L.
WEAK CONVERGENCE OF RAO-BLACKWELL ESTIMATOR
OF DISTRIBUTION FUNCTION
By
B. B. Bhattacharyya and P. K. Sen
North Carolina State University, Raleigh, N. C.
Department of Biostatistics
University of North Carolina, Chapel Hill, N. C.
. Institute of Statistics Mimeo Series No. 942
July 1974
WEAK CONVERGENCE OF RAO-BLACKWELL ESTIMATOR OF DISTRIBUTION FUNCTION
BY B. B. BHATTACHARYYA and
*
P. K. SEN
North Carolina State University, Raleigh and University of North Carolina, Chapel Hill.
ABSTRACT
Under the condition that the minimal sufficient statistics are transitive,
the sequence of Rao-Blackwell estimators of distribution function has been shown
to form a reverse martingale sequence. Weak convergence of the corresponding
empirical process to a Gaussian process has been established by assuming that the
sufficient statistics are U-statistics and utilizing certain results on the
convergence of conditional expectations of functions of U-statistics along with
the functional central limit theorems for (reverse) martingales by Loyns(1970)
and Brown(1971).
AMS 1970 Subject Classification: Primary 60BIO, Secondary 62 B 99.
Key Words and Phrases: Gaussian process, Rao-Blackwell estimator, Reverse
martingale, Transitive sufficiency, U-statistics, weak convergence.
* Work
partially supported by the Air Force Office of Scientific Research,
A.F.S.C., U.S.A.F;, Grant No. AFOSR 74-2736.
1
Introduction.
Let {X
--1.• i ~ 1 } be a sequence of independent and identically
~~_~
distributed random vectors (iidrv) defined on a probability space (Sl.G,P) with
L'ach X
..·i
having a continuous distribution function (dO F(x). x
-
-
P
E R •
dimensional Euclidean space. For every n( ~l). the empirical df F
F (x)
(1.1)
n
=
n -
n
the p( > 1)
-
is defined by
-1
where for a p-vector u
(1.2)
c(u) = 1 if u > 0 and is 0 otherwise.
and by x >y we denote the coordinatewise inequalities x. > y .• 1 < j < p. We
- -J J
-assume that the df F admits of the existence of a complete sufficient statistic.
(vector) T
-n
• where
T 1 = (T 1 •...• T ). for some q > 1.
-n
n
nq
(1. 3)
[When we encounter the asymptotic situation where n +
00
•
we assume that q remains
fixed. Note that the sample order statistics ( for p=l) or the collection matrix
I
,
(for p > 1) constitute a sufficient statistic where q=n. and in that case. our
study is of no real interest.]
Let
& n=
~(Xl' ...• X ) be the
-
~(T ) be the
-n
-n
.
,Q
-
a-field generated by T • so that
-n
a-field generated by
'J(T •T +1 •... ) be the
-n -n
is monotone non-increasing and
(1. 4)
v
=
(x)
n -
process
l 2
n / [ F (x) - F(x)
n -
-
] • x
E:
RP • n
~
1.
Then. the Rao-Blackwell empirical process is defined by
(1. 5)
where for every x
(1.6)
'¥n (x)
=
l.
E[F (x)
n
._
Also. let
~
n
I
~ n(s»)
E[c(x-X
- -1
=
=
{~k' k ~ n}. n ~ 1. so that ~ is
:Fn ~n
e~p1~1tal
Consider the uS4al
-n
(s)
a-field generated by
X l'X +2 •.. · . ' n > 1. so that ~
-n+ -n
~n
the unordered collection
I
i
63~s) C ~n
a-field. Finally. let~ be the
a monotone non-increasing
I
I
•
a-f1eld generated by Xl •...• X • and let ~
)1 ~(s)J
n
2
is the Rao-Blackwell empirical df. Note that when T is a complete sufficient
-n
~
statistic t
n
is the unique minimum variance unbiased estimator (UMVU) of F.
~
Specific expressions for
n
for various F (mostly belonging to the exponential
family) have been worked out by various workers; we may refer to Tate(1959)t
Ghurye and Olkin(1969) where additional references are cited. In the context of
the UMVU estimation of the density function
f(~)
and its various functions, the
recent works of Seheult and Quesenberry(197l) and O'Reilly and Quesenberry(1972)
deserve mention.
