "'h(Sj'soI) - F[, "]h(S"s,,)!
nJJ
J
J~J
J J
This implies
(3.5.14)
sup IF "'h(S"s,,) - F[j ,j]h(Sj,Sj')!
(j) nJJ
J J
~J
s.E:I
J n
(j , )
(j ) (j , )
(j )
(j ')
SolE: I
<
max F .. , h (n
,n
) - F [ . . '] h (n
,n
)
J
n
Irl< b
nJJ
r,n r,n
J,J
r,n r,n
I
-
n
+
Since for each r • O,+l, .•• ,+b ,
-
<
=
-
n
(j)
(j ) I I
(j , )
(j , ) I
IF[j](nr+1,n)-F[j](n
r ,n) + F[j,](nr+1,n)-F[j,](n r ,n )
O(a
n
Ib n ) = O(n- 3 / 4
log n),
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
~
63
by the Bonferroni inequality, the second term in (3.5.14) is
o(n-1/21og~
and consequently we get,
m
I
(3.5.16)
n
h=O
sup
IF .. th(so,Sd)
(j)
nJJ
J J
s,EI
J
s
n
EI (j ')
n
j'
m
<
I
n
h=O
Hence for (305.11), it is only required to show that
m
(
max IF 0" (n (j) ,n (j '»
h=O Irl~bn nJJ h r,n r,n
(3.5.17)
as
n -+
- F[ 0 oI]h (n (j) ,n (j '»
J,J
r,n r,n
I -+
P
00.
Let us now write
* ((j) (jt»
nJoJo'h nr,n,nr,n
F
where c(u,v)
G
(3.5.18)
-c /8
1
cf
=.J
°
L
i=l'
y(j),n(j') - y(j'»
n i r n
n i
"
,
is given by (305 04) and let us denote by
(x y) = P
n[j,f]h'
Then,
r c(n(j)
r n
= n- 1
IRn,i
(j) I <
-
f
x
,
IRn,i+h
(j ') I <
-
1
y
,
O<x<oo
02.y<oo
0,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
64
-c
+ O(n
°
1), by Lemma 20301,
We now choose a °3: 5/ 16 < 03 < 3/4 and as before K
-c
such that c > 40 This implies c 0 > 5/4 ~I n 1
4 < 00 and
1
1 3
n>l
where 0 <
10
<
°3¢.
also c 1 (1 - 03)
>
1
Thus, by using the Chebychev inequality and
0
(305018), we have, as
n
for all r = O,+l,ooo,+b
-
p{
~
n
0
00
,
Hence, by the Bonferroni inequality, as
(' )
(' , )
(j)
.::. K
The last equation implies that as n
(305021)
(.,)
max IF ., (n J ,n J )-F*., (n
,n J )I>K
Irl~bn njJ h r,n r.n
njJ h r,n r,n
- 5
~
°3 -l-e:
n
• e:
5
1
n- }
> 00
00,
max
Irl<b
-n
with probability one o
where we take d
~
Also by Lemma 20302, for all r = O,+l,ooo,+b ,
- n
1/4, by proper choice of Ko Therefore (305021),
(305022) and the Bonferroni inequality imply that, as n
(305023)
1-° 3
n~oo,
m
{
max {I F .. , (n (j) ,n (j'»
h=O Irl~bn
nJJ h r,n r,n
~
~
00,
- F. , (n (j) , n (j' »
[J,j ]h r,n r,n
m
In
max {IF .. , (n(j),n(j'»-F* ., (n(j)
h=O Irl<b
nJJ h r,n r,n
njJ h r,n
-n
I
n(j'~)
r,n
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
65
'"
(j)
(j , )
C)
J ,n C')
J ')
+ IFn[j,j']h(nr,n,nr,n ) - F [0J,J0'] h ( nr,n
r,n
log n) + D(n
I~
-d~
4, log n)
m
I
+
n
IF* 00'
max
h=O Irl<b
nJJ h
-n
with probability one.
(n (j) ,n (j , » r,n r,n
0 0' (n (j) , n(j , ) ) I
n(j,J]h r,n r,n
F
Hence for (3.5.17), it is only required to show
that
m
(305.24)
I
n
h=O
max
b
Ir I<
-
IF* 00'
nJ J h
n
(n (j) ,n (j , »
r, n r, n
0, as n
-+
p
- F
.,
n [j ,J lh
(n (j) ,n (j , » I
r, n r, n
-+ co.
For proof of (3.5.24), we see that for all u,v, we have
(3.5.25)
*
E{Fnjj'h(u,v)}
=
n
-1
n
I
i=l
E c(u - y(j) v - y(j') )
n,i'
n,i+h
and
(3.5.26)
Var{F* o. 'h(u, v)}
nJJ
= n
=n
-2 E [nI {c(u-Y (j)i' v-Y (j')Hh)
i-1
-2
~ [
n,
'"
n,
'"
- F [.j O'l (u,v)}
n ,J h
J2
2",
i~l {l-Fn[j,j~h(u,v)} Fn[j,j~h
J
'" [j,j 'lh(u,v)}
+ "'2
Fn [j,J'lh(u,v){l-F
n
+ 2n-
2
mn n-h
I I
h=l i=l
Since
(j )
(j , )
(j )
(j , )
"'2
I
Ic(u-Yn,i,V-Yn,i+h)c(u-Yn,i+h,V-Yn,i+2h)-Fn[j,j']h(u,v)
.::.
we have
1
a.s.,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
66
E!c(u-y(j) ,v-y(j') )C(U-y(j? ~_y(jl»
n,i
n,i+h
n,1+h' n , i+h
and this implies
*
(3.5.27) Var{Fnjj'h(u,v)}.::. n
"",,2
- Fn [. "]h Cu,v)
J,J
-1 """
I
< 1
"""
[Fn[j,ji]h(u,v){l - Fn[j,j']hCu,v)}]
m
+ 2n-
l
I
n
(1 - h/n)
h=l
+ 2n
'::'1/4n
-1
K log n{l - (K log n + 1)/2n}
O(n- l log n)
=
We see that (3.5.27) holds for (n(j) ,n(j'», for all r
r,n r,n
= l, .•. ,b n .
Hence (3.5.24) also holds, since by the Chebychev inequality, for
every e:
>
O.
IF*.
max
Ir/.::. b
, (n (j) , n (j '»
nJj h r.n r,n
F . "
-
n[J,J]h
Cn (j) , n (j '»
r,n r,n
I
>
-
n
m
b
n
.::. L
L
n
-3/4 (log n) 4 }
=
O{n
+
O. as
n +
(.)
+ c2n(nl/2 log n) where c
+
00.
Q.E.D.
00.
(.)
Let ~ J = np J - c (n
-- n, l I n
Theorem 3.5.2.
n
5
n
h=O r--b
C~(j)
- ~(j»
n. 2
n ,1
e:)
+
00
as n
+
ln
00
and c
but c
'-
2n
1/2
log n)
C')
'
~ J
n,2
=
np
(.)
J
are positive constants such that
and c
both converge to zero as
In 2n ";";;:"""--:"';:':"':"';"';;~--:'''''-_'''':'--'-''-
Also let Sj . be the ~(ji)-th smallest observation among
(Xl ., ...•X .).
d
n,J
,1
n,
Then for each
j
=
l, ... ,q, as n
+
00,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
67
Proof .
Since
~(j)
In = F ,(so ) and ~(j) In = F .(s. 1)' we can write
n,2
nJ J,2
n,l
nJ J,
= Fn,(Sj
J , 2) - FnJ.(s.J, 1)
+ [{F[.](s. l)-P
J
(j)
J,
- [{F[j](Sj,2)-P
(j)
(j)
} + {F .(~ (.)) - F .(s. 1)}]
nJ p J
nJ J,
(j)
} + {Fnj(~p(j)) - Fnj (Sj,2)}]
and by Lemma 2.3.5~ both s, 2 and s. 1 belong to r(j).
J,
J,
n
(2.3.3), as n ~ 00,
Hence by
n-1(~(j) - ~(j)) = F[.](s. 2) - F[.](s. 1) + O(n- 3/4 log n),
n,2
n,l
J
with probability one.
J,
This implies p as n
J
J,
~
00,
.. (/j) - ~ (j)) In (s
- Sj .1)
n,2
n,l
j,2
I
=
lim
n
- f
-
~
(~(j))
[j] '" (j) ,
with probability one o
Now i f we write
A
jj ,
L
00
p
Q.E.D.
A
A «j)) A
« j ') )] . , -1
= \)jj,/[f[j]
~p(j) f[j'] ~p(j') 'AJ,j - , ... ,q,
then due to Theorem 3.5.1 and Theorem 3.5.2, as n
~
00,
each j,j' .. 1, ••. ,q and hence a consistent estimate of
~ L •• , for
p JJ
the variance
L •• ,
JJ
!,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
68
covariance matrix of ~ "'Il
Z tis .....
T , where .1 = ( (Tj j , ) ) j ., =1
,J
, •.• , q'
Again for each i,i' = l, ••• ,t; j,j' = l, .•• ,q, if we write
...
p (j) p ,(j , )
i
i
-
F . . , (s . . ,S. , i' )
nJJ
J ,~ J ,
m
+
+
where j.l*
(P
= np (j)
i
n,~
I
q ')}
n
{F
h=l
m
n
~
{F
l..
h=l
"'h(s. i'S" .,) - Pi(j)Pi
nJJ
J,
J,~
(
nJ"'h
J s,J,~., Sj' ,~.,
) - p (j)
•~
(j')}
Pi'
+ 0 (n 1/2 log n) and s. i is the j.l * (j) -th smalles t
J,
observation among (Xl " ... ,X
.~
n, J
,J
n,i
then by the same arguments as in
Theorem 3.5.1, it follows for every i,i' = l, ... ,t;j,j' = l, .•. ,q,
that
V(jj ')
-'7
P
ii'
)j j
,)
ii'
,
as n ....
00
and consequently we have
where
J
(jj ') =
n~ ii'
v~i~ /f [j) ~p(j ~j f [j , ] (E;p(j '~j '»)
')
i'
i
v~i~')
and
nii~')
and
are given by (3.3.9) and (3.3.12) respectively.
Hence a consistent estimate of the dispersion matrix (3.3.14) is given
~
by!* =
~*
«1 JJ.. ,» J,J
.. ,=1 , .•. , q'
t
=
3.6.
Asymptot~c
~
l..
t
~
l..
where
a (j )
i-l i'=l P
(j ) a
i
~ (j j , )
, 1
(j , ) nii '
, j, j = , ... , q .
(j U )
p
f
Normality of a Linear Compound of Several Quantiles
for Random Sam)21e Sizeo
Let {n } be an increasing sequence of positive integers tending
r
to infinity, and let {N } be a sequence of proper random variables taking
r
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
69
positive integer values such that N In
r r
-+ 1)
in probabi1itYj as r
-+
00.
In
conformity with the notations used earlier, for sample of size N ,
r
~j;l
<0'0 < ~j)N
are regarded as the ordered random variables for
r
r' r
the j-th variate, j = 1, ..• ,q.
Let us suppose that z(j)
N
r~Pi
denotes the sample quantile of
(j)
order Pi (j) for the j -th variate and
population quantile and
(1)
(q) )'
~
(l)t ... ,~
;
Pi
~ (j ~j)
be the corresponding
Pi
~
I
,i = (z (1)
(1)'''' ,Z (q) (q)\'
Nr,P i
Nr,P i
/
r
(q)
~
=
.....£.i
Pi
We want to determine the distribution of the vector of linear
compounds
~
0
1/2 ~
~
(1) (z (1)
(1)
1/2
(q)
t(q)
L. a
(1)
(1)- ~ (1) , ... ,N r ,
L
a (q) N P (q)
i=l Pi
Nr'P
.
1
Pi
~=
Pi
r' i
i
r
- «q~S\~)
Pi
as r
-+
where the
00
Let us assume
a(j~j) 's, i=l, ... ,t;j=l, .•• ,q, are constants.
:~at f[.]/t;(1\»o'
J
~ P.J
i=l, ... ,t;j=l, ... ,q, and let
~
us denote by
I (j)
Nr,P i
where
~
r
(j)
= {x..~(j)(')
P J
i
-1/2
log Nr as r
"'- N
r
-~
-+
r
00.
