•
..
A CHERNOFF-SAVAGE REPRESENTATION OF RANK ORDER STATISTICS
FOR STATIONARY ¢-MIXING PROCESSES
By
PRANAB KUMAR SEN AND MALAY GHOSH
Department of Biostatistics, U. N. C., Chapel Hill
and
Indian Statistical Institute, Calcutta
Institute of Statistics Mimeo Series No. 862
<,
FEBRUARY
1973
A CHERNOFF-SAVAGE REPRESENTATION OF RANK ORDER STATISTICS
FOR STATIONARY ¢-MIXING PROCESSES
•
By
..
PRANAB KUMAR SEN
l
AND MALAY GHOSH
University of North Carolina, Chapel Hill,
and Indian Statistical Institute, Calcutta
ABSTRACT
For stationary ¢-mixing processes, a Chernoff-Savage representation of a
general class of rank order statistics is considered and suitable orders (in
probability or almost surely) of the remainder terms are specified.
These are
then utilized in proving the asymptotic normality, weak convergence to Brownian
motions and Strassen-type almost sure invariance principles for these rank
statistics.
The law of iterated logarithm for these statistics is also
established.
l)Part of the work was completed while the author was visiting the Indian
Statistical Institute, Calcutta, and partly supported by the Aerospace
Research Laboratories, Air Force Systems Command, U. S. Air Force, Contract
No. F336l5-7l-C-1927. Reproduction in whole or in part permitted for any
purpose of the U. S. Government.
2
1.
INTRODUCTION
For the problem of two independent samples, Chernoff and Savage (1958)
considered an elegant decomposition of a rank order statistic into a principal
term involving averages of independent random variables (where the central
limit theorem applies) and a remainder term which converges (at a faster rate)
to 0, in probability, as the sample sizes increase.
Similar decomposition for
the one-sample rank order statistics were studied by Govindarajulu (1960), Puri
and Sen (1969) and Sen (1970), among others.
Hajek (1968) and Huskova (1970)
relaxed the regularity conditions to a certain extent by using a powerful
variance inequality along with the polynomial approximation of absolutely continuous score functions [of bounded variation], and Pyke and Shorack (1968a,
1968b) attacked the problem of asymptotic normality through weak convergence of
certain related empirical processes.
Though these later developments weaken
the regularity conditions on the score functions a little, they may not provide
an order of the remainder term (holding either in probability of almost surely
(a.s.», which has certain interests of its own.
For example, a refined order
of stochastic convergence of the remainder term enables one to study the limiting behaviour of rank order statistics (deeper in nature than their asymptotic
normality) by working with their principal terms which involve averages over
independent random variables and are thereby readily adaptable to refined
probabilistic analysis.
Indeed, in certain weak and a.s. invariance principles
for rank order statistics, to be studied in detail in sections 5 and 6, the CS
(Chernoff-Savage) decomposition along with specified stochastic orders of the
remainder terms are very useful.
The law of iterated logarithm for rank order
statistics also follows quite easily from this decomposition.
~
Finally, when
the observations are not independent, it may be quite difficult to extend the
powerful variance inequality of Hajek (1968), and a comparatively easy approach
3
to the study of the asymptotic normality of rank order statistics can be formulated with the aid of the CS-decomposition and the stochastic order of the
remainder term.
Our study in the present paper centers around the discussion of how fast
the remainder term converges stochastically to 0 for the one-sample problem
(see Section 4).
In doing so, we consider stationary ¢-mixing processes which
include independent, m-dependent
(m~l
average processes as special cases.
and fixed), autoregressive and moving
Under essentially two different
~-mixing
conditions [viz., (2.3) and (2.4)], two different orders of the stochastic
convergence of the remainder term are obtained.
It is seen that less restrictive
¢-mixing conditions can be adopted by paying a premium on the growth condition of
the score function [viz., (2.8)].
•
4It
Under either of the two ¢-mixing conditions,
asymptotic normality of rank order statistics is studied in Section 5, while
certain weak and a.s. invariance principles are established in Section 6.
In
passing, we may remark that for stationary ¢-mixing processes, besides the work
of Serfling (1968) on the Wilcoxon two-sample statistic, the authors are not aware
of any development for unbounded score functions.
The basic study is related to certain stochastic order of fluctuations of
the empirical process, studied in detail in Section 3.
The preliminary notions
are given in Section 2 and the order of the remainder terms are studied in
Section 4.
Asymptotic properties of rank statistics are then presented in
Sections 5 and 6.
2.
PRELIMINARY NOTIONS
Let {X., -oo<i<oo} be a stationary sequence of ¢-mixing random variables
~
defined on a probability space
(~,A,p). Thus, if M~ and Mk +n be respectively
00
k
00
the a-fields generated by {Xi' i.::.k} and {Xi' i~k+n}, and i f E1E:M_00 and E2 E:M k+n ,
then for all k(_oo<k<oo) and
n(~l),
4
(2.1)
where
1>~(1»~(2»
-
-
-
.•. and limn-+oo~(n) = O.
~-mixing
The usual
condition pertain-
ing to the applicability of the central limit theorem for sums of the X. is
~
00
~
= 2n=1 ~ (n)<oo,
(2.2)
On the other hand, for weak convergence of empirical processes (to be defined
later on), we may need either of the following two stronger conditions:
(a) for some k>l,
~(~)
nk~~(n)<oo,
= 2 :
n l
(2.3)
tn
(b) for some t>O, 2 : e ~(n)<oo.
n l
(2.4)
We denote the marginal distribution function (df) of Xi by F(x), x£R,
the real line (-00,00), and assume that F(x) is continuous everywhere.