In view of the fact that the empirical process in (1.4) weakly converges to
an appropriate multiparameter Gaussian process [viz., Neuhaus(197l) for p
~
1
and Billingsley(1968) for p=l], our interest centers here on deriving similar weak
'Ie
convergence results for the process V
n
in (1.5). That, in general, the weak conver+
'Ie
gence of V does not necessarily imply the same for V
n
n
the simple example of the uniform [0,8] df t where
.
~
n
can easily be verified with
(x) = (n-l)x/nX( ) , if 0 < x
n
< X(n) and is 1 for x ~ X(n); X(n) = max{ Xi: 1< i 2n} t and the distribution of
e } is
sup{ I~ (x) - F(x)l:o< x <
n
-
sup
n 1/2{ O<x<l
I~n(x)
probability, as n
-
-F(x)
~
00
•
I}
I
2
/
2 max { 1
n 2\ (n-l)/nX(n) -1 t n 1/ \ l-X(n)
We also note that by definition in (1. 6),
(n+l)~n+l(~)
be an average of iidrv and
pendent of
n~
e=
independent of 8 . Hence, assuming
-
n~n(~)
I}
~
1,
.
~ 0t 1n
(x) ceases to
n -
is no longer stochastically inde-
(x). Thus, the basic technique of deriving the functional central
n -
limit theorems, displayed in detail in Billingsley(1968) and extended to the multiparameter case by others t is not readily applicable. our task is accomplished by
showing that under the additional assumption of the transitivity of T , for every
-n
{~(x),}t:;
n _
n n> l}
is a reverse martingale on which the central limits
theorems of Loyns(1970) and Brown(l97l) can be applied.
The basic results on
~
n
and some properties of U-statistics are studied in
section 2. The main theorem is stated and proved in section 3. The last section
includes ( by way of concluding remarks) certain additional results.
3
2. Sl)me baste tenuul1s. Let
"
'
",.
"
(2.1)
-
,
liS
.
'
{x
B(a,b)
denote by
< b}
-a -< x --
where a < b
......
(2.2)
Also, for every n
(2.3)
~
z:(~,~)
I, let
p{
~1
*
E
E:
B(~,~)I
6~s)}
., 1
(-1)~ - '¥ (j.a.+(l- j i)b.,l .2.i.2. p),
n
1
1
1
0 1 , i = 1 , ... ,p.
where the summation E* exten d s over a 11 J., = (.J " " , J. ), .J.=,
1
P
1
Lemma 2.1. Let C and h be some positive numbers such that
(2.4)
1 < nh <
0
and
Then. for every positive integer s , there exists a constant K
s
(independent of h,
(2.5)
Proof. Let us define
(2.6)
= E* (-1)1. '1- Fn (j:a.+(l-j
.)b , 1 2i2 p),
111
i
Z (a,b)
n - -
so that by (1.6) and (2.3),
Zn* (a,b)
- _
(2.7)
E[ Z (a,b)1
n - -
CB(S)].
n
Therefore, by the Jensen inequality,
Now, nZn(~'~) has the binomial distribution with the parameters (n, PF(~'~»' so
that by Lemma 5.2 of Neuhaus(1971). under (2.4),
(2.9)
<
The lemma follows directly from (2.8) and (2.9).
Lemma 2.2. If { T • n > 1 } is transitive sufficient. then for every x
-
fIji
(x).
n .-.
Jrn ;
-n
n > I}
-
-
is a reverse martingale.
Proof. 13y definItion in 0.6),
~,
t
c;
E:
RP ,
_
• .-....~~~ .• __ •
.. - . -
_'~"' __ - " " . __ . . . . _ , . . . . . . . . _
-0 . ., 0 _ "
(2.10)
'1'
Since
{T}
(2.11)
(8»]
0+1
I ~ n+l
(8)
} •
n
are conditionally independent given GO(s). Therefore,
n
E[C(~-~l)1 a(t;(,(S).~(s»]
n
n+l
== E[c(x-X
- -1
)I~(s)] ==
-
-
_
'¥n+l (~) == E{ '1' n (x)
16:l n+l
(s)
}
'1' (x) a.s.
n -
n
Hence. from (2.10) and (2.11). we have for every n > l.x
(2.12)
,et.>
is transitive. by the Wijsman theorem [viz .• Zacks(1971.p.84)]. ~
-11
and ~n+1
-~
(x) == E{c(x-X ) I ~ (5) } == E{r:[c(x-X ) 10(~(S)
~
- -1
n+1
- -1
0
n+1
(s)
.__ .~
E{'¥n (x)
£
I :rn+ l }
RP •
a. s .•
and the lemma follows.