< x < ~(j)
(j)
Pi
+ ~}
i=l, ••. , t
r
j=l, ..• ,q
By Lemma 2.3.5, Z (j)
for i = 1" .. ,t;j = l.",q, with probability one, as r
the condition that
E:
N p. (j)
r'
-+
~
00
I (j)
N P (j)
r' i
and under
sup sup f[.] (x) is finite, Lemma 3 of Bahadur
l~2.q x
J
(1966) extends directly to our case.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
70
By (302.5) and Lemma 3 of Bahadur (1966) we get that for every
E >
0 there exists an n
This implies that as r
>
0 such that as r
~ ~,
~ ~,
= (U(l)
N
(3.6.2)
u(q) )'
, ... , N
r,i
r,i
where
(3.6.3)
u(j)
N
'"
r,i
v (j) ~ (j ) .\
u, i \/:; (j ) )
Pi
III
c ~ (j )
- X . ) - P (j)
\"
(j )
u ,J
i
Pi
By (2.2.7) and (2.2.8) the sequence {X , u=0,+1,+2, ••. } satisfies the
""U
--
E ~1/2 <~. Again for any
n n
.
j-l,
... ,q, v (j)i. ~~ (j»)
(') be1ng
a Borel-measurable function of X . is
u,
J
u,J
Pi
also a ~-mixing process. Moreover since X ,decreases exponentially
condition of a
~-mixing process with
U,J
we shall still have
L~
n
1/2
n
<~.
Also
E v (j )(~ (j) \ = 0 and varfv (j) ~ (j) ,~
u,i
(j»)
u,q' (J»)
Pi
Pi
For any vector
~
=
Pi (j) (1 - Pi (j» .
= (dl, ... ,d )', we have by (3.6.2) and
q
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
71
L, say,
=
and similar to (3.307), (3.3.9) and (3.3.13}, here we also have
(3.6.5)
where
2
_
aL
~
~
t
L L
~
j =1 j 1=1 i=l i
n(il:)
(30606)
var(L) -
=
is defined in (3.3.12).
YN
r
(~p)
""'
=
-1/2 -1
N
aL
r
~
L
t
(j )
=1 J J
P
L d.d.,a
I
(.)a
i
J
(j , )
P
f
I
(j j , )
(.,)n. i ,
J
1.
Let us denote by
N
'/
L
f
f
•L dj
j=1 u=1 1.=]
~
~
~
a (j )(j / f [ . ] E; (j)('), v u(j . t; (j )(j) )
p.
J p. J
'Pi
1.
1.
Then by Theorem 20.3 of Billingsley (1968),
(3.6.7)
Hence by (3.6.4), as r
N1 / 2 It a(l)
r i=l p (1)
i
O
~
00, the distribution of the vector
(zN(1) p
-E; (1) ), ... ,N 1 / 2 It a (q)
(1) p (1)
r i=l p (q)
r' i i i
is multivariate normal with mean vector
as given in (3.3.14) and (3.3.15).
(zN(q)
r,Pi
-E; (q)
(q)
~~I
P (q)
i
Q and same dispersion matrix
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
CHAPTER IV
ASYMPTOTIC PROPERTIES OF THE WILCOXON SIGNED RANK
STATISTIC AND RELATED ESTIMATORS FOR MULTIVARIATE
STATIONARY AUTOREGRESSIVE PROCESSES
4010
Introduction
In this
chapter~first~
the asymptotic multinormality of the
Wilcoxon signed rank statistic (vector) for multivariate stationary
autoregressive processes is
derived~
Since the Wilcoxon signed
rank statistic can be expressed as a particular case of Hoeffding's
(1948) U-statistics, in the next section we extend the above theory for
a class of U-statisticso
As an application of the earlier results
of the chapter, in the last section t the asymptotic distribution of
the median of the
mid~range
estimator(vector) is deduced from the
asymptotic multinormality of theWilcoxon'signed rank statistic
(vector) and a straight" forward multivariate extension of the
Hodges and Lehmann's ,(1963) techniques
4020
0
Asymptotic Properties" of the Wilcoxon Signed Rank Statistic
Let
{~l,ooo,~J
be the" chance variables associated with the
sample" of size on from the process" defined in (1. 1. 1)
0
Let
Rl,
too,R ,be the ranks of IX l jlpoo,lx ojl where Xl j'OOO'X .
jn
,J
p
n,
,n,J
are the sample observations for" the j-thvatiate and let Sij
or 0 according as Xi . > or < 00
,J
=1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
73
Then let" us define the vector of Wilcoxon signed rank statistic
as ~n
T
=
(Tn l~ooooT
• nq )' where
=
{n(n + 1)}
n
-1
~ Ri·S .. ,
j
~J J..J
i=l
If the empirical codofo for the absolute value of the j-th variate
is defined by
=
then the Wilcoxon signed rank statistic for the j-th variate can be
written as
00
T.
nJ
n(n + 1)-1
=
J HnJ.(x)dFn j(x)
o
~
For the m -dependent stationary process, where m
n
n
K log nand K
is a positive number, if the empirical codof o for the absolute value
of the j-th variate is defined by
n
n
-1
~
i=l
c(x - Iy(j) I)
n,l '
j
then the corresponding Wilcoxon signed rank statistic for the j-th
variate can'be written as
00
*
T
nj
="
Let the vector T*
""'n
,
*
*.
r .(x)dF .(x),
n(n + 1) -1 ,H
0 ' nJ
nJ
j = l,.oqqo
=
The true codofe of
IX i ,J. I
=
p{ Xi
I
is defined by
.1 .::.
,J
x}
0 < x <
,
co,
and for the m -dependent process the true cod.f. of Iy(j) I is
n
n,i
denoted by
=
I
P{ y (j)
n,i
I :5..
x},
o
< x <
co,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
74
For the derivation of the asymptotic normality of the Wilcoxon
signed rank statistic J let us first prove some lemmaso
n l / 2 !H .(x) - H*.(x)1
nJ
nJ
sup
x
= O(n- l / 4 log n).
with probability one o
Proof o
Upon noting that H .(x) =F .(x) - F j(-x) and H*j(x) =
~
n
~
n
F*.(x) - F*.(-x). the proof follows from Lemma 3 2 2
nJ
nJ
0
For any j = 1'0
0
0
Jq~_
nl/2lT , _ T*.
nJ
nJ
with probability one
0
Proof o
1,ooo~qJ
For any j =
as n
I
-+
0
0
OOJ
= O(n- l / 4 log n)~
we can write by (40205) and integration
by parts.
Tnj = n(n + 1)-1 [J(Hnj(X) +
H~j(X»dFnj(X)
7o H*,(x)d(F
j(X)-F*.(X»
nJ
n
nJ
+
j H*.(X)dF*.(x~
nJ
nJ ~
0
00
• T* + n(n + 1)-1 J (H ,(X)-H*j(x»dF ,(x)
nj
0
nJ
n
nJ
+ n(n+l)-l[H*.(x)(F j(X)-F*j(X»]ooo
nJ
n
n
00
- n(n+l)-l
- n(n+l)-l
so that
r (Fn j(x) - F*.(X»dH*j(X)
nJ
n
o
J
fo
(F .(x) - F*j(X»dH*j(X)
nJ
n
n
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
75
00
3 2
n / (n+1)-1
<
J
o
IH .(x) - H*j(x)ldF .(x)
nJ
n
nJ
00
+ n 3/2 (n+1) -1
~
By Lemma 40201. as n
JIF
o
n
I *
j (x)-F*. (x) dH .(x)
nJ
nJ
.
the first term in (402012) is of order
00.
1 4
(n- / log n). with probability
one~and
by Lemma 30202. as n
~
00.
1 4
the second term is of order (n- / log n). with probability one. QoE.D.
Lemma 4 2 3 0
0
0
For large n.
where
(~n l'
~nq )
q
H
I,
r: = «yJ. J' »j ., =1 .0 00 • q •
i
~J
00
(402013) )lnj
=
(4 02.14) Y . ., =
JJ
(4 o 2 015) ~U j j 'h =
~
Hn[j](X)dFn[j](X).
m
n
m
h'"
n
h'"
o"I+I (l--)o"'h + I
(l--)o"'h. j5j'=l p o q q.
JJ
h=l
n JJ
h=l
n J J
'"
~(11) _ ~(12) _ ~(21) + ~(22)
U
j j 'h
00
(402 016)
j = 1 p oo.q.
U
jj ih
U
j j Ih
U
j j 'h.
hal 2
. . . . . 0 00 •
00
8~~~h)=
JJ
f0 f0
8j}~ =
U(F
{F [. "]h(x,y)-F [j](X)F [.,](y)}dF [.](-x)dF[,'f- y).
n Jd
n
n J
n J
n J
0000
(4.2 17)
0
'" (21)
(4.2018) O. "h=
JJ
h
[j ~ j ']h(X.-y)-Fn [j ]<JQ Fn[j , ] (-y) }dFn [j] (-x) dF
nU
' ](y).
0000
JJ
{Fn [.J
00
j'] h (-x, y )
j
- Fn[j] (-X)Fn[j'] (y) }dFn[j] (X)dFn[j'] (-y) •
"'(22)
(4 2 •19 ) 0., 'h ..
0
JJ
00 00
U
fFn[j.j l]h (-x.-y)
- Fn[j] (-X)Fn[j'] (-y) }dFn[j] (X)dFn[j'] (y) •
Proofo
We can write for j =
1.coo~q.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
76
00
T*
nj
= J H*~x)dF*,(x)
+
nJ
nJ
0(n- 1 )
o
00
=
00
iln [j](x»dF*,(x)
+ 6(iln [,](X)d(F*,(X)-F
['lex)
nJ
J
nJ
n J
!(H*,(x) nJ
o
00
+
Jo Hn [,](x)dF
['lex)
J
n J
+ 0(n-
1
)0
For every j = 1,,0.,q, if we write
00
(402020)
Rnj
=
[(H~j(X) - Hn[j](x»d(F~j(X) - Fn[j](X»'
then by (402013), integration by parts and the relations Hn[j] (x)
F ['lex) - F [j](-x) and H*,(x)
n J
n
nJ
=
=
F*,(x) - F*j(-X) , we have
nJ
n
(4 02021)
- J (F~j(X)
- Fn[j] (X»dHn[j] (x)
o
1
+ !(H*,(x) - H [.](X»dF ['lex) + R , + 0(n- )
o nJ
n J
n J
nJ
00
= ~nJ,+ J(F*.(x)
- F [j](X»dF ['](-x)
0 nJ
n
n J
+ Rnj + 0 (n
-1
)
so that
(402022)
= n -1/2
n
l:
i=1
where
(402 023)
(402024)
and
00
(4 0 2 0 25)
(j)""
= !{c(-x-Y
o
n,1,)
,..,
-Fn [j](-x)}dFn ['lex)
J
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
77
To show that the remainder term n
1/2
R, converges in probability
nJ
to zero. we write by (4,2,20).
n
n
n
00
L L f{c(x-ly(j~l)
-3/2
i=l Q,=1 0
n.1
-II [.](x)}d{c(x-y(j!)-'F
n J
n~)(,
['](x)}
n J
which implies by Fubini's theorem and the fact that y (j) I S are
n.i
ID
n
-dependent.
d{c(x-y(j»
n.Q,
+ n-
3
n
I
IDn
I
-
Fn ['](x)}]
J
n-h'oooo
I JJE[{c(x-ly~j~I)-Hn['](X)}
i=l h'=l Q,=1 00
J
•
(j) I
i )-H ['](y)}
n.
n J
{c(y-\y
+ n-
3
m
n n-h
I
I
m
n
I
n-h'
I
h=l i=l h ' =l Q,=1
N
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
78
where we note that for any
ClO
i,i',~,~'
= 1,2'000 and h,h' > m ,
n
ClO
(j) .....
(j)
!E[{c(x-Iy i!-H [.](x)}{c(y-Iy i+h l ) JD
On,
n J
n,
.....
H [.](y)}
n J
Using the fact that the absolute value of each of the integrands
in (402027) is bounded by 1 and the inequalities
(4 2 28)
0
0
and
...
..........