•
~
For a
sample (Xl""'X ) of size n, define the empirical df Fn by
n
Fn (x) = n
-1
21=. n 1
u(x-Xi ) ,
-oo<x<oo,
where u(t) is 1 or 0 according as t is > or < O.
(2.5)
The weak convergence of the
empirical process
{n~[Fn (x)
- F(x)],
-oo<x<oo}
(2.6)
to an appropriate Gaussian process has been studied by Billingsley (1968, p. 197)
under
A2(~)<00
and by Sen (1971) under
Al(~)<oo.
In passing, we may remark that
(2.4) holds for m-dependent and a general class of autoregressive processes,
while (2.3) is much more general.
Let Rni = 2j~1 u(\xil-Ixj
I)
be the rank of Ix.1
I
among Ixll, ... , Ixn I,
l2i2n, and consider the usual one-sample rank order statistic
Tn
-1
= n -1\L.1=n 1 u(X.)J
1 n «n+l) Rn1.),
(2.7)
5
where J n (i/(n+1»
= EJ(U n1..) or J(i/(n+1», l<i<n,
Un l<
- - ..• -<Unn are the ordered
random variables of a sample of size n from the rectangular (0,1) df, and
J(u) = J*«l+u)/2),
score function.
O~u<l,
is an absolutely continuous and twice differentiable
As in Chernoff and Savage (1958) and Puri and Sen (1969), we
assume that there are positive (finite) constants K,
a(O<a~~)
and
o(O<o~a),
such that for O<u<l,
(2.8)
for r=0,1,2.
Note that (2.8) implies that
\J(r)(u)
where K*<oo.
or (2.4).
2(2k+1),
I~
K*[l_u]-a-r+o, O~u<l,
r=0,1,2,
(2.9)
The choice of a depends on the ¢-mixing conditions (2.2), (2.3)
In fact, under (2.4), we will let
k~l,
or (k-2)/2k,
k~3,
a~,
while under (2.3), a=(2k-1)/
depending on whether we desire to have a bound
for the remainder term holding in probability or a.s.
We write H(x) =l?{lxil~x} = F(x)-F(-x), x>O and Hn(x) = Fn (x)-Fn (-x-) ' x>O.
Then, we have by (2.7),
00
f
T =
J (nH (x)/(n+1»dF (x)
nOn
n
n
= ~+n
~ n
-1
Li=l B(Xi ) + Rn ,
(2.10)
where
00
~ =!
o
J(H(x»dF(x),
(2.11)
00
B(X.)= u(Xi)J(H(lx. I»+! [u(x-lx1.. I)-H(x)]J'(H(x»dF(x),
1.
1.
0
(2.12)
6
00
R = J [J (nH (x)/(n+l»-J(nH (x)i(n+l»]dF (x) +
nOn
n
n
n
00
(2.13)
00
J [J(nHn (x)/(n+l»-J(H(x»]dFn(x)-J
[H (x)-H(x)]J'(H(x»dF(x).
On
o .
We term n
-1
Sn =
~+n
-1 '\ n
Li=l B(Xi ) and Rn as respectively the principal and
remainder terms of T.
n
Limiting behaviour of these terms and other properties
are studied with the aid of certain convergence properties of the empirical
process in (2.6), which we consider first in Section 3.
3.
ASYMPTOTIC BEHAVIOUR OF THE EMPIRICAL PROCESS
It is known [cf. Sen (1971)] that if
Al(~)<oo
then the empirical process
in (2.6) weakly converges to an appropriate, Gaussian process and
1
sup
n~IFn(X)-F(x) I =
-oo<x<oo
o
P
(1).
(3.1)
We are interested in the following related results where we define Y = F(X ),
i
i
l
i~l, so that P{Yi~t} = t: O~t~l, and we let Gn(t) = n- Li~l u(t-Y i ), O~t~l,
n>1.
LEMMA 3.1.
1i (2.3)
holds for some
k~l,
then for every £>0,
1/2(2k+l»
o
n , as n-+oo;
(
p
if (2.3) holds for some
sup
O<t<l
k~2,
(3.2)
then as n-+oo
n~IG (t)-tl{t(l-t)}~+£ = 0(n l / 2k ) a.s.
(3.3)
n
Finally, under (2.4), for every £>0,
sup
O<t<l
O(log n) a.s.
(3.4)
7
[Note that for k=l, (3.3) always holds, but is of little practical interest.]
Proof.
First note that minl<i<n Yi = Yn,l (say) i s ' in n, so that for every
r>l,
p{y
00
= p{
[Y
j=l
< \ 00 p{y
- L.j=l
m,l
< m-r for some m>n}
j-l
-r
< m for some Z n < m < Zjn] }
m,l -
< (nZj-l)-r}
j-
nZ ,1
nZ j
J'-1 -r
= Lj=l p{Li=l u((nZ
)
- Yi ) > l}
00
(3.5)
Let us define
O<t<l.
(3.6)
Then, note that G (t)-O, O~t~n-r, implies that h (t) = O(n-(r-l)/Z) ~ 0 as
n
n~.
n
Thus, by (3.5), for every r>l,
. sup
O<t<n- r
h (t)
n
~
0 a.s., as
n~.
(3.7)
Similarly,
sup
l-n
-r
<t<l
h (t)
n
~
0 a.s., as
n~.
(3.8)
8
Next, we note that if for some positive integer k,
~(¢)<OO,
then [cf.