For a large class of df's (including the exponential family). T can be
-n
equivalently expressed in terms of a set of V-statistics. Thus. we write;:,"
T' == [ V(l) •...• V(q) ]. for n
(2.13)
-n
n
n
~ m ~ 1.
where m is the maximum of the individual degrees of the kernels for the V(j) .
n
From the results of Hoeffding(1948). it follows that if the kernels are all
square integrable. then
(2.14)
E{(T
-n
~.
where the
k
.
-1
-2
- ET )' (T - ET )} == n A + n ~2 + ....•
-n
-1-n
-n
~
semi~definite
1 are positive
matrices. For proving a few results
on V-statistics. to follow in the lemmas 2.3-2.6, we assume for simplificity that
e
q==l. and let
¢ is the symmetric kernel for V . Let then
n
(2.15)
(2.16)
so that
h==O ••••• m.
O.
A-.
'+'0
Lemma 2.3. I f
(2.17)
So == 0 and for q==l. Al
E{¢2(~1' ... '~m)}
n(n+l)E{(T n -T n+1 )2
<
00
•
I)~~n+l}
then
-+
2
m sl a.s .• as n
Proof. Following Miller and Sen(1972). for all n >
(2.18)
T
J . .. J ¢ (xl' •..• x)
n
Rpm
e+
where
.-
-m
-+
00
•
m,
m
II d [ c( ~j - X. )]
j==l
-1
j
5
(2.19)
T
(2.20)
T
nt1
*
n -1
EC
~h(~i t" 't~i );
nth
1
h
(h)
nth
h
~h(~1,···t~) - Ej=1~h-1 (~1t··'~j_1t~j+1,···t~)
(2.21)
+ E1'::'j <k<h
~h-2(~1t ... t~j-1t~j+1'···t~k-1t~+1" "t~)
k
... + (-1) 8 t for h=2 t ··. t m.
To avoid notational complexities t we shall take the case of m=2; the extension to
the case of m
~
3 can be made ina similar but more laborious way. For m=2 t
(2.22)
Hence t
Now t
(2.24)
T
nt1
- T
n+1 t l
so that
(2.25)
n(n+l)E{ (T
= n
-1
nt1
1,~n+1
- Tn+
)2
1t1
(n+1) [(n+1)
-1
}
n+l
i
E =l { ~1 (~i) -
.
.
-1 ,n+l{
}2
By the Klntchlne strong law of large numbers t (n+l)
L =l ~1(~i)-8
i
as n
+
00
{Tn+ltlt ~n+1} is a reverse martingale sequence which converges
whereas
,
+
2
a.s. to its expectation Ot and hence t T +
also a.s. converges to 0 as n
n 1t1
(2.26)
T
n,2
T
n+1 t 2
~
------::--1J.
1 2
n- n+,
+
00.
Again
_ (n)-l~ n ~*(X X
)
2
u i =1 'f'2 -l
. t -n +1'
Hence t
(2.21)
n(n+11(T
nt2
- Tn+ 1 ,2)
2
.::. (n-l)
n -2
n(n+1)2(2)
-2
2
8n(n+l)T n+ 1t2 +
n
*
2
{E i =l ~2(~it~n+1) } .
By the Berk (1966) reverse martingale propert y of U-statistic
a reverse martingale ,with expectation 0t so tffit
converge s to
° as n
+
00
Wherea s
S,
{Tn + 1 t 2 t An+l}
E(T~+lt21 ~n+1)
=
T~+1t2
a.s.
i s
e
n ¢2(X"X
'* _1. _n+1» 2 1... ' n +1}
E{( 1:'-1
1.-
(2.28)
n( n+ 1)-1
2
~
*2()
(1) (n+1)-1
¢2 ~i'~j + n n3
u
1..s.i<j..s.n+1
'*
E
¢*(X X)¢ (X X)
1..s.i<j<k<n+1 2 _i'_j 2 _i'_k'
By (2.28), the 2nd term on the rhs of (2.27) is giverLliy:
~(1)
n
Now,
is a U-statistic, and hence, converges a.s. to its expectation which is
-(2)
finite, while Un
'*
and hence, it a.s. converges to 0 as n
n
~
00
•
'*
is also a U-statistic with expectation E¢2(~1'~2)¢2(~1'~3) = 0,
~
00
•
Thus, (2.29) a.s. converges to 0 as
Finally, [E{n(n+1) (Tn ,l - Tn+1 ,1) (Tn ,2 - Tn +l ,2)I
a.s. to 0 [ by the Schwarz inequality and
(2.29)~
~n+1}
]2 converges
(2.29)]. Hence, the proof is
complete.