~ =1, ° ° • , n-h ' ,
2. dFn[j,j]h' (x,y)+3dFn (j] (x)dFn(j] (y), h l =l
,o •• ,m ,
n
we have,
Hence (402.30) and the Chebyshev
(4.2.31)
n
1/2 Rnj
p°
inequality imply that, as n+ oo ,
°
Now using the definitions (2.309), (203.11) and the Fubini
theorem we see that for every i
= l,eo.,n;
j , j ' = 1,ooo,q,
E{B
which imply
(402032)
and
s
(y(j») •
n,i
0,
s = 1,2
h = O,l,ooo,m , and
n
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
79
s.s'
00 CIO
=
= 1.2
...
s-l
- Fn[j] «-1)
x)}
E !!{c(_l)s-lx _ y(j»
00
n.i
{c«_l)s'-ly _y (j'»
n.i
CIO
=
CIO
...
Jo 0f{Fn ['JtJ"]h«-l)
..... (ss')
= 0jj'h
t
s-l
x.(-l)
by (4.2.16) -
s'-l
y)
(402019)~
so that
=
......
0jj'h t by (4 02 015)0
Here note that if h = 0 and j = j' then t
(4 2 34)
0
0
Since B(y(j~is are m -dependent t the dispersion matrix of
n
n.
n n
{n- l/2 L B(y(l»tOootn-l/2 L B(y(q»} is given by
i=l
nti
i=l
n.i
......
r
.....
where
= «Y'J"»'
J
Jt J"=l to oo • qt
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
80
mn
n-h
l:
+
l:
h=l i=l
m
=
8J• j , +
n
L (1
h=l
Now for any j =
q) )}]
cov{ B(Y (j , »B(Y n,i
n,~+h
m
h""
h ""
n
- -)o'j'h + I (1 - -)OJ'jh
n J
h=l
n
l~ooo~q, the BCy(ji»'S
are bounded valued random
n,
variables and they satisfy the condition of a
~
n
= 0 for n/10g n > Ko
~-mixing
process with
Hence by a multivariate extension of the
central limit theorem for strongly mixing processes by Rosenblatt
(1956 b), the lemma follows 0 QoEQDo
where
r
""
and
-
~.
=
and Y.. , are defined in the same way as
J -
JJ
""
~
. and ""Y.. , in
nJ -
JJ
-
(402013) and (402014) - (402019) with Hn[j](X)' Fn[j] (x) and
Fn(j~j']h(x,y) replaced by H(j](X)' F(j](X) and FU,j']h(x,y)
respec ti ve1y
0
Proof o Upon noting that H[j](x) = F[j](X) - F[j](-x) and Hn[j] (x)
= In[j](X) - Fn[j] (-x), under (2 0208) and (2 02010), we have by
(30201), as n
+ ~~
sup IH[j](X) - ""
Hn[j] (x) I =D(n -d ),
x
Also by a straight forward extension of (2 03016), similar to the
extension of (203015) to (30201), we have under (202.8) and
I
I
I
I
I
,
I
I
I
I
I
I
I
I
I
I
I
I
I
81
{sup IF[, "]h(x.y)-F [. "]h(x. y )!} • O(n-d).
J.J
n J.J
x.y
= l,oootq,
For every j
we can write
00
00
~nj = ~ Hn[j](X)dF[j] (x) + ~ Hn[j] (X)d(Fn[j] (x) - F[j](X»
(402039)
First integrating by parts and then using (3.201). it is
seen that the second term in (4 2.39) is of order (n- d ), d ~ 1 t
0
as n
n +
+
00
0
By (4.2037) and the dominated convergence theorem, as
00,
""
""
J Hn [j ] (x) dF [j ] (x)
o
for every j = 1.ooo.qo
as
n +
+
~H[j](X)dF[j](X) = ~j'
Hence by (402039) and the above arguments.
00.
where for each j
= 1.00o.q,
~,
J
is defined similar to
~
. with
nJ
Hn[j] (x) and Fn[j](X) replaced by H[j](X) and F[j](x) respectively.
Again for every j,j'
00
= 1 t ooo.q
we can write
00
~ ~{Fn[j,j']h(X,Y)-Fn[j](X)Fn[j'](Y)}
QO
+
QO
f J
o
0
{FnJtJ
[' "]h(x,y)-FnJ
[,](X)FnJ
["~ley)}
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
82
Integrating by parts and then using (30201), it is seen that
each of the second and third terms in (402.41) is of order (n-d) •
d
~
1. as n
~ ~o
Also by (30201). (402038) and the dominated con-
vergence theorem. as n ~ ~. the first term in (402041) tends to
(11)
(11)
~(11)
~
~
0jjih where 0jj'h is defined as 0jjih with Fn[j](X) and Fn[j.j']h(xty)
replaced by F[j](x) and
FIj.j']h(x~y)
respectively 0 By similar
~(2l)
(21)
~(22)
0jjih ~ 0jjih and 0jj'h ~
~(12)
(12)
arguments, as n ~ ~, 0jjih ~ 0jjih t
(22)
(12)
(21)
(22)
0jjih. where 0jj1h. 0jj'h and 0jj'h are defined similarly 0 Therefore.
l~ooo.q~
for every j.ji •
= 0.1.000.
h
as n
~ ~.
(402042)
where 0jj'h is defined similar to 6 jj 'h with Fn[j](X) and Fn[j.j']h(x.y)
replaced by F[j](x) and F[j.j']h(x.y) respectively 0 For h
= j'.
j
=0
and
~ ~.
also similarly. as n
(402043)
where
~j
~
is defined similar to 0,. with the above replacement 0
JJ
Further. the fact that IFn[j.j']h(x.y)-Fn[j](X)Fn[j'](Y)! 2. 1.
for all x.y,gives us 16jjihl ~ 4. for j.j' • l.ooo.q. h • 0.1.000.
m
n
~which
in turn implies
m
(4 2044)
0
n
IIhal
for every j,jl
~
(h/n)ojj'h l
=
<
-
2n
-1
m (m +1) ~ 0.
n
n
l~oco,qo
Hence (402042) - (4 02044) imply that
r'"
~
[.
as n
~ ~~
as n ~ ~.
I
I
I
I
I
I
I
I
83
where
r
is defined similar to
r
with the above replacement 0
Now the theorem follows readily from (4 2 040).(402045). Lemma
0
Remark:
In Lemma 4 2 3 and Theorem 4 02 1 we have proved the
0
0
0
asymptotic normality for fixed
vector X. o
~l.
F~
the marginal distribution of the
If we consider a double sequence U
defined for each n, where ~n~i
U
=U
I
I
I
I
I
I
U
= 1.•00•
o.n
(Xi)' i
~n~'
distribution depending on n then
2Pooo~Y.
'-n~n
•
and ~,
U i has
Lemma 4 2 03 and Theorem 40201
0
also hold provided the dispersion matrix r
defined on U
~n
I
I
I
I
I
1~
~n.~.
~pi
is
positive definite in the sense that
(402046) lim inf{minimum eigenvalue of r }
since then for a vector
c
>
~
n
>
O.
~ ~ Q~
so that if
EI~'!!n'i jl3=
.:s. ~ I9"J.3IElu~:~1
l
•
then Liapounov's condition (see
satisfied o
Lo~ve~
<
ex>
~
1963. p. 275) is automatically
In our case B(y(ji»' s being bounded valued random
n.
variables, Liapounov's condition (see Loeve~ 1963~ po 275) is
automatically satisfied whenever the dispersion matrix is p.d. in
the above sense and this implies the condition (2) of Theorem 3 of
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
84
Gnedenko and Ko1mogorov
(1967~
ppo 101-103)0
Hence the lemma and
the theorem hold whenever the dispersion matrix is podo in the above
sense.
4,30
Asymptotic Properties of a Class of U-statisticso
It is well known that the Wilcoxon signed rank statistic
(vector) can be viewed on as a Hoeffding's (1948) U-statistico
Under
certain regularity conditions, the asymptotic normality of the Wilcoxon
signed rank statistic obtained in section 4 02 is extended here to a
class of U-statistics for stationary autoregressive processeso
These regularity conditions are somewhat more restrictive than those
in Hoeffding (1948) dealing with independent observations, and appear
to be necessary in view of the infinite chain of dependence in the
series of observations.
It is intended to follow up the general case
in the near future o However. for the sake of completeness. we will
summarize the results on U-statistics obtained in this chapter o
We may note that in (2 207)
0
pound of
E
~
t -r ~ r
= O.l.oo.~~o
~t
is expressed as a linear com-
Essentially Hannan (1961) and Eicker
(1965) considered a similar truncation of this linear compound as is
done in (20305) and approximated the linear function of the
by the corresponding function of Y tlSO
~J
~t'S
The latter function is
again linear in the independent errors £t over a certain number of
terms depending on n. and thereby they were able to use the CoL.T o
But the kernel
~(~lJo.OJ~r)
of a U-statistic is not. in general. a
linear function of its arguments 0 Therefore the decomposition
(20305) is not adaptable.
The non-linear nature of the U-statistic
and the chain of dependence of the successive observations makes it
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
85
quite complicated to reduce the autoregressive process to an m n
dependent<process in which the results of Sen (1963) can be incorporated and extended to prove our desired resu1ts o
This. however, is
done here through the following conditionso
i)
E > O~
For every
we can find a positive number A such that if
we define the qr-dimensiona1 rectangle
c
qr
and A = R - A*, ioee> the complement of A* and I
C
is the
A
indicator function of AC~ then
and
11)
< 00 0
sup
n
For a finite time interval Tn = {t:1.:.. t..:. n}, let {~POOq~n}
be the ramdom variables associated with the sample of size n from
the process defined in (10101)0
In addition to satisfying the above
¢(~~ tooo~~~
assumptions, let the statistic
1
arguments X
~~
,ooo.X
1
.~~
be symmetric in the
)
r
r and let us define the U-statistic
;~1<000<~
r
based on the sample of size n as
U(Xl ~ 00o.X ) = (nr) -1
~,
.~
L,,(X.
'+'
S
~~.
0
0
o.X
1
where the summation S extends over all 1..:.
)
.",~
r
~1
<coo<~r":'
U-statistic for the m -dependent process, where m
n
K is a positive
number~
n
is defined as
~
n.
The
K log nand
I
I
I
I
I
,
86
Let us define
g(F )
n
I
I
I
I
I
I
I
>
for i
m
n'
= l,ooo,r-1}
and
I
I
I
I
I
I
=
v.,O<v.<m
1.
~
remaining r -
Obviously g(F )
n
1.-
n
~ i~,
for i
-1 values of it a + - a
i 1
i
>
m for i
n
while for the
= l,2,000,r-lo
= g(Fn !O)o
The symmetric estimator UO(Y
Y 1'000'
-n, l'ooo,Y
"'n,n ) based on -n,
Y
~n,n
is defined by
where the
The following two assumptions are made concerning the kernel
Assumption
(A)~
It is assumed that the kerne1,j, (x too 0,x ) is
't'
"'a
"'a
1
r
continuous everywhere °
Assumption (B):
Further assume the following Lipschitz type condition,
qr
namely, there exists a 00 > 0, a function g(o) on R
and a constant
M such that if x = (Xl ,ooo,x' )' and x*
-a
"'a
"'a
- Oi.
1
r
'"
two points on Rqr with I~a - ~~I I <00 then
,~
!'oJ
= (x*'
000 x*')' be any
-a'
''''a
1
r
I
I
I
I
I
I
I
I
I
I
I
I
87
and
Eg(~~) ~
M,for all ~, all n
Lemma 4 3.1.
Under (2 2 6), (202.8), (2.2.10) and the assumptions (A)
and (B), as n
+
0
(4.309)
n
ProoL
0
1/2
0
00,
{U(X
~oo,X ) - U(Y l' ••• 'Y
~1
""n
~n,
.....n,n )} +0.
p
Under the notations defined above,
<j>(X
~al
p , qX
~ar
)
and <j>(:t;
,00. ,:t;n,a )
. n~a1
r
can be viewed on as functions of the rq x 1 vectors x
~£
respectively.
every
£
(4.3010)
>
and v
'""'n,£
Hence by (4 3.3), (403.4), (4.3.8) and (2.3.5), for
0
0,
p{ n 1/2, U (~1 ' •• o,~
n)
- u(Xn,a ,
0
0
l
•
,Xn,a ) I
> £}
r
1/2 lu(xl,ooo,X )-U(Y
, •• 0,Y
)I
-< P{n
""
.....n
~n a
~n a
, 1
> E:,
' r
I
I
I
I
I
I
I
I I <j>(~a)- <P(Xn
S
"'"
a)
' .....