Sen (1972b)]
n
2(k+l)
¢(n)
~
0 as
n~,
(3.9)
(3.10)
for every
O~t~l, n~l,
where T=t(l-t) and K¢«OO) depends only on {¢(n)}.
note that G (t) and t are non-decreasing in t
n
(O~t~l),
Also,
and hence, if we define
(1) = h (-r)
1·... ,n = [nr-l]+l,
V.
jn
,j=,
nJ
n
0
0.11)
it follows by some routine steps that
sup
n
_
-r
where by (3.10),
<t<n
V(l) + O(n-(r-l)/2)},
[h (t)] < 12{ max
n
nj
-1
l<j<n
--0
~(¢)<OO ...
for every
l~~no'
r 2E(k+l){ l+T -1.+... +T -k}
E[(V (1.))2(k+l)] < KA-(J./ n)
. ,
nJ
nJ
nJ
- ~
and T . = n[j(n r _j)/n
nJ
2r
0.12)
r
] < l+O(n- + l ).
(3.13)
Thus, for every n>O and a>O,
(3.14)
= KA-[n- 2a (k+1) /n 2a (k+1)] [O(n )+O(n
~
0
0
k
-2E(k+1)
log n )+ ... +O(n )][O(n
].
o
0
Thus, on letting r=(2k+3)/(2k+1) (>1) and a=1/2(2k+1), we obtain from (3.12)
and (3.14) that if
~(¢)<oo
for some
n
<t<n
then
[h (t)on- 1 / 2 (2k+1)] =
sup
-r
k~l,
-1
n
0
p
(1).
0.15)
9
Also, if
~(¢)<oo
for some
k~2,
2 -1
on letting a=1/2k and r=1+(2k) ,we obtain
from (3.12), (3.14) and the Borel-Cantelli lemma that as
sup
n
The case of l-n
-1
-r
~
< t
0(1) a.s.
<t<n
l-n
n~,
(3.16)
-1
-r
,r>l, follows similarly.
h (j/n),
n
Let us now define
j=l, •.. ,n-l,
(3.17)
and note that as in (3.12),
(3.18)
Also, by (3.10), for every
l~j~n*
[ n (k+l)/(2k+l)] an d n> 0 ,
p{v(:) > nn l / 2 (2k+l)} < n-(k+l)/(2k+l)n- 2 (k+l)E[(v(2))2(k+l)]
nJ
nj
(3.19)
so that
1/2(2k+l)} <
p{ max
V(2)
nJ. > n n
ls..js..n*
-or
0 .as
n+oo,
I '1
n*
J=
p{v(2)
n l / 2 (2k+l)}
nJ· > n
for every E> 0, n> o.
(3.20)
Similarly, for every n> 0,
p{
max
(3.21)
n-n*~.5.n-l
Thus, to prove (3.2), it remains only to show tha t for every n> 0,
p{
V(2) >
1/2(2k+l)}
max
nj
n n
n*<j<n-n*
-or
0
as
n~.
For proving (3.22), we note that the left hand side is bounded by
(3.22)
10
L~-n*-l p{V(~) > n n1 / 2 (2k+l)}
J=n*+l
nJ
< (n-2n*-1)[
max
p{V(~) > n
n*<j<n-n*
nJ
n
1/2(2k+l)}] ,
(3.23)
so that it is enough to show that for each n*<j<n-n*,
(3.25 )
Now, by (3.6) and (3.17),
n
p{
IL
i=l
[u(i - Y.) - i]
n
1
n
I
> h*.},
(3.26)
nJ
where
h* . =
nJ
~
n
*
n (2k+2)/2(2k+l)[.(
J n-J')/ n 2]~-£ ,n *<j< n-n.
(3.27)
Since {u(i - Y ), i~l} is ¢-mixing where·
P{u(j/n-Y.)=l}
= l-P{u(j/n-Y.)=O}-j/n,
1
1
i
n
following the proof of Lemma 4.1 of Sen (l972a) and choosing (in his notations)
. that for every t>O.
kn = n1/2(2k+l) , we obta1n
E[exp{tLi~l[u(~
- Yi ) -
~(¢)<oo
for some
~]}] ~E[exp{tknL~:~[u(~ - Yi+~k
k~l,
)
n
(3.28)
< {l+[i + 0(n-(k+l)/(2k+l»] [exp(tk )_l]}mn+l,
n
n
where mn is the largest integer for which l+k nmn<n (i.e., mn~n
l-l/Z(Zk+l»
.
By (3.26), (3.27), (3.28) and the Markov inequality. we have
p{\,n [u(i _ Y,) _ i] > h*.}
L 1=1
n
1
n
nJ
< inf{(exp[-tj-th*,])E[exp(t\·~l{u(j/n-y.)-j/n})]}
1
nJ
L1 t>O
1[.(
2]2£ o
+ (l)}
2. exp {-nn1/4(2k+l) + "2
J n-J')/ n
,
J~-l/4(2k+l)
2 -~+£
where in the last line we use t=t ,=n
[j(n-j)/n ]
. A similar
nJ
bound holds for p{L,nl[u(j/n-y.)-j/n] < -h*.}. Now. for every n>O and k~l,
1=
1
nJ
(3.29)
11
nn
1/4(2k+l)
~
as
n~t
and for every s(>l) there exists an n (stn) such that for
o
nn
1/4(2k+l)
> slog n +
21
(3.30)
+ log 2.
Consequently, by (3.26)t (3.29) and (3.30)t the left hand side of (3.25) is
O(n-s+l )~O as
Hence t the proof of (3.2) is complete.
n~.