Lemma 2.4.Under the assumptions of the previous lemma,
(2.30)
Proof. Since
C
7-:J1n+l
0
~n+1'
n(n+1)E{(T n - Tn +1 )
(2.31)
2
I ~+l
Now, by Lemma 2.3, E{n(n+l) (Tn in nand n(n+1)E{(T
n
-
} = n(n+l)E[E{(Tn -
Tn+1)2~n+l} ~ m2~1
Tn+1)21~n+1}
£
Tn+1)2~n+1}1 ~n+l]'
a.s., as n
L . Hence, by
1
~
00
,
~
is
+
Loeve(1963,p~409), the
result follows.
Lemma 2. 5 • If
; •"
i
X }
--m
, then for every k
~
m,
(2.32)
Proof-Here also, for slmplicity, we take m
= 2. It suffices to show that
for each
~~~ n(n+l)E{(Tn,j- Tn+l ,j)2 1J:n+l} is integrable. Note that by (2.24),
sup
2 I Vn>k n(n+1)E{(T ,1 - Tn+l,l) ~n+1 }
j(=1,2),
n
(2.33) ~ sup
-1 n+l
2
- n> k [n L_
T
+1
1}
]
1{¢1(X,)
1. -.1.
_1
n,
sup M
n+1' say.
~
= n- 1,
7
Nl)'-', by definition, { M , tJ' : n :-- 2}
11 /,-,n
~
Kolmogorov inequality, for every t
(2.34)
p{
Is 11 reverse martingale, so that by the
0,
_< t- 2E[M 2 ]= t- 2s '
k
l
sup M > t}
n >k n
sup M for every k>
n>k n'
and this implies the integrability of
lengthy proof applies for j
1. A similar but more
= 2. Q.E.D.
Lemma 2.6. Under the conditions of Lemma 2.5,
(2.35)
I
n(n+l)E{( Tn - T + )2
n l
~n+l}
E[E{(Tn -
Proof-Since E{(Tn - Tn+1 )2 1 }rn+l}
is
~in
2
m sl ' as n
~s.
+
Tn+l)2~n+l}I~+1
00.
,:£n
] and since
n, the lemma follows directly from Lemma 2.5 and Loeve(1963,p.409). Q.E.D.
3. The main results.To motivate the main theorem, we require to introduce certain
assumptions. Note that by the Rao-Blackwellization, for every n> 1,
(3.1) nE['!' (x) - F(x)]2 <nE[F (X)-F(x)]2 = F(x) [1-F(x)],
n -
~
-
n -
-
-
-
V x-
E RP •
But, as in the example of the uniform df, the 1hs of (3.1) may tend to 0 as n
+
00.
So, in order to avoid this degeneracy, we assume that for every x, yE RP ,
nE[{'!'n (x)
- F(x)}{
(3.2) n+lim
00
_
'!' (y) - F(y)}]
n -
-
= h(x,y),
where there exists a subset 3C C RP such that for every xi l;: ~., 1=;<\..:; ~ ....in, the
matrix ( h(x.,x.) ) is positive definite for every m > 1. We also assume that
-1
(3.3)
-J
E{ n(n+l)[ '!'n (x)
_ -
converges to 0 as A
+
00
, uniformly in n( > n ), and
-
0
for every ~l' ~2 £ RP , where I(A) stands for the indicator function of a set A.We
may note that (3,4) implies that
whereas the uniform integrability condition in (3.3) implies that whenever. h(x,x»
0,
8
for every [ > 0,
co
2 -1
co
2
1/2,
[Lk-=n E [ '¥k(~) - 'JIk+l(~)]] [{Lk=n{ E[('JIk(~) - 'JIk+l(~» .l(n
'JIk(~)-
'JI +
k l
(3.6)
(~) I >[hl/2(~,~)
I 'fk + l )]}
° a.s.,
] -..
as n -..
00.