1>£,
sup
II ~a -Xn a II <co}
1<a <. o.<a <n
"'"
.,~
- 1
:r-
+ p{
sup
1<a <oo.<a <n
- 1
:r-
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
88
< p{n 1 / 2
sup
Ilx.....a -y.....n,g II (n)-l
\' g(X) > d
r
t..
.....2
1<a <000<a <n
.....,
S
- 1
r-
< p{ (nrq)
-
1/2
(') n 1
sup
max
IR Jil ()- L g(~ » E}
l
'
1
n,
r
S
a
i- ,0 00•n J = '0 •• ' q
.....
+ p{(rq)1/2
sup
max
IR(j)1 > 00}
'-1 ,o.o,q n,i
i - 1 ,ooo,n J-
n -1 \'
The first thing to be noted is that, since E[(r)
L
S
g(~a)]
<
,...
M < 00, there exist a small n(>O) and a large K such that
n
(403011) p{(n)-l \' g(X) > K} < M < nfor K > K ,
r
L .........a
- K
n
S
n -1 \'
t.. g(~ ) is bounded in probability.
which implies that ()
S
by Lemma 20301, a constant c
2
r
(403012)
(')
I~
n,ai
P{IR J
c n
2
1
Secondly,
(>0) can be so chosen that
-c 1 /o
} < c n
3
-c 1
,
where 0 < 0 < 1 and c , c are also positive constants and this
2
3
implies that.
(403.13)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
89
<
In (4.3.13) let us choose c
implies clio> 10
> 1 and. since 0 < 0
l
Then by (403.13).
is also bounded in probabilityo
..
'(nrq)l/2
~
1. this
sup
max
IR(j~1
i-l ••••• n j=l ••••• q n.
This and (4.3.11) imply that the first
O. as n ..
00 0
Again. if 00 is kept fixed then also by (4.3.13) the second term in
(4 • 3 • 10). 1. e ••
P{(rq)1/2
sup
max
'-I ••••• q
i =l ••• o.n J-
IR(j?1
n.1
> cO} .. O.as n"
This completes the proof of the lemma.
Lemma 4:3';2';
Under (2'.2.6) and (2.2.8). as n ..
""
E{U(Y 1.' ••• Y )}. g(F ) + O(n
""n.
""n.n
n
(i)
-1
00.
log n)
and
(it)
Proof.
(4.3.14)
n
1/2
{UO(Y 1.0 ••• Y ) - U(Y 1 ••••• Y )} ..
""n.
""n.n
""n.
""n.n
p
Similar to (2.4) in Sen (1963). here we have
E{U(X
n.
1.' ••• Y ) - g(F )}
""n.n
n
O.
00.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
90
-g(F )]
n
By the Liapounov's
g(Fnlt.vl.ooo.Vt)-g(Fn)
a i + l -a i
= vi
~
I is bounded for all (al ••• o.ar) with
mn+l. for all i
g (n) (F
s
inequality of moments and (4.3".1).
n
) =
= l.ooo.t-Io
Let
sup
Ig(Fnlt.vl.o ••• vt ) - g(Fn ) I
al··o •• a r
Then similar to (206) in Sen (1963). we have by (4.3.2).
(403015)
IE{U(Y
""n,
1.000.Y
~n.n
)-g(F ) I<fn )n -={r
<
=
n
-1
10/h) _
(-(r-l)mn~ (n)(F)
r
s
n
~r
r(r-l)m g
(n)
n s
O(n
-1
~
(F)
n
log n).
which proves (i)o
Also
(403016)
n
1/2
n _l(n-(r-l)mn)
{U(Y
l.ooo.!
UO(!n. l'o •••! n.n )}
""n.
n.n l r
r
)-1)
= rn
-1/2(n-l)-1
r-l
\'
<P
L.
S-SO
(Y
n.a l
,
0
•
0
,
Y
n.a r
)
)1
l
n
where the summation extends over all possible {((nr )_(n-(rr- )m )
terms with a i + l - a i ~ mn , for at least one i - l ••• o.r-l.
from Lemma 4.7 of Sen (1963). it follows that
n-l)-lfl(n) ( r- 1
r
t-
(r-l)mn)~
r
<
-
m (r-l).
n
similar to (208) of Sen (1963). here we get
Since
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
91
which by the Chebyshev inequality and a well known convergence theorem
due to Cramer (1946. pp. 253-254) implies (ii). Q.E.D.
Let us now put
,/,(n) (y
~'....
't'a
~o
•
= E{ '/'(y
't' n
••••• y
)
a.'
• ""'"0
-.... • a. a
1
••••• v
"" , a.'
1
.Y
•
'''''n ' ""'" a '''''n ' ""a+
'" 1 •
for a = O•• oo,r; where a. + - a. > m for all i = 1,2,.0 •• r-l and let
i l
i
n
1;;(n)
a.O
(4.3019)
(4.3.20)
for a
=
l'" (n)
{ (n) (
) (n)
(h
h ) • E ¢
Y
, ••• , Y
¢
(!n'~l
Q
,
ao 1'0." a
a
""n,a. 1
""n.a. a a
'"
O,l,ooo.r; where Ia.i - eil
but Ia. i - a. t
l,oo.,n.
=
I
>
= hi;
0
<
••••
! n'~a ) }
Q
hi ~ mn for i - l •• o.,a,
mn , Ie.1 - e t I > mn and Ia..-e
1 t
I
>
mn for all i
+t
=
Finally. let
m
(4.3021)
=
l'"(n) +
"'1.0
\,n
2 L
h=l
(n)
1;;l.h
Then we have the fo11owingo
1/2
Under (2.2 6) ~nd (2.2.8). n
{UO(Y 1" • .,Y
)}
.
""n.
""n.n
is asymptotically normally distributed ,with zero ,mean and
Lemma 4.303.
-g(Fn )
.
0
2 (n)
var1ance r 1;;1 .
Proof.
Y
NIl, n
By Lemma 4.3.2. it is sufficient to show that n
1/2
{UO(In,l •••• '
) - g(F ) } has asymptotically the normal distribution with zero
n
mean and variance r 2l'"l(n).
'"
BY t h e same tech
n quesiasd
use i n Sen
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
92
(1963) to obtain the variance of a U-statistic for an m -dependent'
n
process, here we have
•
Y )}
Var {U0 (Y 1'000'....
""U,
n,n
(4.3022)
n- 1 r 2r... (n) + O(n- 2 log n) •
l
Again, if we define
...
Y
n
rn
-1/2
n
L
i=l
where
Var{Y }
n
...
2 (n)
-1
r 1;;1
+ O(n log n),
then by exactly the same procedure as in (2022) of Sen (1963), we
get
(403023)
Yn - n
Now
1/2
....
{UO(Y
""U, l'o",Y
""U,n ) - g(Fn )} ~
p
a
0
{~(n)(y
)} form an m -dependent stationary stochastic
1
""U,i
n
process, whose third (absolute) moment is finite by the assumption
(40302)0
Hence by the CoL.T o for strongly mixing processes by
Rosenblatt (1956b), it follows that Y has asymptotically the normal
n
distribution with zero mean and variance r 2I;;i n ).
The above arguments
along with (403.38) complete the proof of the lemma.
Under the assumptions (A), (B). (2.2.6), (2.2.8),
Theorem 403.10
and (2.2010), n
1/2
{U(~l",o'~n)
- g(F)} is asymptotically normally
2
distributed with mean zero and variance r 1;;1' where
(4.3 24)
0
g(F)
...
JoooJ ~(~l""'~r)dF(~l)o.odF(~r)'
Rqr
(403025)
I;
Q.EoD.
1
...
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
93
=
JoooJ{~(~1'X2,000'Xr)-g(F)}{~(~1,X~,000'X~)-g(F)}
Rq (2r-1)
dF(X2)·0.dF(Xr)dF(X~)000dF(X~)dF(~l)'
(403027)~loh = fooo!{~(~1'X2,ooo,Xr)-g(F)}{~(~2,X2,00o,Xr)-g(F)}
2qr
R
*
dF(X2)·00dF(Xr)dF(X~)ooodF(Xr)dFh(~1'~2)
Proof o If we denote by Fn(~l) and Fnh(~l~~2)' the joint codof o of
Y . and (Y
.~Y
i+h) respectively, then by straight forward
~n,~ ~n,
~n,~
~
~
generalizations of (302 1), it is easily seen that F and F 1. con0
n
n.1.
verges weakly to F and Fh respectively 0
Since
~(o)
is continuous,
= Joo! ~(xl'ooo,x
g (F)
n
~.
qr
'~r
)dFn (xl)ooodFn (x
)
~r
R
and (40301) holds, using a result in Cramer (1946, po 74), we get,
(403028)
lim
n-+
oo
g(F)
n
=
~
g(F),
("foJ
by (403024)0
f"W"";
2''''''''
dFn (X 2)ooodFn(Xr )dFn (X*2)ooodF(X*)dFh(xl'~2)-g~)
nr
n~
n
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
94
by the same arguments as given above»
I;;(n) =
1.h
lim
=
Similarly»
lim (;(n)
n -+ 00 1'0
=
where 1;;100 is defined in (403026)0
Now under (202 08) and (40302)p
00
sup { \' II;; (n)!} is finite
n
h~O 1 h
0
and
where
00
<
00
Hence by above and bounded convergence theorem» we have
00
(4 03031)
lim
n
-+
00
(;(n)
1
=
1;;1.0 + 2
I
h=l
1;;1.h
= (;1»
by (4.3.25).
Therefore the proof of the theorem follows readily from (4.3.28)>>
(403031)p Lemma 4.3.1» Lemma 4 03 02 and Lemma 4.3030
4.4~
Asymptotic Distribution of the Median of the Mid-range Estimator.
Let {~l»ooo»~n} be a sample of size n from the stationary auto-
regressive process defined in (10101)0
codof. of
~i
We assume that the marginal
is diagonally symmetric about its location parameter.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
95
Then the median of the mid-range estimator (vector) for the location
parameter is 8
for j
1 ..... 8
)' where 8 . =
med
{t(x t ,+Xt ' .)}.
n.
n.q
n.J
l<t<t'<n
.J
.J
-From the asymptotic normality of the Wilcoxon
=(8
-n
= 1 ••••• q.
signed rank statistic (vector). the asymptotic distribution of the
vector
1n8""'Il
is derived here by a straight forward multivariate
extension of the Hodges and Lehmann's (1963) techniques.
Using the relation H[j](X)
= F[j](X) - F[j](-X). j = 1 ••••• q.
we see that if the c.d.f. F is diagonally symmetrical about
implies F[j](x) + F[j](-X) • 1 for every x and for every j
Q.
which
= 1 ••••• q.
00
(4.4.1)
and making use of the substitutions -y to Y. -x to x and -x to x and
-y to y in the second. third and fourth integrals respectively.
(4.4.2)
8(11) _ 0(12) _ 8(21) + 0(22)
jj'h
jj'h
jj'h
jj'h
8
jj'h·
0000
JJ
=
I
{F[. j']h«-l)sX.(-l)S'y)
o0 s. s ' =0 • 1
J•
dF[j,]«-l)
00
00
=_L _L{F[j.j']h(X.y)
- F[j](X)F[j'](y) dF[j](X)}
dF(j'] (y)
and for h
= 0.
j
s'+l
= j' = 1 ••••• q.
it is equal to
y)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
96
00
0jj
={
00
F[j](x)(l-F[j](x»dF[j](-X)+ ~F[j](-X)(l-F[j](-X»
dF [j] (x)
00
.[' (s.s' ) • a,s ' 1
2
d e fi ne d after (4 ••
2 36) in t h e same
h
were
Ujj'h
.. "are
manner as in (4.2.15) - (4.2.19).
Hence for symmetrical population. Y , is as usual given by
jj
(402 036), where for every j.j' • 1.00 •• q. h
= 0.1.2 •••• ,
~j'h
and
0jj are given by (40402) and (4.403) respectively.
For each j
X~
~,
j
= Xi , j
= 1 •• 0•• q.
let us now consider the variables
= l,.o •• n
- a., i
J
where a. is a number chosen according
J
Le t us t a ke a
j
to our convenience.
where uj's are some fixed numberso
-1/
2.
u ' J
• n
j
= 1 •.•.• q.
Then the Wilcoxon signed rank
statistic (vector) based on the variables Xi. • i • 1 ••••• n.
j
j .. 1 •• 0.,q is given by
(4.404)
T (a)
""11 ,..,
..