To prove (3.3)t by virtue of (3.7), (3.8)t (3.16) and (3.l8)t it suffices
to show that as
n~
l 2k
[h (j/n)/n / ]
max
0(1)
n
l~j2.n-l
k~2,
3 4k
For this purpose t we define n* = [n / ],
(3.31)
a.s.
proceed as in (3.19) through
(3.21), and obtain that for every n>Ot
p{ max
V(:) > nn l / 2k }
l2.j2.n* nJ
2. K<jln
K<jln
-(k+l)/k
-(k+l)/k
n
n
-2 (k+l)
n*
Lj=l{l+O(j /n)}
-2(k+l) n3/4k[l+O(n-l+3/4k)]
= K<jln- l - l / 4k n- 2 (k+l) [l+o(l)]t
(3.32)
so that by the Borel-Cantelli Lemma t
p{ max
1~2.Jn*
2. K*<jl n
V(:) > nm1 / 2k for some m~n}
mJ
-2 (k+1) -1/4k
n
where K;«oo) depends on K<jl and k.
p{
max
m-m*~2.m-l
~
0
as
Similarly,
(2)
1/2k
V
> nm
for some
mn
(3.33)
n~t
m~n} ~
Ot as
n~.
For n*<j<n-n*t we repeat the steps in (3.22) through (3.30)t where we take
k
n
1/2k
= n a n d s>2 t so that
(3.34)
12
p{
(2)
1/2k}
-s+l
max
V.
> nn
= O(n
) as
nJ
n*<j<n-n*
n~.
(3.35)
~O a.s., as n~,
(3.36)
Thus, again by the Bore1-Cante11i Lemma and s>2,
[n- 1 / 2k h (j/n)]
max
n
n*<j<n-n*
which completes the proof of (3.3).
Finally, to prove (3.4), by (3.7) and (3.8), it suffices to show that for
every r>l, and some
K(l~K<oo),
sup
n
-r
[(log n)
<t<l-n
-1
hn(t)]
~
K, a.s. as
Here, we let r=l+£, £>0, where £ is defined in (3.4).
(n
-1-£
, 1-n
range (n
-1-£
-1-£
, n
) into (n
-1
n~.
-1-£
, n
-1
Then, we divide the range
-1
-1
-1
-1-£
), (n , 1-n ) and (l-n , 1-n
).
.
), we aga1n use (3.12) and show that on defining n
max
l<j<n
(3.37)
-r
o
For the
as in (3.11),
1
[h (j/n +£)/10g n] ~ 0 a.s. as n~.
(3.38)
n
--0
Since (2.4) holds for some t >0, we can select a C«oo) such that Ct
o
Then on choosing k
n
0
> 1+£.
= clog n
(3.39)
Thus, proceeding as in (3.26) through (3.30), we have for every n>O,
P{h (j/n 1+£) >
n
n log n}
£
= 2[exp{-n(10g n) ~ n £ + o(l)}], V l<j<n
~n •
--0
Again, for every £>0 and n>O, there exists an n (£,n), such that
o
(3.40)
13
~ £
n(log n) n
~
slog n, where s>2,
V n~no(£,n).
(3.41)
Consequently, by the Bore1-Cante11i Lemma,
1
[h (j/n +£)/10g n] ~ 0 a.s. as n~.
max
n
l<j<n
--0
A similar proof holds for the range 1-n
n
-1
~t~l-n
-1
-1
~
t < 1-n
,we again use (3.18) for each j
-1-£
.
(3.42)
Finally, for the range
(1~j~n-1),
and note that (2.4)
-1 -ton
t n
implies that as n~, ne 0 ¢(n)~O for some t >0 i.e., ¢(n) = o(n e
) for some
o
-Ct log n
o
-1
-1 -1
t >0. Thus, ¢(C log n) = o(e
(C log n) ) = o«log n) 'n ), where
o
Ct >1.
Thus, if we repeat the steps (3.26)-(3.29) where we choose k
o
2 J...,+£
-!-.<
we obtain on choosing t=t .=n 2[j(n-j)/n ] 2
nJ
p{
max
,that as
n~,
n
-
~ 2(n-l) [exp{-K log n + t(C log n) [j(n-j)/n 2 ]2£+0(1)}]
by choosing
Clog n,
h (j/n) > K log n}
1~j~n-1
< 2 n
n
-s+l
(3.43)
,s>2,
K(>~C+s)
(1~j~n-1)
adequately large.
Thus, again by the Bore1-Cante11i Lemma,
h (j/n)/log n < K a.s. as
max
n
1,Sj-S.n-1
and the proof of (3.4) is complete.
n~,
(3.44)
Note that (3.4) extends a lemma of Ghosh
(1972) to a class of ¢-mixing processes.
STOCHASTIC ORDER OF THE REMAINDER TERM
4.
We rewrite R in (2.13) as
n
C + ..• + C ;
6n
1n
(4.1)
Jo [J n (nH n (x)/(n+1»-J(nH n (x)/(n+1»]dFn (x),
(4.2)
Rn
00
e
C1n =
00
C
2n
=
a
J [J(nHn (x)/(n+1»-J(H(x»]dFn (x),
n
(4.3)
14
a
n
f
[J(nHn (x)/(n+l»-J(H(x»-(nH n (x)/(n+l)-H(x»J'(H(x»]dFn (x),
o
a
= _(n+l)-l
a
f
(4.4)
n
Hn (x)J'(H(x»dFn (x),
o
(4.5)
n
C = J [H n (x)-H(x)]J'(H(x»d[Fn (x)-F(x)],
5n
0
(4.6)
00
C6n = -
J
a
[H (x)-H(x)]J'(H(x»dF(x),
n
(4.7)
n
where we define a
O
n
by H(a ) = I_n- l + , 0(>0) is defined in (2.8).
n
Note that
dF < dH , so that
nn
(4.8)
by Theorem 2 of Charnoff and Savage (1958); for a proof of (4.8) under less
restrictive regularity conditions, we may refer to Theorem 3.6.6. of Puri and
~
Sen (1971).