First, we present the main theorem. Later on we shall show that (3.3) and (3.4)
hold under general conditions on
Theorem 3.l.If
{T}
n -
W = [W(x) ,x
n
is transitively sufficient, then under (3.2), (3.3) and
n
(3.4), V* = [V *(x), x
n
'l'
Rp ] converges weakly to.a
£
-
RP ] whose mean (function) is
£
° and
1hultiparaineter'Gaus~ianl:function
covariance function is h(x,y).
Proof.Let F[j] be the marginal df of the jth compon nt of Xl' j=l, ... ,p. Let then
Y.. = F[.](X .. ), l<j<p, i> l,and let Y.
1J
J
(3.7)
1J
- -
Gn (:) = n
-1
-
-1
n
Li=lc(: - ~i)' :
(Y'l""'Y' ),
=
1
1p
p
£
E , n ~ 1,
where EP denotes the unit p-cube
{t:O<t<l}.
Also, let
.
..... .....- ........---.......
(3.8)
(3.9)
=
G(t)
p{ Y. < t }, t £ EP ;
V (t) =
n -
-1 -
-
_
E{ n l / 2 [Gn (t)
_ - G(t)]
I ~(s)}
n
Then, it suffices to show that V converges weakly in the Skorokhod Jl-topology on
n
DP[O,l] to a Gaussian function W = [W(t),t £ EP ] . For this, we require to show
(a) that the process
V
-
n
is tight and (b) that the finite dimensional distributions
(f.d.d.) of V converge to the corresponding ones of W, as n -..
n
00
•
For the proof
of (a), we principally follow the treatment of Section 5 of Neuhaus(197l) where the
original empirical process has been considered. We only need to replace his Lemma
5.2 by our Lemma 2.1, and the rest follows on the same line.
The proof of (b) is a little more complicated. By virtue of our Lemma 2.2, for
{
m ~ 1, ~
.2!: 1
Vn (t),
_
=I •••
~n , n ->
I- ~m 2.
1 and
1 }
A
is a reverse martingale. As such, for every
(AI' ... ,A )
m
I- 0, { L. mIl. ,~
-
J=
(t,),
J n -J
1r,
n
n > 1 }
-
is a reverse martingale. This tempts us to use the recent functional central limit
2002
theorems of Loynes(1970) and Brown(1971). For this, we let sn = E[Lk=nZk] ! where
(3.10)
k > 1.
Then. by the Brown(197l) modifications of Loynes' (1<:l70) results. we are only to
sho\" that as n
00
•
E~=n E(Z~ I ~k+1 )]/s~
(3.11)
[
(3.12)
S~2[
E >
for every
+
2
1, in probability,
+
'
E~=,n E{Zk I(lzkl > E sn) l1'k+1 }]
+
0, in probability,
O. NoW, (3.11) follows from (3.4) and (3:5) while (3.12) follows
from (3.3), (3.6) and the inequality
(3.13)
(E :
ui)2I(\Ei:1uil> mE)}
i 1
<
Let us now examine the conditions (3.3) and (3.4). Note that
If (x)
. n
~
= Ifn (x,T)
-n
~
is a bounded function and for a broad class of df's (including the exponential
family having a se of minimal complete sufficient statistics), we may by the usual
Taylor's expansion write
If (x)
n
If (x +8 + (T
n
~
~
~
-n
-8»
~
1
= F(x) + n- v (x) +
0
~
~
q
v.(x,8) [T j
J= 1 J ~ ~
n
E.
(3.14)
+
where
<;"
q
1E ..q 1v ... (x,8)[T .- 8. ] [T .. - 8.,]
J= J =' JJ. ~ ~
nJ
J
nJ
J
2
1< cn- , O<c< ooand v.,j=O,l, ... ,q, v .. "
IRn
+ Rn
t...
J
j,j'=l, ... ,q are all bounded and
JJ
continuous functions of x and 8 .For example, consider the case of the univariate
normal df with mean
(3.15)
where ~n
~ and variance
0
2
; the corresponding If (x) [cf. Lieberman and
n
*
n (x)
= (-Xn' S2)
-1 n
-1 n
2
n = (n Ei =l Xi' n f. i =l (Xi-Xn ) ) and Gn _ 2 is the Student t distri-
button with n-2 degrees of freedom. Now, from Johnson and Kotz(1970), we have
0.16)
C (t)
n
¢(t ) -
Z ( t) [
1
2
-1
L;t
(t +1 ) n
+
so that (3.14) is valid in this case.