(Tn l(a ) ••• o.Tnq (aq »'.•
l
where
00
(4.4.5)
T (a ) .. n(n+1) -1
nj j
f
o
H .(x-n
-1/2
nJ
uj)dF j(x). j .. 1 ••••• q.
n
Similar to (4.2.39). here we get
00
..
~
00
l 2
H[j](X)dF[j](X)'-n-1/2uj{HbJ(X)dFU] (x) + a (n- / )
f f[j](x)dx
2
00
1
.. 4 so that
n
-1/2
uj
_00
-1/2
+ o(n
)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
97
rn E{Tnj (aj )
(4.4.6)
co
i} - - uj_L f~j] (x) + 0(1)
-
If we denote the corresponding 0jj'h by ojj'h then by (4.4.2) and the
Taylor series expansion. we can write
(4.407) 0Jj'h
co! !co
-1/2
-1/2
-1/2
--co _~F[j.j']h(x-n
uj_y-n
uj)-F[j] (x-n
uj )
F[j,](y-n
=
0jj'h + O(n
-1/2
-1/2
uj,)}dF[j](x)dF[j](Y)
)
Hence
(4.408)
where
=
Ynjjl
(4.4.9) implies. as n
-+-
r
(4.4010)
"'n
Yjjl + O(n
-1/2
)
co.
-+-
r.
'"
Without any loss of generality we take the location parameter
to be Q and suppose
~o
value of the parameter.
denotes the probability when Q is the true
Then by a straight forward multivariate
extension of Theorem 4 of Hodges and Lehmann(1963). we have
(4.4.11)
n
lim
-+-
• <
co po{v'Ue
'"
n.J -
u •
j
j - 1 ..... q}
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
98
lim
n-+ co
=
p
1
vh"[T ,(a )- -] -
t
nJ
o.
'"
4
j
00
J
_co
f 2[ ,] (x)dx
J
By (404010) and the remark in the end of Theorem 402.1. as
n -+
co)
the right hand side of
(4~4~11)
converges to the codof. of a
multivariate normal distribution with mean vector
Q and dispersion
matrix
(404012)
1:** = «Lj*J~»'J. j'-l )0.0. q
where
T~~,
JJ
=
Y 'j'
J
Ie}
-co
2
f [,]
J
(X)d)(j
It_co
fij']
(X)dX)
Hence by (404.11) and the above arguments. the vector I:n~n is
asymptotically mu1tinorma11y distributed with mean vector
dispersion matrix
1:**.
where
1:**
Q and
is defined in (404012) and (4.4013)0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
CHAPTER V
COMPARISON OF THE
PERFO~~CES
OF SEVERAL
ESTIMATORS OF LOCATION FOR STATIONARY
AUTOREGRESSIVE PROCESSES
5.1.
Introduction
In this chapter we consider several rival estimators of
location parameter for stationary autoregressive processes and
compare their relative performances by their asymptotic relative
efficiencies.
In section 5.2, A.R.E. of the median (vector) with
respect to the mean (vector) for the general stationary multivariate autoregressive processes are considered.
The details of
this A.R.E. value for the univariate case are studied in section
5.3.
In section 5.3, the A.R.E. of the 27% mid-range estimator
(vector) with respect to the mean (vector) and the A.R.E. of the
median of the mid-range estimator (vector) with respect to the mean
(vector) for the general stationary multivariate autoregressive
processes are considered in a nut shell.
However, emphasis is given
on the study of the A.R.E. values for the univariate case.
In the
end of the section the A.R.E. values for univariate Gaussian autoregressive processes for the above estimators are tabulated.
We
conclude this chapter with a description of some related problems
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
l!)J
not considered in this dissertation and is expected to be studied
later.
5.2.
Asymptotic Relative Efficiency of the Median with Respect to
the Mean.
In this section first, for samples from general stationary
multivariate autoregressive processes the A.R.E. of the sample median
(vector) with respect to the sample mean (vector) is obtained and
then some bounds for the A.R.E. are worked out for some particular
cases.
The A.R.E. values for samples from univariate (normal) auto-
regressive processes are studied in the next section and some values
of it are also tabulated there.
Let X , t = 1,2, ... ,n be the sample observations (q x 1 vectors)
"'t
where ~t has the representation (2.2.7).
Here we assume (2.2.10) to
be true with some 0 _>2 and let us suppose that var(E ) = L =
"'t
«0' ESS ,»
-
S,S
-
I
=1 •.•. ,q
for all t and 0
n
I
-
(X l' ...• X
) , where X . = n
n,
n,q
n,J
In'
<
I"'EL I
<
00.
Define X =
LX., j = l, ... ,q and denote by
t=l t,J
bY~1/2'
the population median
vector.
By (2.2.7).we have
00
(5.2.2)
E(X )
"'t
=
E(X X' )
"'t....t+h
=
I
r=O
00
=
~rE<'~t-r)
E(I
r=O
I
=
Q,
B E
~r£t-r)(I
"'r"'t-r+h)
r=O
00
I
r=O s=o
B E(E
E'
)B
"'r "'t-r"'t-s+h "'s
00
=
I
r=O
",n
-1 '\
the sample median vector, and
(5.2.1)
"'E
B LB
....r"'E....r+h
h = 0,1, .•. ,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
101
and (5.2.1) and (5.2.2) imply
(5.2.3)
E(X )
=
~n
0
'"
and
(5.2.4)
E(X X')
= n-
If we write J(h) =
finiteness of
max
o ..
h=l j=l, ••• ,q JJ
1
fI
I
<
n
00
]
B L: B + 2 I (1-!!')
B L: Br+h
n I -I"""E:~
h=l
r=O
r=O
~I"""E:~r
~~E:~r+hl~ it
I
'"L:
(h)
I -<
(h)
q
II OJ'J , for
j=l
is sufficient to show that
Let us write
00.
I E(~t~~+h)
h=l t=l
'\L ~r~E:~
B L: ~r+h' then since
r=O
II
(h)
L
t=l
J
n-h
n
I E(~t~~) + 2 I
00
h=l r=O
00
[ n
= n-2
'-n""""'n
0
0
max
=
s= 1 , •.. ,q
0
E:SS.
Then by
Schwarz's inequality and (2.2.8),we have for any j = 1, ... ,q,
=
(5.2.5)
I [I ~ ~r)2 J s'=ll
I (b JS~~h)l20
,J
r=O s=ll Js 5 E:SS
5 E:S S
1 2
/ [
20
< q
1/2
,
00
C2 q 30
<
\'
/.,
0
r=O
so that
00
(5.2.6)
I
max
o~~)
h=l j=l, ... ,q JJ
which proves finiteness of
~ c 2q 400
I ~L ~L
h=l r=O
(5.2.7)
00
I
00
I
rg(r + h)g(e*)2r+h <
00,
h=l r=O
B LB
hi.
~r"'E:"'r+
Again by (5.2.5),
I ~n r=OI ~~E:~r+hl
Ih=l
<
~
h
ri o~~)
h=l n j=l JJ
n
00
I I
h=l r=O
hr g (r + h) g (e * ) 2r+h
~
0 as n
~
00,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
102
since the double series
co
co
I
I
hrg(r + h)g(e*)2ri-h
h=l r=O
converges for 0
nE(X
x')
-n-n
(5.2.8)
If ~ =
...
<
e*
Therefore as n ~
1.
<
~ [~I... ......Br . .~. e:.B. . .r
+ 2
=0
~I... ~I...
h=l r=O
co,
~
Br-e:......
Bri-hJ
......
= -]; , say.
«O""»j
., 1
then we assume that
JJ
,J = , ••• ,q
(5.2.9)
); is non-null.
By (2.2.8) it follows that
(5.2.10)
where
II ~r 11 2
an d
~
=
sup
11~112=1
is a q x 1 vector with
condi don
II ~ I 12
.. 1.
II ~~ II ,
__
L
11~112·
1~=lx2i}1/2
For our case let us take
~*
=
Q,
a q
x
and satisfying the
1 null vector, and Yt = 1
in the model (1.1.2), considered by Hannan (1961).
(c4) and (c5) conditions are also satisfied.
C.L.T.,
Inx-n
Then Hannan's (c3),
Hence by Hannan's (1961)
has asymptotically a multivariate normal distribution
with mean vector Q and dispersion matrix ;
Also by (2.4.2),
ten 1/2 [.tu -~1/2])
~ Nq(Q,:V,
as
n~
co,
where! is given by (2.4.1) with p(j) = 1/2 and
F[j](~(1~~=
p
for each j = 1, ..• ,q.
1/2
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
103
In the above model we have E(X )
""'!l
00
say. If however, ~ = k l / 2 then the
r=O
sample mean (vector) and the sample median (vector) estimates the
ME
then E(~n)
= "'"O.
= L ~~E = ~,
same parameter.
then
~
In particular if the c.d.f. G is symmetric about
= kl / 2 .
In that case the
A~R.E.
Q,
of the sample median (vector)
with respect to the sample mean (vector) is the ratio of the reciprocal
of their generalized variances as considered by Wilks.
definition of
A.R~E.
For a detailed
see chapter 6 of Puri and Sen (1971).
Hence the
A.R.E. of the sample median (vector) with respect to the sample mean
(vector) is given by
(5.2.11)
Now we use the following well-known result, known as Courant's
theorem.
If
~
and
~
be two square matrices of same order and B is non-
singular then
maximum eigenvalue of
sup
-1
(~§
)
a :f 0
and
minimum eigenvalue of (AB- l ).
inf
.e ;& Q
,.."...,
By the above theorem, the expression (5.2.11) is bounded below
and above by the q-th root of minimum and maximum eigenvalues of
We consider, now, some particular cases.
a)
We assume
E
.
t,)
's are independent for j
are diagonal matrices for r
= 0,1,2, ....
= l, ..• ,q
Then X
and B 's
""'r
. 's are also
t,)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
104
This gives us a .. , = 0 for j ; j' and also
independent.
for j :/: j'.
JJ
T •• ,
JJ
= 0
In this case the efficiency (5.2.11) is given by
q
{III/I:f1 }l/q
(5.2.12)
(ITa .• h .. )
j=l JJ JJ
1/q
.
Actually a .. /T .. correspond to the A.R.E. of sign test with respect to
JJ
JJ
t-test for the univariate case o Hence by an application of Hodges and
Lehmann's (1956) lower bound for the A,RoE o of sign test with respect to
t-test~the AoR.Eo~
as given by (5.2012) , is ~ 1/3.
1
'11
0\("11
U- 1 =
all
0
1'11
Again since
0
=
1 It'·
...!lS.
0
a
l'
0
-qq;
- /\ 0
qq
qq
which has eigenvalues a11/T11"'" aqq/1'qq' by Courant's theorem, the
a
A. R. E. (5.2.12) lies between the q-th root of minimum and maximum
= 0,1,2, ... ,
If .....
Br =Ar-q'
I
r
b)
where Ar 's are constants and g t,]. 's
are exchangeable random variables for j = 1, .. ,q, i.e.,
a
2
g
[(1 - p )1 + P J ) then we can write for h
E -q
E-q
~
~g
=
= 0,1,2, .•. ,
00
=
=
I
r=O
C1
00
2
\L [(1 - p E)B
B' + PE.....rPq.....r
B J B']
.....r .....r
r=O
E
00
=
C1
g
I·
2(
r=O
Hence in this case X
Ar Ar+h) [(1 - p E)1
~q
+ P g-q
J )
. 's are also exchangeable random variables for
t,J
j=l, ... ,q.
By (5.2.8) here we have
00
(5.2.13)
«a
k.
=
jj
,)) =
I
r=O
00
[B
L B' + 2
""r.....g....r
ag 2S [(1 - pE) lq + p g~q ) ,
I
~rlE}l~h)
h=1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
lJ)
where
S
_(~L
r=O
.2
A
r
+
2
oe-
00
'I
\'I
L
"
h=l r=O
The variables X . being also symmetrically distributed, from
t ,J
(2.3.23) and (2.4.1) we see that
1, ... ,q and
j
=
T. ••
JJ
1. _,
JJ
is same for all j
is also same for j = 1, ... ,q.
~
j' =
Here for p(j)
1/2,
1" .• ,q, let us write
for j
(5.2,14)
1
<
,
j'
,
JJ
for j
~
j'
and define the function
1 for F(x) .:..