THEOREM 4.1.
If for some
k~l,
n~ R =
n
If for some
k~3,
~(¢)<oo
~(¢)<oo
and (2.8) holds for a=(2k-l)/2(2k+l), then
op (n-n) for some n>O.
(4.9)
and (2.8) holds for a=(k-2)/2k, then
~ Rn = O(n-n) a.s. as n~, for some n>O.
If (2.4) holds for some t>O and (2.8) holds for
a=~,
(4.10)
then
n~ R = O(n-n) a.s. as n~, for some n>O.
(4.11)
n
Proof.
We only prove (4.11) as (4.9) and (4.8) follow on similar lines.
virtue of (4.8), we only show that \cknl = O(n-n) a.s. (as n~) for 2<k<6.
consider C2n '
Using the fact dFn< dH n , we can write
By
First,
·15
00
00
IC2nl~IJ [J( n+n11In (x»]dH n (x)I+IJ
a
a
n
J(H(x»dH n (x)1
(4.12)
n
Integrating by parts,
00
J
a
00
-J
J(H(x»dHn (x)
J(H(x»d[l~H (x)]
n
a
n
n
00
[l-Hn (a n )]J(H(a n » +
J [l-Hn (x)]J'(H(x»dH(x).
a
n
Hence, by (4.12),
00
00
IJ( n+n1un (x» IdHn (x) +
J
a
(l-H n (x» IJ'(H(x» IdH(x)
n
+ [l-Hn (a n )]\J(H(an »
e
I.
(4.13)
By the same arguments as in Lemma 4.1 of Sen (1972a), it can be shown that under
(2.4),
-1~6
log n) a.s. as
Hn (a n )-H(a n ) = O(n
•
(4.14)
n~.
In the sequel, we assume that n is so large that H(an )~~.
Now, by (2.9) with
and (4.14), the first term on the right hand side of (4.13) is bounded by
00
J
a
-<
-<
[l-nH (x)/(n+1)]-~+6dH (x)
n
n
n
[l-nH (a )/(n+1)]~+6
n n
[l-nH (a )/(n+1)]~+O
n n
00
J
a
[l-nRn (x)/(n+1)]
-1
dH n (x)
n
00
J [l-nH
o
n
a.s.
Thus, the right hand side of (4.15) is
(x)/(n+1)] -1 dH (x)
n
a~
16
O(n
where
O<n<~6(1-6).
J~-~6(1-6)
log n) a.s.
Now, by (2.9) with
a~
= O(n J~-n ) a.s.
(4.16)
and (3.4), the second term on the
right hand side of (4.13) is bounded by
00
J
a
(l-H(x)) IJ'(H(x))
IdH(x)+[O(n-~(log n))]J
a
n
00
~J
a
n
00
[l-H(x)]~-£IJ'(H(X)) IdH(x)
a.s.
n
00
(l-H(x)) J~+o dH(x)+[O(n J~log n)]J [l-H(x)] -1-£+0 dH(x)
a
(a. s.)
n
= O([l-H(a n )]~+O)+[O(nJ~log n)][O([l-H(an )])6-£]
(a.s.)
(4.17)
a.s.
where we choose 0<£<6/2 and
a~,
0<n<~6(1-26).
Finally, by (4.14) and (2.9) with
the last term on the right hand side of (4.13) is bounded by
I [l-H(a n )+{Hn (a n )-H(a n )}]\ IJ(H(a n )) I
(a.s.)
(a.s.)
(4.18)
as
O<6<~. Hence, Ic2nl = O(nJ~-n) a.s.
that dF
Next, we consider C ,
3n
Using the fact
< dH and the mean value theorem, we obtain from (4.4) that
nn
a
n
[ +n 111 (x) -H (x) ] 2 1 J" (H e (x) ) I dH (x),
n
n
n,
n
(4.19)
17
-1+0
,0>0, and
Since H(a n ) = 1-n
where Hn, 8 = 8nHn (x)/(n+1) + (l-8)H(x), 0<8<1.
by our choice
O<£~
0~8~1, O~x<an'
, by (3.4), for every
1-H ,8(x)
n
~
1-H(x)-[O(n
J~
log n)]{H(x) [l-H(x)]}
~_£
(a.s.)
~ {l-H(x)}[l-[O(n~logn)]{l-H(X)}~-£]
> {l-H(x)}[l-[O(nJ~log n)]{l-H(a )}J~_£]
-
n
= {l-H(X)}{l-[O(n~log n)][O(n-(~+£)(l-0»]}
= {1_H(x)}{1_0(n-(O/2-£)-0£log n)}
{l-H(x)}{l-o(l)}, a.s. as
n~.
n~.
{l-Hn (x)}{l-o(l)},
a.s. as
.
4It
•
(4.20)
[Putting 8=1, the same inequality applies to 1-Hn (x).] Using then (2.9) with
a~,
(3.4), (4.19) and (4.20), we obtain that for large n,
a
-12
n
1-2£
< [O(n (log n) )]/ {H(x) [l-H(x)]}
[l-H
-
0
a
3
n
< [O(n- 1 (log n)2)]/ [l-H (x)]
o
=
n
- -
+
~
u -
o±
O(n
0/2
5
2
+ 0
dH (x)
n
(a.s.)