From (3.14), we have
31.1• . '-'
0 (n
-2
) ].,
10
(3.17)
}:J~lVjj(~,~)(Tnj - Tn+ lj ) (Tnj + Tn + lj - 2 OJ) +
hl .2jf j'.2q Vjj,(~,~)[(Tnj-Tn+lj)(Tnj'- ej') + (Tn+lj-ej)(Tnj~-Tn+lj')] +
+ O(n
-2
a.s.
),
Thus, if the {T } form a U-statistics sequence, by the a.s. convergence results of
~n
Berk(1966), T
~n
(3.18)
~
e a.s., as n
~
~ n (x)~n +l(x)
=
~
~
with probability one, as n
~
00,
and hence, by (3.17), we have
L·ql(T.- T +1') [v.(x,e)
J=
nJ
n J
J ~ ~
~
2
+2L,~
IV, .,(x,e){o(l)}] + O(n- )
, J = JJ
~
Consequently, (3.3) and (3.4)
00.
~
follow~
from (3.18),
Lemma 2.5 and Lemma 2.6.
Remarks. Srinivasan(1970), Lilliefors(1967) and others have used the studentized
Kolmogorov-Smirnov statistic ( in the univariate case)
n1 / 2
(3.19)
{s~PI
x
~ (x) n
F (x)l)
n
as a goodness of fit statistic when F is an exponential or normal df, and in those
cases, the distribution of (3.19) is known to be independent of the population
parameters. Condition for this statistic to be independent of nuisance parameters
in the general case has been given by O'Reilly(197l). We should note that if
~
n
~
(x,T ) is absolutely continuous then
~n
n
(X. ,T ), i=l, ... ,n are ancilliary
~
~n
statistics, each having marginally the uniform(O,l). df[ viz., O'Reilly and
Quesenberry(1973)]. Let us now assume that the probability structure of the problem
is such that Stein's Theorem [cf. Zacks(197l,p.79)] is applicable, and let
~
denote the group of transformations operting on the sample space. Then, if g £ ~
is a monotone increasing transformation, it is easy to show that
(3.20)
(3.21)
X
= (Xl' ... ,Xn )
~
~
n (gX.,T
~
_n (gX»
~
Hence, noting that
if
~n
- Fn (gX.)
=
1
~
(X»
n (X.,T
1 ~n ~
sUPI~ (x T ) - Fn(x)
x
n ''''n
I
- Fn(X ).
i
occurs at one of the jump points of F (x)
n
is a continuous function, and by (3.21) we conclude that (3.19) is an invariant
statistic and that the distribution of (3.19) depends on the
11
maximal invariant funct ion in the parameter space. Thus, 1 f the maximal invariant
fllth'tlnn I:; l'lH1stant, then (3.19) hUH
11
distribution Independent of the
purU1fil~terH.
gVl'n wlll'n this distribution Is not free of nuisance parameters, under faIrly
~n (x,T
l 2
general conditions, n / [
.
~n
) - F (x)], X E R
n
will converge to a Gaussian
process and the percentage points of the original Kolmogorov-Smirnov statistic[ viz.,
sUPI F (x) - F(x)
x
n
I ]
will give a crude but distribution free upper bound for the
corresponding percentage points of the asymptotic distribution of (3.19).
such that
(4.1)
fg(x)dF(x)
=
and
]J
fg(x)d '¥ (x)
W
n
so that W
n
n -
~
2
fg (x)dF(x) <
= E[
Let then
00
g(~l)1 ~~s)}
,n
> 1.
By Lemma 2.2, { W , ~ , n > I} is a reverse martingale, and hence, under the
n
Sn
-
conditions of Brown(197l) or Loynes(1970), n
1/2
(W
-]J)
n
is asymptotically normal.
Consider then the general V-statistic
(4.2)
g (Xi , ..... ,X. )
- 1
-1
m
V
n
By Berk(1966), {V
n
, n':' m.
} forms a reverse martingale sequence. Loynes(1970) cites this
and conjectured that the functional central limit theorem is applicable to the
V-statistics. He, however,fails to provide the necessary verification of the
underlying conditions needed to apply his main theorem.By virtue of our Lemmas
2.2-2.6, we immediately conclude that the assumptions of Loynes' main theorem are
all met for V
n
,and hence, the results hold good. In fact, we may proceed one
step further and define a Rao-Blackwell V-statistic
(4.3)
V
*
n
tt>(s)
n
Under the assumption of transitivity of {Tn } , {V n*
], n .:. m.