J(F(x»
=
"21
f
0 otherwise,
Then we can write the correlation coefficient between the medians for
any two variates
Sen (1968 a)
by OJ = u/, and by Lemma 4.4 and Lemma 4.5 of
we get,
(5.2.15)
>
- (q - 1)
where the equality holds iff J(F(X»
-1
is a linear function of X with
probability one, in which case the distribution of X is degenerate,
Hence in the non-degenerate case we can write
(502.16)
>
-(q - 1)
-1
By use of (5.2.13) and (5.2.14), in this case the A.R.E. (5.2.11)
can be written as
(5.2.17)
{ifl/III} 1/ q = [
(02S)q{1+(q-1)P }(l-p )q-1J1/ q
e:
e:
_~
h + (q-1) ll) (r -jJ) q
= (0 2 SiT.)
e:
u
[
+ (q-l)p }l/q (l-p ) (q-1) /q
e:
e:
J
{1 + (q-1) p }l/q (l-p ) (q-1) /q
J
J
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
106
e and
We denote the efficiency (5.2.17) by
consider some particular
cases.
i)
Let p
""
e
O.
=
e:
Then
=
=
>
>
>
Thus in this case we get different lower bound of the efficiency
depending on the value of P •
J
11)
= -(q-l) -1 •
Let P
e:
Then as P
J
>
_(q_1)-1 by (5.2.17),
e=
0
so that our proposed procedure is not at all suitable for the
degenerate case P
iii)
-(q-l)
=
e:
-1
•
In general for given p" and -(q-l)
2 .
e >
(0
e:
< P
J
2
1, we have
{I + (q-l)p }l/q(l_p )(q-l)/q(q_l)(q-l)/q
S/r.)-
e:
E
q
(02 S/L) [(q-1) (q-1)!qU
=
-1
E
+ (q-l)p }l!q(l_p ) (q-l)!ql!q
e:
E
so that whether we have a larger or smaller bound of the efficiency
than in case (i) depends on the value of p .
e:
In this case· if the number of variables q increases
indefinitely and PJ
e=
~
Os then the efficiency
«q_l)-l + p }l/q(l _ p )(q-l)/q
(02 S/ T ) lim
E
q
-+
00
E
e:
J
J
{(q-l) -1 + p }ljq (1 _ p ) (q-l)!q
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
107
2
=
(aES/T)(l - PE)/(l - P )
>
(a
J
2
E
s/T) (1 - P )
E:
2
and this lower bound is (a S!T) if P
E
E
~ O.
Hence if the number of
variables increases indefinitely, the efficiency can be bounded
below by a quantity which tends to a fixed limit irrespective of the
c)
If X has a q-variate normal distribution then X . and X .,
t ,J
~t
t,J
have jointly a bivariate normal distribution.
correlation coefficient between X
0
t,J
correlation coefficient between X
If we write P
jj
and X ., and Ph'"
t ,]
JJ
. and X + ."
t h ,J
, =
=
then using the
td
relations
(See Cramer (1946), p. 290) we have the following simplifications.
In fact then
(
\
V •• ,
JJ
~n
[Sin
-1
I 1
lZ;
p,.
JJ
=1
00
00
,+
\"
Sin
L
h=l
00
[2!. + 2
2
)'
Sin
L
-1
h=l
-1
Phjj , +
P
] , for j
hjj
\/.,
Sin
-1
Ph' , .] ,for j;t=j'
J J
h=l
j'
and
=
1
where Oj2 • var(X
f27T cr.
.)
t,]
J
5,3,
Asymptotic Relative Efficiency for Univariate Stationary
Autoregressive Processes
For univariate autoregressive processes of order k, we can
simplify the results quite considerably
can write the observation X as
t
0
In this particular case, we
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
108
I
r=O
where b
(2,206»
(r)
=
k
\'
L.
r
y .e .•
i=l 1. 1.
are the roots of the auxiliary equation of
=
(5.3.2)
and yi's are constants,
max
e* = l<i<k
<
we can write
e* =
e -p* ,where p*
'>
O.
Then
=
where A is a finite constant independent of r.
This shows that in
this case the assumption (2.2.8) is obviously satisfied.
Here all
the other results can be obtained much more easily due to the
simplicity of the expressions (503,1) and (5.3.3),
But we shall
study, in particular, the details of the A.R.E. values,
Let
Xt~
t =
l"o.~n
be the sample observations where X has
t
2
the representation (503,1), We assume var(€t) = 0 for all t and
Define X = n
n
-1
n
I
X and denote by t , the sample
n
t=l t
median and by \/2' the population median.
Since X has the representation (5.3.1), we have
t
E(X )
t
var(X )
t
=
0
00
=
\' b (r\ (s)E(€
L
t-r €t-s) =
r,s=O
0
00
2
I
r=O
00
cov(Xt,X t +h )
which imply
=
I
b (r)b (s)E(€
r~s"O
=
t-r €t+h-s)
0
2
b
(r)2
00
I
r=O
b (r)b (r+h)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
109
=
E(X)
n
0
and
(5.3.4)
n var(X )
n
as n
-?- "",
since
n
I!!.n ""I
h=l
b ( r \ (r+h)
r=O
=
* n+1 (1 - e*) -1 ]
-- n(e)
-?-
0 as n
-?- "",
By (50303) the right hand side of (5.304) is of the order
(1 + e*)(l - e*)-1{1_(e*)2}-1 which is non-nUll for 0
<
e*
<
1.
<
e*
<
1.
Also by (5 0303),
""
L Ib(r)1
r=O
Now taking B*
~ A
=0
""
L
(e*)r = A(l - e*)-l
for 0
and Yt = 1 in the model (1.1.2), considered by
Hannan (196l),we see that his (c3),
satisfied.
< "",
r=O
(c4)~
and (c5) conditions are also
Hence by Hannan's (1961) Central Limit Theorem, ~X is
asymptotically normal with mean
2 ""L ""L b(r)b(r+h)]o
n
o
""
2
and variance cr 2 [ L b(r) +
r=O
h=l r=O
Also by (2.4.2), I:n(t n - ~1/2) is asymptotically normal with
2 2
mean 0 and variance v If (~1/2)' where
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
110
(5.3.5)
\)
2
In the above model we have E(X )
n
= 0.
Instead of
0, if
00
then E(X ) = L b (r)].IE: = ].I, say. I f however, ].I = 1;;1/2
n
r=O
then the sample mean and the sample median estimate the same
E(E: t )
= ].IE:
parameter.
In particular if the c.d.foG is symmetric about 0, then
= ~1/2'
In that case the A.R.E. of the sample median with respect
].I
to the sample mean is given by
(5.3.6)
eff(t/X)
=
00
If the underlying parent distribution'is normal then f(~1/2)
/
..,rr-:{;)2)-1
1-1
~
I_b'
and Fh(~1/2'~1/2) = 4+ (Sin Ph )/2TI (See Cramer
r=O
(1946), p. 290) where Ph is the correlation coefficient between X
t
or
and X +h .
t
Hence for a univariate Gaussian autoregressive process
the A.R.E. of the sample median with respect to the sample mean is
given by
00
1 + 2
(5.3.7)
eN (t/X)
=
(2/TI)
I Ph
h=l
00
1 + (4/TI)
I Sin
h=l
-1
Ph
For first order Gaussian autoregressive processes where
=
lal < 1 and E:
t
aX t- 1+E:,
t
are independent normal variables with mean 0 and
variance-0 2 , we have Ph = a h and b(r)
e
lN
= a r . Hence if we denote by
, the A.R.E. of the sample median with respect to the sample mean
for a first order normal autoregressive process then from (5.3.7) we
=
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
111
00
get.
[1 + 2
e
(503.9)
lN
=
h=1
(2/7r)
00
1 + (4/rr)
=
ah )
I
Sin- 1 (ah )
I
h=1
2[1 + 2a(1 - a)-l]
00
rr + 4
I
h=l
Sin-l(ah )
00
=
2(1 + a)(l - a)-I{rr + 4
I
Sin- 1 (ah )}-1
h=l
We now observe some limiting values and bounds of the effici.ency
, First. the series L Sin-l(ah ) being uniformly convergent for
1N
h=l
lal < 1. e -+2lrr as a-->-O. Secondly. using the infinite series expanlN
h and summing term by term, we get
sian for Sin -1 (a)
e
I
00
(5.3.10)
-1
h
Sin (a ) = a(l-a)-l[l + (a 2 /6)(1 + a + a 2 )-1
h=l
+ ".]
which consequently imply as a -+ 1,
(503.11)
As a -+ -1. the expression (5.3.10) also implies that e
(5.3.12)
-1
x
~
e 1N ~ (2/rr) (1 + a) (1 - a)
-1
by using the inequality (2/rr) Sin
x for x
{l
+ 2
~
'i'
L
lN
-+ O.
Again
O. we get for a
=
2/rr
h=l
so that the efficiency is bounded below by (2/rr) for non-negative
values of a.
~
0,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
112
For a second order Gaussian autoregressive process where
X + a X_ + a X_ =
2 t 2
1 t 1
t
E
t
, aI' a
2
are constants and
normal variables with mean 0 and variance
i)
Case of unequal roots:
If e
1
(-1
e
<
0
1
2
<
E
t
are independent
, two cases may arise.
1) and e
are the distinct roots of the auxiliary equation x
2
2
(-1
<
e
2
(5.3.13)
(r)
00
(5.3.14)
I
b (r)
=
(e
- e )
2
1
-1
(e
r+1
r+1
- e
).
1
Z
Therefore
z
r=O
=
=
-1 - e (1-e e ) -1 }]
+ e 2{e 2 (1-e Z
)
1
1 2
Z
-1
Z
-1
-1
(e -e ) [e (1-e ) (1-e e )
1
2
1
1
1 Z
Z -1 (1-e e ) -1 ]
1 Z
- e (l-e )
Z
Z
..
and
=
Z -1 (l-e Z) -1 (1-e e ) -1
1 Z
(1+e e ) (1-e )
1
1 Z
Z
1)
+ a 1x + a = 0,
2
then we can write
which gives us b
<
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
113
=
2 -1)
- e 2 (1-e) -1 (1-e)
222
=
Hence if we denote by e
2N
, the A.RoE c of the sample median with
respect to the sample mean for a second order Gaussian autoregressive
process with unequal roots of auxiliary equation, then by (5,3.14)
and (5.3.15) we get,
00
TI
+ 4
\'
Sin
L
-1
Ph
h=l
where
(5.3.17)
Since the series
\'
L
Sin-1 Ph is uniformly convergent for
h=l
le11 <1, le21 < 1 and ph-+-O for e -+-O, e 2 -+-0 with e 1;'e 2 , e 2N 4- (2/TI)
1
as both e -+-0 and e -+-0 with the restriction e1;'e2o If Ph is
2
1
given by (5.3017) then
Sin- 1P = (e1-ez)-1(1+e1e2)-1{ei(1-e;)(1-e1)-1-e;(1-ei)(1-e2)-1}
h
h=1
-3
-3 6
2 3
3 -1
+ (1/6) (e -e ) (1+e e ) {e (1-e ) (1-e )
1 2
1
2
1
1 2
f
6
2 3
3 -1
- e (1-e ) (1-e) } +
Z
1
2
and as e
2
-+
•
0
•
0,
... ]
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
114
which is exactly similar to (5.3.10).
Therefore from above and
(503.16) we conclude that the efficiency e
same conclusion can be arrived at for e
1
2N
~
00
~
e
1N
as e
2
~
00
The
This means that the
A,ReE. of the second order Gaussian autoregressive process con. verges to
tha~
of first order process if anyone of the roots of the
auxiliary equation converges to zero.
for e
1
~
0, e
2
~
0 and e
- e
1
2
O.