2£
2
-1
2
-1, n*
[O(n (log n) )][n Lj=l[(n- j )/n]
Now, by (4.14), n* = n-n
n,
8(x)]
-
dH (x)
(4.21)
n
3
2+
0 - 2£
] (where n*=nHn(a n
».
log n) a.s., while for every k<n,
(4.22)
(1-0) ) a.s., for k=n*=n-n 0 ± o(n- 0 / 21og n) a.s.
= O(n (L-0+2£)
~
Thus, the right hand side of (4.21) is
18
0(n- 1 (10g n)2)0(n(~-0+2E)(1-0)) a.s.
O(n
=
Thus,
with
IC3n I
a~,
=
J~-n
) a.s. for some n>O.
O(n-~-n ) a.s., as n+oo.
(4.23)
Next we consider C .
4n
By
(4.5), (2.9),
(4.14) and (4.20), we have
a
IC4n l -<
(n+1)-1
J0
a
-<
=
(n+1)-1
O(n
-1
=
O(n
J0
)-O(n
1
n
-I-
[l-H(x)]
n
3
--+ 0
2
dH (x)
n
3
- - + 0
[l-H (x)] 2
(l+o(l))dH (x)
n
n
1
-(- - 0)(1-0)
2
)
a.s.
n
) a.s., where n>O.
(4.24)
For the study of the stochastic order of C , for the convenience of
5n
manipulations, we assume by virtue of the probability integral transformation
that F is rectangular (0,1) df, and we write t
o
=
F(O), so that O<t <1.
-
0-
Then,
a*
n
=
J
t
where H(a*) = 1-n
n
-1+8
[H (u)-H(u)]J'(H(u))d[F (u)-u],
n
(4.25)
n
o
,so that 1-n
-1+0
-1
< a* < l(a* = F (a )).
- n
n
n
Since, for the
rectangular df, the density function is a positive constant for the entire positive
support of the df, proceeding as in Theorem 3.1 of Sen (1972a) and using Theorem
4.2 of Sen and Ghosh (1971) [namely, their (4.13)], it follows that as n+oo,
sup
O<u<l
sup
t:
It-u In
J/-
{IFn(t)-Fn(u)-t+u\}
2
= 0(n- 3/4 log n)
Now, we can rewrite C as
Sn
(4.26)
a.s.
19
[Hn (u)-H(u)]J'(H(u»d[Fn (u)-u],
(4.27)
where
I . = {u: t +(j-1)/1U < u < t +j/v'U}, j=l, ... ,n*-l,
nJ
0
0
(4.28)
I nn * = {u: t 0+(n*-l)/vn<
- u -< a*},
n
and n* is the largest positive integer such that t
by definition n* = O(n~).
t
no
=t
o
+ (n*-l)/1n < a*.
n
Note that
Let us then write
J
t .=t +jn ~, l<j<n*-l, t
=a*.
0' nJ 0
- nn* n
Then, by (4.26), for every uEI ., as
nJ
(4.29)
n~,
H (u)-H(u) = H (t .)-H(t .)+O(n- 3/ 410g n) a.s.
n
n nJ
nJ
(4.30)
F (u)-u = F (t .)-t .+O(n-3/4 log n) a.s.
n
n nJ
nJ
(4.31)
and by (2.9) (with
a~),
for UEI .,
nJ
J'(H(t .»+[O(nJ~)][O({l-H(t .)}-S/2+0)]
nJ
nJ
J'(H(u»
(4.32)
Consequently,
III
[H (u)-H(u)]J'(H(u»d[F (u)-u]1
n n
nj
<
I [Hn (tn j)-H(tnJ.)]J'(H(tnJ.»
3 4
.+ O(n- / 1og n) \J'(H(t j»
n
+ O(n~)1
[F (t .)-F (t . 1) n nJ
n nJ-
II I . Id[Fn (u)-u]\
_.2. +
I .
nJ
[l-H(t .)]
nJ
(4.33)
+
nJ
IH (t j)-H(t j) I [l-H(t .)]
Inn
n
nJ
nj
+ o(n- S/ 41og n)I
~] I
y"
2
- -S + 0
2
Id[F (u)-u)]1
n
0
Id[F (u)-u]l,
n
a.s.
20
By (2.9) with
a~,
Lemma 3.1 and (4.26), the first term on the right hand side
of (4.33) is bounded by
•
a.s.
(4.34)
Also, by (4.26),
J
I
Id[Fn(u)-u]I ~
.
J
I
nJ
dFn(u) +
.
nJ
= F (t .)-F (t j 1) + ~ <
n nJ
n n yn -
J
I
du
.
nJ
2nJ~ + 0(n- 3 / 410g n) a.s.,
(4.35)
so that the second term on the right hand side of (4.33) is of the same order
as of (4.34).
Similarly, the third and the fourth terms are bounded by,
respectively,
[0(n- 3/ 2 log n)][l-H(t .)]-2+6
nJ
a.s.
[0(n- 7/ 4 log n)][l-H(t .)]-5/2+0
nJ
(4.36)
a.s.
(4.37)
Now, for 8>1,
n*
. l[l-H(t .)]
LJ=
nJ
-8 < L. n*l[F(t ,)] -8 = n 8/2 . n*let vn+J)
2J= 0 r . -8 = O(n8/2 ).