~ ; n > m} is a reverse
n
-
martingale, and by using the results of Miller and Sen(1972), the asymptotic
*
normality and tightness of the process based on {V}
n
can be established. Similar
results hold for the von Mises differentiable statistical functions.
12
,
If for n > n ,
o
~
,
~
(x,T ) has a continuous derivative
n
-n
(x,T ) such that
n
-n
~
n
(x, T )
-n
< h (T ) is integrable, then under the assumption of transitivity of {T } ,
,
n -n
n
-n
n
{~ (x,T ),~, n > n
n
0
} is a reverse martingale. Hence, under !conditions
tially similar to those in Theorem 3.1, asymptotic normality of n l / 2{
essen-
~'(x,T
)
n
-n
- f(x)}
may be established.
It has been shown in Sen, Bhattacharyya and Suh(l973) that the bundle strength
of filaments under suitable conditions may be represented as
x >
oJ.
(4.4)
In view of the Rao-Blackwell estimator
*
Z
n
sup{
x[l- ~ (x)] : x >
Then, it is easily seen
(4.5)
p{
sup n 1 / 2 1
x
n
Z
n
sup{ x[l-F
.. n (x)]:
~ (x) of F(x), we consider
n
o}.
*
that {Z } is a reverse submartinga1e sequence. Also,
n
~ (x) _ F(x)1
>
n
K
€
} <
€
,K
€
<
00.
Hence on using our Theorem 3.1 and then proceeding as in the proof of the main
theorem in Sen, Bhattacharyya and Suh(1973), it follows that the
normality results apply to
as~nptotic
{Z *} - as well.
n
*
Finally, the weak convergence of V
n
for random sample sizes can also be
established by using our lemmas 2.1 - 2.6 along with the results of Sen(l973)
on the weak convergence of V for random sample sizes.
n
REFERENCES
[1]
r21
Berk, R. H. (1966). Limiting behavior of posterior distribution when
the model is incorrect. Ann. Math. Statist. 37 51-58.
Billingsley, P. (1968).
Hiley, New York.
Convergence of probability measures.
Brown, B. M. (1971). Martingale central limit theorem.
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[4J
John-
Ann. Math.
Ghurye, S. G. and Olkin, I. (1969). Unbiased estimation of some
multivariate probability densities and related functions. Ann.
Math. Statist. 40 1261-1271.
Johnson, N. L. and Kotz, S. (1970). Continuous univariate distributions -2.
Houghton Mifflin Company, Boston.
[6J Lieberman, G. J. and Resnikoff, G. J.
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(1955). Sampling plans for
J. Amer. Statist. Assoc 50 457-516.
Lilliefors, H. W. (1967). On the Kolmogorov-Smirnov test for normality
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399-402.
Loeve, M.
(1963).
Probability theory.
3rd. Ed. van Nostrand, New York.
Loynes, R. M. (1970). An invariance principle for reversed martingales.
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Miller, R. G., Jr. and Sen, P. K. (1972). Weak convergence of u-statistics
and von Mise's differentiable statistical functions. Ann. Math.
Statist. 43 31-41.
O'Reilly, F. J. (1971) On goodness of fit tests based on Rao-Blackwell
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University.
O'Reilly, F. J. and Quesenberry, C. P. (1973).
transformation. Ann. Statist. 1 74-83.
The probability integral
Sen, P. K. (1973). On weak convergence of empirical processes for
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Phil. Soc. 73 139-144 .
[J)']
f~p.n, P. K.,
nhattacharyya, B. B. and Suh, M. W. (1973). Limiting
behavior of the extremum of certain sample functions. Ann.
iJtatist. 1 297-311.
Seheult, A. H. and Quesenberry, C. P. (1971). On unbiased estimation
of density functions. Ann. Math. Statist. 42 1434-1438.
[16J
Srinivasan, R. (1970). An approach to testinr.:s the goodrless of fit of
incompletely specified distributions. Biometrika 57 605-611.
Tate, R. F. (1959). Unbiased estimation: Function of location and
scale parameters. Ann. Math. Statist. 30 3)+1-366.
[18J
Zacks, S. (1971).
New York.
The theory of statistical inference.
John Wiley,
Department 'of Statistics
North Carolina state University
Raleigh, North Carolina
Department of Biostatistics
University of North Carolina
Chapel Hill, North Carolina
~'~,,,,,,a
_