>
Again here Ph
~
0 for all h,
Hence, similar to (5.3.12), for
2
= -TI
(5,3.18)
so that the A.R.E. e
of e
i)
l
and e
2
2N
is bounded below by (2/TI) if both the values
are non-negative and e
Case of equal roots:
the auxiliary equation x
If e(-l
2
+ alx +a
<
2
> e •
l
2
e
<
1) is the double root of
= 0, then we can write
=
r
which gives us' b (r) = (r+1)e for r = 0,1"0'
00
(5.3.20)
I
r=O
b(r)
2
00
=
=
=
and
I (r+l)
r=O
2
e
2r
{r(r-l)+r}e
00
=
I
r=l
2r-2
Therefore,
2 2r-2
r e
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
115
00
(5,3 21)
0
I
00
00
I
h=l r=O
00
I
b (r\ (r+h) =
I
(r+l) (r+h+l)e 2 r+h
h=l r=O
00
=
00
I I
h=1
r(r+h)e
2r+h-2
r=1
00
=
00
I I
h=1
2
{r(r-l) + (h+l)r }e r+h-2
r=1
00
=
I
h=l
00
I
=
{2e 2 (l_e2) -3 + (h+l) (l_e 2 )-2}eh
{(1+e 2 ) + h(1-e 2 )} (1_e 2 )-3 eh
h=l
2 -3
+ e (He) (I-e) -1 (I-e)
Hence if we denote by e;N t the AoRoE, of the sample median with
respect to the sample mean for a second order Gaussian autoregressive
process with equal roots of auxiliary equation. then by use of
(5,3.20) and (5.3.21) in (5,3.7) we get
00
TI
+ 4
\'
L
Sin
-1
Ph
h=l
where
(503.23)
Let us noW observe some limiting values and bounds of the
efficiencye;N'
First. we see that by (5.3,23).P
h
+
0 as e
+
0 and
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
116
00
\L Sin-1 Ph is uniformly convergent for
the series
<
Hence
I,
h=l
as e
-+
O.
e*
2N
(5,3,24)
-+
(2/7T)
Secondly. using the infinite series expansion for Sin
-1
Ph with Ph
given by (5,3,23) and summing up term by term. we get
00
(5,3 0Z5)
2
(1 -" e) (1 + e)
\
L
-1
Sin Ph
h=l
+ 3(1+e)Z(1+e 3 ) (1+e 2 )-1(1+e+e Z)-3
+ (1+e)3(1+4e 3+e 6 )(1+e Z)-Z(1+e+e Z)-4}
+
which consequently imply as a
-+
1.
= 4/40493
=
As e
-+
Again e
,890
-1. the expression (5,3,17) also implies that e~N
~
0 implies Ph
here we get for e
I
I
I
I
IPh I
~
~
0 for all h,
-+
0,
Therefore similar to (5,301Z)
O.
7T(1 + Z
I
Ph)
h=l
so that the efficiency e~N is bounded below by (2/7T) for non-negative
values of e,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
117
For visualizing the nature of the efficiencies in details; the
values of the efficiencies e lN , e;N and e
are tabulated at the end
2N
of this section (TABLE 5.3.1) for values of the roots lying between
-.9 and .9.
For independent Gaussian processes the 27% mid-range estimate
being optimum among the class of mid-range estimates,various authors
have compared'the performance of it with that of other similar
statistics.
Here for autoregressive processes we want to compare the
performance' of the 27% mid-range estimate with that of the process average.
For a multivariate autoregressive process if the parent distribution'is symmetrical then the 27% mid-range estimate (vector)
for the location
U (j) = (Z (j)
n
n p (j )
, 1
p~rameter
+
Z (j )
n p (j)
' 2
is £n
) /2
,
= 1,,(1)
n
y.J
, •••
,U
(q) i
). where
n
J' = l, ••• ,q, with P{j) = .27,
By section 3.4, ~ U has a multi"'n
variate normal distribution with mean
and dispersion matrix
where
(5.3.29)
=
Q (population
median vector)
I
I
I
I
I
I
I
I
I
118
(' • I )
where ViI~
Hence in the same situation described earlier in connection with
AoRoE o of the sample median (vector) with respect to the sample mean
(vector)p the AoR.E. of the 27% mid-range estimate (vector) with
respect to the sample mean (vector) for q-variate stationary autoregressive processes is given by
{if 1/1.1:* I}1/ q
As in section 5 02, here also we can have some bounds of the efficiency
(503030) and we
can consider those particular cases o However, to
avoid repetition we omit the details o
I
I
I
are given by (30309)0
For univariate autoregressive processes if the parent distribution is symmetrical then the 27% mid-range estimate for the location
parameter is U = (2
+ 2
)/2 with PI = 027, P2 = 073 0 By
n
n,Pl
n,P2
section 304, In U is asymptotically normal with mean 0 (population
n
median) and its variance is
I
I
I
I
I
I
I
where
~
PI
(5 03 032)
and ~p
are the 27% and 73% quantiles respectively and
Z
vII = PI (1 - PI) + 2
ao
v Z2 = PZ(l - PZ) + 2
L {Fh(~P '~p
h=l
2
ao
) - p;}
Z
I
I
I
I
I
I
I
I
I
I
I
·1
I
I
I
I
I
I
I
119
Using the facts of symmetry the expression {50303l) can be simplified
quite' a lot o
First we see that for symmetrical distributions
) = f{~ )0 Again the variances'ofthe sample quantiles of
PI
P2
lower 27% and' upper-27% being equal for such distributions we must
f{~
have v
11
= v 220 Using the above facts the expression (503.3l) can be
simplified to
Hence in the situation described earlier. the AoR.E. of the 27% mid-range
estimate with respect to the sample mean for univariate stationary autoregressive processes with symmetric distribution is given by
(5.3.36)
where vII and \)12 are given by {5.3032) and (503034) respectively with
PI
= 027 and P2 = 0730
If'the underl in
and f(~
) =
(
arent distributiop is normal.~' = -.6128
2)-1
PI
b{r)
exp{- ;<'6l28)2}. Hence,
L
00
I2'TI
Pl
r=O
for a univariate Gaussian auto.regressive process, the AoRoEo of the 27%
mid-range estimate with respect to the sample mean is given by
00
~(U/X)
=
~
(1+2
Ph)exP{-{06l28)2}/~{Vll + \)12)
h=l
where
00
Ph
•
I
r=O
00
I
r=O
brbr+h l
b
2
r
00
V
11
(5.3.40)
=
01971
I
+2
{~h(-06128.-o6128) - .0729}
h=l
=
00
+ 2
I
h=l
{~h(-06128~o6128) -01971}
I
I
I
I
I
I
I
I
I
I,
I
I
I
I
I
I
I
I
I
120
and
~h(x,y)
is the codof, of standard bivariate normal distribution
with correlation coefficientPho
For studying the nature of the efficiencies in details, the
values of the efficiency
(5~3037)
for-first and second order Gaussian
autoregressive processes are tabulated at the end of this section
(TABLE 50302) for values of-the roots lying between -09 and .90
If the-parent distribution is symmetrical about its location,
the-median of the mid-ranges provide another alternative estimator
for the location parameter o
The asymptotic distribution of the
median of the mid-range estimator is-studied in chapter 40
We now
want to compare its performance with that of the process average.
For a multivariate autoregressive process if the parent distribution- is symmetrical then the median of the mid-range estimator
(vector) for the location parameter is
where
Ine"'n
e (j)
=
med
{t(x t .+ Xt ' ,)}, for j
l<t<t'<n
,J
,J
=
n
e
"'0
has multivariate normal distribution with mean
Q (population
median vector) and dispersion matrix
'1**
=
**
«'r JJ
.. , »j ,J., --1 '000' q
where
**
T •• ,
JJ
where for h
= 0,1,2'000;
=0
=
j,j'
= l,ooo,q, C. "h is given by (40402)
JJ
= l,ooo,q,
C.. is given by (40403)0
JJ
Hence in the same situation described earlier in connection with
and for h
and j
j'
AoRoE, of the sample median (vector) with respect to the sample mean
(vector), the A,R.E o of the median of the mid-range vector with respect
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
121
to the sample mean vector for q-variate stationary autoregressive
processes is given by
Similar to section 5.2. here also we can have some bounds of the
efficiency (5.3.43) and we can consider those particular cases.
However~
to avoid repetition we omit the details.
For univariate autoregressive processes with symmetrical
distribution, the median of the mid-range estimate for the location
~
1
~
parameter is e = med
{'2(X + X I)}. By section 4.4,
e is
n
l<t<t'<n
t
t
n
rn
asymptotically normally distributed with mean 0 (population median)
and variance
**2
T
(5.3044)
where for h = 1,2, •• "
oh
(5.03.45)
00
=
(1 + 24
I
h=l
00
0h)/12(J f 2 (x)dx)2
_00
00 00
=
f J{Fh (x,y)
- F(x)F(y)}dF(x)dF(y)
_00_00
hence in the same situation described earlier in connection with
A.R.E. of the sample median with respect to the sample mean. the A.R.E.of
the median of the mid-range estimate with respect to the sample mean for
univariate stationary autoregressive processes is given by
00
20000
00
(5.3.46)
eff(S!X) = 12a 2 (
b(r) +2
b(r)b(r+h»(! f 2 (x)dx)2
r=O
h=l r=O
_00
00
I
ch
where
I
I
is given by (5,3.45).
If the underlying parent distribution is normal then
(21TIa )-1 and
x
00
J
_00
2
f (x)dx =
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
0h =
lZZ
00
00
f
!{CPh(X,y) - tP(x)tP(y)}dcp(x)dtP(y)
_00_00
where
~(x)
is the c.dof o of standard normal distribution and
~h(x,y)
is the c,d.f. of standard bivariate normal distribution with correlation coefficient Pho
By Kendall (1948, p 118) and integration by
-1
parts, 0h = (l/Z1:) sin -(PhiZ).
Consequently the expression (5.3.44)
is simplified to
00
2
,**2 = (nox /3) [1 +(lZ/n)
(503,48)
I
h=l
Sin- 1 (P /Z)],
h
Hence for a univariate Gaussian autoregressive process the
~median
AoR.~af
of the mid-ranges with respect to the sample mean is given by
~
~
l
eN(slx) = (3/n)(l + ZI P ){l+(lZ/n) L Sin- (P /2)}-1
h
h
h=l
h=l
(503049)
where
=
If we denote by E
1N
• the A.R,E, of the median of the mid-ranges
with respect to the sample mean for- a first
order Gaussian autoregressive
process. then similar to (5,3,9). here we get
00
(5,3.51)
=
3(1 + a)(l - a)-l{n + 12
L
Sin- 1 (ah /2)}
h=l
Similar to the case of A,F,E o of the sample mean with respect to the sample
median for a univariate
first order Gaussian autoregressive process.
here also E + 3/n as a + 0, E +0 as a + -1 and E +
1N
1N
1N
1
3
-1
(1 + 72 + 3200 + ".)
=. 985 as a+ L
In this case. from TABLE
5.3.3 9 we see that for a
~
.5. the efficiency E
is very close to its
1N
limiting value.
Similar to
(5,3~16)~
the efficiency· E of the median of the midZN
ranges with respect to the sample mean for a second order Gaussian
I
I
'I
I
I
I
I
I
I
I
,
I
I
I
I
I
I
I
I
123
autoregressive process with unequal roots of auxiliary equation is
given by
3[1 +
QO
TI
\'
L Sin-1 (P h /2)
+ 12
h=l
where Ph is given by (5 3.17).
Here also E
2N
0
and e
Z
0 with the restriction e
+
1
+
3/TI as both e
1
+
0
# e Z"
Similar to (5.3 22). the efficiency E;N for the case of equal
0
roots of auxiliary equation is given by
=
3 [1 + Ze (Z + e +
QO
'IT
+ 12 /.,\' Sin-1 (PhIZ)
h=l
where Ph is given by (5,3,23),
Similar to the corresponding case of
AoRoE. of the sample mean with respect to the sample median. here also
E*
ZN
+
+
3 I TI as e
-1
000)
.
=
+
* . 0 as e
0) E2N'T
0979 as e
+
10
+
-1 an d E*
ZN
+
In fact. from TABLE
(1'"1"7'2+
1. 162
1 +3Z000
3
5.3.3~
we see that
for e ~ oZ) the efficiency E;N is very close to its limiting va1ue o
For studying the nature of the efficiencies in details. the
values· of the efficiencies E )E
and E;N are tabulated in
2N
lN
TABLES.3.3 for values of the roots lying between -.9 and .9.