- J=
nJ
On summing over j(=l, .•• ,n*), we have from (4.33) through (4.38), as
ICsnl
(4.38)
n~,
~ [0(n- S / 4 (10g n)2)][O(n 3/ 4- 6 / 2 )] +
[0(n- 3 / 210g n)][0(n l - 6 / 2)] + [O(n- 7 / 410g n)][O(n S/ 4- O/ 2)]
a.s.
(4.39)
where O<n<%6.
Thus, C = O(nJ~-n) a.s. as n~.
Sn
Finally,
21
00
IH (x)-H(x) I IJ'(H(x» IdH(x)
n
_1+ 0
~ [O(nJ~log n)]f {H(x)[l-H(x)]}~-E[l-H(X)] 2
dH(x)
..
00
a
< [O(n
(a. s.)
n
00
_k
2 10g
[l_H(x)]-l+o-E dH (x)
n)]J
a
n
[O(n~log n)][O(n-(1-0)(0-E»]
=
O(n ~-n ) a.s.,
n>O.
Hence, the proof of (4.11) is complete.
(4.40)
The proof of (4.9) and (4.10) follow
on parallel lines; for intended brevity, the details are therefore omitted.
Remarks.
For independent random variables, the proof of C S
n
2
-1
more easily by showing that E[CnSl = o(n ).
=
1
0
p
(n~) follows
However, for cjl-mixing processes
and a.s. convergence, evaluation of higher order moments of C becomes cumbrous,
nS
and the present proof appears to be simpler.
For independent random variables, the assumption on JlI(U) in (2.8)-(2.9)
has been dispensed with [See Govindaraju1u, LeCam, Raghavachari (1967) and Puri
and Sen (1969)].
These results depend on the property that for independent
random variables,
(4.41)
sup
O<t<l
On the other hand, for cjl-mixing processes, our results (3.2) is not as strong as
(4.41), and this necessitates the assumption on JlI(U).
~
In any case, for a.s.
convergence in (4.10) or (4.11), (4.41) does not suffice and we may need the
condition on J"(u).
22
We also note that by the same technique as in Lemma 3.1, it can be shown
that if
Al(~)<oo,
then for every n>O,
I
sup {n~-n IGn(t)-t}
O<t<l
=
a.s., as
0(1)
(4.42)
n~.
Hence, if J(u) has a bounded first derivative for all u£(O,l), we may repeat the
steps for the proof of (4.11), use the boundedness of J'(u) and obtain more easily
that IRnl
= O(n~
THEOREM 4.2.
then'\Rnl =
S.
-n) a.s. as
n~. Hence, we have the following.
li J(u) has a bounded first derivative inside (0,1), and
o(n~)
a.s., as
Al(~)<oo,
n~.
ASYMPTOTIC NORMALITY OF RANK ORDER TESTS AND ESTIMATES
Let us define B(X ) as in (2.12) and let
i
(S.l)
where we know [viz., Puri and Sen (1971, Section 3.6)] that under (2.8),
1
Further, Ao(~) = Ln:l ~~(n)<oo ~02 <
00
[viz., Billingsley (1968, p. 172)].
In
the sequel, we assume that
o<
02 <
(S . 3)
00
Now, by Theorem 21.1 of Billingsley (1968) under A
o
('
~(n
n
[L.i=l B(Xi)]/O)
J-1 \
On the other hand, by Theorem 4.1, A
l
~
have the following.
-+
(~)<oo ~
(~)<oo,
as
n~
)«0,1).
R
n
o
p
(n-~).
(S.4)
Consequently, we
23
Ii
THEOREM 5.2.
under (5.3), as
~(¢)<oo
and (2.8) holds for a=(2k-l)/2(2k+l) for some
k~l,
then
n~
.
(5.5)
Under more restrictive ¢-mixing conditions, Philipp (1969) has studied
the order of the remainder term in the central limit theorem for sample averages.
By virtue of our Theorem 4.1 and (5.4), under similar ¢-mixing conditions, we
1
have analogous expressions for sup Ip{n~(Tn-~)/a 2 x} - ~(x)
x
standard normal df.
I,
where ~(x) is the
Our next result concerns the asymptotic distribution of point estimators
based on rank order statistics.
i~l},
process {X"
1
Here, we assume that for the stationary ¢-mixing
the marginal df F(x) is
(5.6)
where
e(~<e<oo)
is a location parameter and F
o
interest is to estimate 8.
~<b<oo
Our
With this end, define T (b) similar as in T
n
but the Xi being replaced by Xi-b,
is defined for
is symmetric about O.
l2i~n,
n
Then T (b)
where b is a real variable.
and when J(u) is non-decreasing, T (b) is
n
inf{b: T (b)<~ }
n
0
+ in
in (2.7)
n
b.
Define
(5.7)
(5.8)
where
11
1"'0
=
~
n
Jol J(u)du[=(2n) -lI 1'=1
J n (i/(n+l»
when J n (i/(n+l»
As in Hodges and Lehmann (1963) and Sen (1963), we consider
of 8.
= EJ(U n1,), l<i<n].
--
e
n
as an estimator
In view of our Theorem 5.1, an argument analogous to Theorems 1 and 5 of
Hodges and Lehmann (1963) leads to the following .
.
~
THEOREM 5.2.
If for a non-decreasing score function (2.8) holds and Al(¢)<OO,
then under (5.3),
24
p{n~(e n -6)B(F 0 )/0
< x}
-+
(2n)
-1: X
J e -\<t2 dt,
2
(5.9)
2
-00
for every -oo<x<oo, where
00
B(F ) =
o
and it is assumed that as
f
[dJ*(F o (x»/dx]f 0 (x)dx, f 0 =F'0'
d J(F (x»
dx
0
(5.10)
is bounded.
A particular estimator not covered in (5.7) is the smmple median for which
J(u)=l \I
O~u::.l.
However, in this case, or in the general case of sample quantiles,
results similar to (5.9) are already derived in Sen (1972a).
So far, we have
assumed that {X.} is stationary ¢-mixing; non-stationarity can be handled in the
1
same manner as in pages 179-180 of Billingsley (1968), provided the probability
measure governing the X. is dominated by a stationary measure.
1
6.
•
INVARIANCE PRINCIPLES FOR RANK STATISTICS
Here we consider a Donsker-type invariance principle for rank statistics
as well as a law of iterated logarithm for these statistics.
Consider the space C[O,l] of all real, continuous functions on the unit
interval [0,1], and associate with it the uniform topology
sup Ix(t)-y(t)
O<t<l
where both x and y belong C[O,l].
I,
(6.1)
For every n~2, define a process Z ={Z (t),
n
n
(6.2)
where TO=O, Tk , k~l, are defined in (2.7), ~ in (2.11) and
0
2
in (5.1).
By
linear interpolation, we complete the definition of Z (t) for tE[k/n, (k+l)/n],
•
n
25
THEOREM 6.1.
If
~(¢)<oo
holds, and in (2.8),
k~3,
for some
a~],
and in (2.8), a=(k-2)/2k, [or if (2.4)
then under (5.3), Z converges in law to a standard
n
Brownian motion W = {W(t), O<t<l} in the uniform topology on C[O,l].
Proof.
Let us define B(X.),
i>l,
as in (2.12), and let
1
Z*(k/n)
= [\jk
B(X.)]/aln,
k=l, ... ,n, Z*(O)=O,
n
L =l
1
n
(6.3)
and by linear interpolation, we complete the definition of Z*(t) for tE[k/n,
n
(k+l)/n], k=O, .•. ,n-l.
Let Z*={Z*(t), O<t<l}.
n
n
Then, by (2.10), (2.11), (6.1),
(6.2) and (6.3),
p(Z ,Z*) < {max I~ I}/aln.
n n
l<k<n --k
(6.4)
Therefore, by (4.10), under the hypothesis of the theorem,
p(Z ,Z*)
n
n
P
+
0 as n+oo.
(6.5)
On the other hand, by Theorem 21.1 of Billingsley (1968),
Z*~
W, in the uniform topology on C[O,l].
n
The theorem follows from (6.5) and (6.6).
(6.6)
Q.E.D.
Note that by virtue of Theorem 6.1, for every E>O and n>O, there exist a
0>0 and an n0
(E,n), such
that for n>n (E,n),
- 0
p{ sup
Iz (t)-Z (s)
It-sl<o n
n
I
> E} < n.
(6.7)
Consequently, by Theorem 5.1, (6.7) and by Theorem 2 of Mogyorodi (1967), we
•
obtain the following.
~
THEOREM 6.2.
If {N , r~l} be a sequence of positive integer valued random variables,
r
such that as r+oo
26
r
-1
N
r
P
-+
A.,
(6.8)
where A. is a positive random variable defined on the same probability space
(Q,A,p), then under the conditions of Theorem 6.1, as r-+oo
1
t{N;[T N -~]/a)
-+
X{O,l).
(6.9)
r
On defining B{X.), i>l, as in (2.l2) and noting that these form a
1
stationary
~-mixing
-
sequence, so that under (5.3), by the results of Reznik
(1968),
~
\' n
lim sup n -~ [Li=lB(X
)]/a{2 log log n) ~ = 1 a. s. ,
i
n-+oo
(6.l0)
-1 a.s.
(6.11)
On the other hand, under the hypothesis of (4.10) or (4.11),
•
(6.l2)
Consequently, from (2.l0), (6.10), (6.ll) and (6.l2), we arrive at the following.
THEOREM 6.3.
Under the hypothesis of Theorem 6.1, the law of iterated logarithm
!.:
!.:
1
!.:
P{lim sup n 2{T -~)/a{2 log log n)2
n
n-+oo
P{lim inf n'2 (Tn -~) / a ( 2 log log n)2
n-+oo
Remark.
l}
I,
(6.13)
-11
= 1.
(6.14)
For independent and identically distributed random variables, Sen and
Ghosh (1973) have established Theorem 6.3 (under the hypothesis (5.6) with 8=0),
~
and Sen (1972c) has considered Theorem 6.1.
Both of these results are based on
a fundamental martingale property of {n[T -ET l, n~I}.
n
n
For ~-mixing processes,
27
the martingale property does not hold.
Nevertheless, our theorem 4.1 and the
decomposition (2.10) provide the parallel results.
For independent random
variables or martingales, Strassen (1967) has established an a.s. invariance
principle.
By virtue of (3.4) [always holding for independent processes], under
(2.8) (for a=~), IRnl = O(n
~n
2-
)
a.s. for some n>O (see (4.11)).
On the other
hand, for independent processes, Strassen's (1967) Theorem 4.4 holds for
{B(X.), i>l}.
1
-
Consequently, if ~(t) be a standard Wiener process on [0,00), then
by Theorem 4.4 of Strassen (1967), as
n~,
n k
2
. 1B(X.)
I 1=
1
= ~(n) + o(n
)
a.s.,
(6.15)
and hence, for independent and identically distributed random variables, under
(2.8) for
a~
(6.16)
Under (5.6) with 8=0, a similar result has been deduced earlier by Sen and
Ghosh (1973) .
•
28
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