---.-----~-----_.--
TABLE 50301,
A.R.E, of Sample Median/Sample Mean for
First and Second Order Gaussian Autoregressive Processes
-,9
-.8
-.7
-06
-.5
-.4
-,3
-.2
-.1
0
,1
.2
.3
,4
.5
,6
.7
,8
.9
-.9 1.0031
- ..8
.OU].022]
-.7
,019 .034 ].051[
-.6
0029 .049 .070 1.0941
-,5
0041 .067 .093 .116 [i531
-,4
,056 ,077 .120 .150 ,189 ~
-.3,074 c101 .151 .190 .230 c280 \.298\
-.2
.095 .143 .187 .230 .276 .331 .373!.427l
-.1
.132 .179 .230 .279 .326 .387 ,430 .482 ~
,152-t .221+ .280+ ,335+ .387+ .438+ ,490+ .540+ ,589+ 637 II
o
Il
.1
.189 ,272 .338 .397 .451 .503 .552 .597 .641 .681+1.6971
.2
.235 .332 .405 .467 .522 .571 .615 .655 0691 .724+ 0752\.7781
.3
.290 ,401 .481 .544 .596 ,639 .677 .709 .738 .762+ 0783 ,802 ~
.4
.358 .482 .564 .639 .670 ,707 0736 .760 .780 .797+.724 .722 0853 10957\
05
.6
.438 .573 .653 .706 .743 .770 .790 .806 .818 .828+.835 .841 .847 .851 !.856l
+
,~
.534 .671 .742 .783 .809 .846 .738 .846 .852 .855 .857 .859 .860 ,.860 ,.861 ~621
.7
.646 .772 .825 .851 .865 .873 .877 .879 .879 .878+ .876 .875 0872 .870 .868 .866 1.865[
.8
.773 .865 .893 .903 .907 ,906 .905 .903 .900 .897+.893 .890 ,792 .776 ,876 .872 .868 1.866\
.9
.899 .931 .934 .931 .928 .925 ,921 .918 .914 .911+ .907 .904 .900 ,895 .890 .885 .879 .871 k866l
---+ First order process.
c==J---+ Second order process
+
Ii
If ~
Independent process.
f-'
with equal roots.
N
.p-
_
---.---~e_-_-
TABLE 503.2.
A.R.E. of 27% Mid-Range/Sample Mean for
First and Second Order Gaussian Autoregressive Processes
-,9
-.8
-,7
-,6
-.5
-.4
-.3
-.2
-.1
0
-.9
j.003!
-.8
,012 1.0351
-.7
.024 .0541,0811
-.6
.039 .078 .114~158l
-.5
.058 .108 .155 .210 1.2701
-.4
.080 .144 .204 .269 .335 ].4011
-.3
,108 .189 .261 .333 .402 ,467 ].5291
-.2
.142 .242 ,325 .401 .469 .530 .587 1.6421
-.1
.184 .304 .395 .470 ,535 .592 .644 .692 J.7361
. +~
.235+ .374+ .468+ .541+ .601+ ,652+ ,698+ .739+.776~
o
,I
,2
,3
,4
,5
06
.7
.B
.9
.1
.297 .450 .542 .610 .664 .709 .748 .782 .812 0838+1.8601
.2
,370 ,529 ,617 .678 .724 .762 .794 ,821 ,844 ,863+,880 1.8951
,3
.4
,5
.453 .610 .689 .742 .780 ,811 .835 ,855 ,872 .886+,897 .908 \. 917 1
.544 .690 ,757 .800 .831 .854 ,871 .885 ,896 .905+ ,913 ,919 .925 1.931l
.638 .765 .820 .853 .875 ,890 ,901 .910 .916 .922+ .926 .930 .934 .937 1.941[
.6
,732 .833 .875 .897 .911 .920 .926 ,931 .934 .936+.938 .940 .941 .943 ,945 10947\
.7
.819 .893 .920 .933 .940 .944 .946 .947 .948 .949+ .949 .949 .949 .949 .949 .949 ~
.8
.897 .941 .954 .959 .960 .961 .961 .960 .960 .959+ .959 .958 .957 ,955 .954 ,953 .951 1.951\
.9
.973 .986 .987 .987 .986 .985 .984 .983 .982 .981+ .980 .979 .977 .976 .974 .972 .970 .970 1.9921
+
~First
order process.
r===J~ Second order process with equal roots.
Ii
11- Independent
process.
.....
N
\JI
------------------TABLE 5,3.3.
A.R.E. of Median of the Mid-Ranges/Sample Mean for
First and Second Order Gaussian Autoregressive Processes
-.9
-.9
-.8
-07
-.6
-.5
-.4
.827 j.859[
-.7
.849 .855 J.8501.
-.6
,853 .852 .846 1.8421
-.5
.855 .850 .844 .841 1.842l
-.4
-.3
0855 .850 .845 .844 .847 j.8541
-01
o
-.2
-.1
0
.1
.2
.3
.4
.5
06
.7
.8
.9
~
-.8
-.2
-.3
0857 .852 .849 .850 .855 .865 ~
.859 .856 .856 .860 .868 .879 .892 ~
.862 .862 .866 .873 .882 .894 .907 .921 1.9341
.867+ .871+ .878+ .888+ .899+ .911+ .923+ .935 + .946 .955
"'I!
Il
.1
.873 .883 .893 .904 .915 .927 .938 ,947 .956 o963~J.9681
.2
.881 .896 .909 .921 .932 .942 .951 0958 .964 0969+ .973 ~
.3
.891 .910 .925 0937 0947 0955 .962 0967 .971 0974+ ,976 .978 1.9791
.4
0902 .925 .941 0952 .961 ,967 .971 ,974 .976 .978+.979 .979 .979 1.980l
.5
.915 .941 .956 .966 .972 0976 ,978 0980 .980 .981+ .981 .980 .980 0980 ].980l
.6
.929 .956 .969 .976 .980 ,982 .983 .983 .983 .983+.982 .981 .981 0980 0980 1.9791
.7
.946 .971 .980 .984 .986 0986 .986 .985 .985 .984+.983 .982 .981 .981 .980 .979 ].979[
.8
.965 .983 .988 .989 .988 .988 .987 .986 .986 .985+.984 .983 .982 .982 .981 .980 .979 J0979[
.9
.985 .990 0990 .989 .988 .987 .987 .986 .986 .985+ .985 .984 0984 .983 .982 .981 .981 .979 ~
+
~
First order process.
c===J ~
Second order process with equal roots
]1
Independent process
II
~
......
N
0\
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
127
54,
Proposed TopiCS of Future Research,
In this dissertation we have solved several problems in connection
with stationary autoregressive processes.
However, there are still some
other problems in connection with it which we did not consider here.
In connection with a stationary autoregressive process the problems
which come to our mind immediately and which are not considered in this
dissertation are the following,
i)
For stationary autoregressive processes we have derived in Chapter III,
the asymptotic distribution of linear combination of a fixed number of
order statistics but we did not consider the problem of finding the asymptotic distribution of a linear combination of all order statistics or a
subset of that,
This problem includes the derivation of asymptotic dis-
tribution of Trimmed mean, Winsorized mean, etc.
for stationary auto-
regressive processes,
ii)
For stationary autoregressive processes we have developed in Chap-
ter IV, the asymptotic distribution of Wilcoxon scores which is a particu1ar case of Hoeffding's (1948) U-statistics.
The problem of deriving
the asymptotic distribution for the more general form of U-statistics
for stationary autoregressive processes remains still open.
iii)
In Chapter IV, we have considered only a particular case of U-sta-
tistics, namely the Wilcoxon scores,
Wilcoxon scores can also be ex-
pressed as a particular case of rank statistics of the following type:
R
n
T
(5.4.1)
n
=
L
i=1
.
J (~) Sgn(X,)
n
n+1
1
where Sgn(X.) is 1 or 0 according as X. is >0 or
1
Ixil among Ix11 "",Ixnl and
1
In(n~l)'
1
~
i
~
~O,
R
. is the rank of
n,1
n, are suitable rank scores.
Such statistics are knwn as Chernoff-Savage statistics,
Hence for
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
128
stationary autoregressive processes similar extension, as referred in
(ii), is also possible for more general form of Chernoff-Savage
statistics, defined in (50401)0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
,
REFERENCES
[1]
BAHADDR, R. R. (1966).
Ann. Math, Statist.
[2]
A note on quantiles in large samples.
12,
BILLINGSLEY, P, (1968).
577-580,
Convergence of Probability Measures.
John Wiley, New York,
[3]
CHERNOFF, H, and SAVAGE, I.R. (1958).
Asymptotic normality
and efficiency of certain nonparametric test statistics.
Ann, Math. Statist.
[4]
CRAMER, Ho (1946).
~,
972-994.
Mathematical Methods of Statistics.
Princeton
Dniv, Presso
[5]
DOOB, J. L, (1953),
Stochastic Processes.
[6]
EICKER, F.
Limit theorems for
(1965).
and dependent errors.
John Wiley, New York.
regre~sions
with unequal
Proc. 5-th Berkeley Symposium. Vol
I,
59-82,
[7]
GNEDENKO, B. V. and KOLMOGOROV, Ao No (1967).
for Sums of Independent Random Variables.
Limit Distributions
Addison-Wesley Pub-
lishing Company, California.
[8]
GRENANDER, D, and ROSENBLATT, M. (1957).
,
[9]
Stationary Time Series.
HAJEK, J. (1968).
John Wiley, New
HANNAN, E. J. (1961),
regressions.
[11]
Yor~.
Asymptotic normality of sample linear rank
statistics under alternatives.
[10]
Statistical Analysis of
Ann. Math Statist.
Proe. Cambridge Philos. Soc. Vol
324-335.
~,
583-588.
The efficiency of
some nonparametric competitors of the t-test.
12,
325-346.
A central limit theorem for systems of
HODGES, J, L. Jr. and LEHMANN, E. L. (1956).
Statist,
~,
Ann. Math.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
130
[12]
HODGES, J. L. Jr, and LEHMANN, E. L. (1963)0
Estimates of loca-
I
tion based on rank tests.
[13]
HOEFFDING, Wo (1948)0
Anno Mathe Statisto
HOEFFDING, Wo and ROBBINS, Ho (1948)0
for dependent random variables,
[15]
598-6110
A class of statistics with asymptotically
normal distributione
[14]
li,
Anno Mathe Statisto
KENDALL, Mo G. (1948)0
~,
293-325.
The central limit theorem
Duke Math, J, 15, 773-780,
Rank Correlation Methodso
Charles
Griffin & Company, Londono
[16]
quan~ileso
[17]
On Bahadur's representation of sample
KIEFER, Jo (1967)0
Anno Matho Statisto 38, 1323-13420
KIEFER, J. (1970),
Deviations between the sample quantile pro-
cess and the sample DFo
Inference,
[18]
Nonparametric Techniques in Statistical
299-319, Cambridge University Press.
"
LOEVE,
Mo (1963)0
Probability Theory (3rd Edition) Van Nostrand,
Princeton.
[19]
On some useful lIinefficient ll statistics.
MOSTELLER, Fo (1946)0
!I,
Anno Math, Statisto
[20]
PURl, M. Lo and SEN, P. Ko (1971)0
Multivariate Analysis.
[21]
371-4080
ROSENBLATT, Me (1956a)0
Nonparametric Methods in
John Wiley, New York,
On the estimation of regression coeffi-
cients of a vector-valued time series with a stationary residual.
lI,
Anno Math. Statisto
[22]
ROSENBLATT, M. (1956b).
mixing condition.
[23]
99-121.
A central limit theorem and a strong
Praco Nato Acd o SCo Volo 42, 43-470
SARHAN, A. Eo and GREENBERG, Bo Go (1962).
Statisticso
John Wiley, New York o
Contribution to Order
I
I
131
[24]
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
SEN, P. Ko (1961).
On some properties of the asymptotic variance
of the sample quantiles and mid-ranges 0 JRSSB Vol Q, 453459.
[25]
SEN, Po Ko (1963).
On the properties of U-statistics when the
observations are not independent; part oneo
Assoc. BulL
[26]
~,
SEN, Po K. (l968a) ,
two-way layouts.
[27]
SEN, Po Ko (1968b) ,
69-920
On a class of aligned rank order tests in
Anno Math. Statist,
39, 1115-1124.
Asymptotic normality of sample quanti1es for
m-dependent processes Anno Math, Statist,
0
[28]
Cal Statist.
SEN, P. K. and GHOSH, M. (1971).
~,
1724-1730,
On bounded length sequential
confidence intervals based on one-sample rank order statistics.
Ann. Matho Statisto 41, 189-203,
[29]
USPENSKY, Jo Vo (1937).
Introduction to Mathematical Probability.
McGraw Hill Book Company, New Yorko
[30]
WHITTLE, P. (1953).
serieso JRSSB
~,
The analysis of multiple stationary time
125-1390