RAO-BLACKYffiLL ESTIMATOR OF DISTRIBUTION ~ICTION AND
ITS USE IN DETEffi~INING BUNDLE STRENGTH OF FILAMENTS
by
B. B.
Bhattacha~rJa
Department of Statistics
North Carolina State University, Raleigh, N. C.
Institute of Statistics Mimeo Series No. 1196
December 1978
RAO-BLACKWELL ESTIMATOR OF DISTRIBUTION FUNCTION AND
ITS USE IN DETERMINING BUNDLE STRENGTH OF FILAMENTS
ABSTRACT
Small sample and large sample properties of the estimate of the
bundle strength of filaments based on Rao-Blackwell empirical process
have been studied.
The sequence of statistics has been shown to be a
reverse submartingale process.
Martingal convergence theorems have
been used to establish the properties of the statistics.
AMS Classification:
Key Words & Phrases:
Primary 60BIO
Secondary 62B99
Gaussian process, Rao-Blackwell estimator, reverse
martingale, transitive sUfficiency, weak convergence,
bundle strength
1.
Let X
n,l
~
X
n,2
•••
~
~
INTRODUCTION
X
be the ordered values of Xl' X , ••• X
n,n
n
2
representing the strengths (nonnegative random variables) of individual
filaments in a bundle of n parallel filaments of equal length.
If we assume
that the force of a free load on the bundle is distributed equally on each
filament and the strength of an individual filament is independent of the
number of filaments in a bundle, then the mimimum load B beyond which all
n
filaments of the bundle give way is defined to be the strength of the bundle.
Now if a bundle breaks under load L, then the inequalities nX
(n - l)X
n,2
~
L
'
•••
,
X
n,n
~
L are simultaneously satisfied.
n,
1 ~ L,
Hence the bundle
strength B can be represented as
n
Bn =maxLnX n, l'
(1.1)
(n-l)Xn, 2' ···,Xn,n}.
The small sample and large sample statistical properties of B*
n
= BnIn
have
been investigated by Daniels (1945), Suh-Bhattacharyya-Grandage (1970),
Sen-Bhattacharyya-Suh (1973), Phoenix and Taylor (1973) and recently by
Sen-Bhattacharyya (1976).
We observed that if F (x) is the empirical distribution function of
n
Xl' X ' ••• X then n-1 B can be represented as supLx(l - F (x))}. In our
2
, n'
n
x;z;O
n
previous papers this representation was used to establish the large sample
properties of the distribution of mean bundle strength.
In the present
paper we assume that LX., i ;z; I} is a sequence of iid random variables having
J.
an absolutely continous distribution function F(x,
E)
where ~ belongs to a
subset of a finite dimensional Euclidean space and that there exists a
complete sufficient statistic (vector) T where
-n
I
T - (T
T
... T )
-n nl' n2'
'nq
for some q ;z; 1, q finite and independent on n •
. (1.2)
Therefore the Rao-Blackwell empirical distribution function
*n (x) = E[Fn (x)\Tn}
is the unique minimum variance unbiased estimator of F(x, ~).
The object of the present paper is to study the small sample and large
*
sample properties of the statistic sup[x(l- (x) )}. Section 2 deals with
x;;::O
n
the basic assumptions and preliminary notions. In Section 3 a few basic
lemmas and convergence and moment convergence of sup[x(l-
(x) )} have been
n
The asymptotic normality has been proved in Section 4 and a
x~O
established.
1lr
few concluding remarks have been made in the final section.
2.
ASSUMPTIONS AND NOTATIONS
Let us define
h (~)
o
(1)
= sup[x(l -
F(x, ~) ) } •
(2.1)
We assume that there exists a unique x 0(8)
such that
h (8) = x (8)[1-F(x (8), 8)}
0-
0-
0--
and for every 0 < e < h (8), there esists a 0(8) > 0 such that
0-'
-
suptx(l-F(x, 8)); \x-x (8)\ >0(8)} <h (8) _e.
-
(II)
0-
-
0-
Moreover for Ix - xo~) I :s: 0 (8),
(2.4 )
where c ' c «
2
l
constants.
00
but may depend on 8) and k1 , k2 , 1 :s: k 1 , k2 <
00
are postive
(III) We also assume that
o<
F(x (8), 8) < 1
0-
-
and F(x, ~) has a continuous and finite density function f(x, ~) for all
Ix -
X
o
(~)
I<
0 (.§.), where
(2.6)
(IV)
In Bhattacharyya and Sen (1977) sufficient conditions for asymptotic
1
normality of n2[~ (x)
F(x, 8)} have been given.
'n
In the present paper we
1
shall assume n2[~ (x) - F(x, ~)} converges in distribution to a Gaussian
n
random process with an appropriate topology and the Gaussian random process
Moreever there exists a subset XC Rl
is continuous with probability one.
such that for every x.]. eX, i = 1, 2, ••• , m,
limnE[~ (x.) - F(x., 8)H~ (x.) - F(x., §.)} = h(x., x., .e.)
n~
n].
].
-
n
J
J
].
J
and the matrix h(x., x., 8) is positive definite for every m ~ 1.
,
(V)
].
J
-
x (8) eX for every _8 •
o -
(VI)
3.
Lemma 3.1.
SOME BASIC LEMANS
Let [X.,
i ;;:: l} be an arbitrary sequence of non-negative random
].
variables (not necessarily stationary or independent), such that for some
p
>
0,
s~p
].
f t xPdP(Xi : :; x)
00
= ~(t)
... 0
as t-too •
Then for every e
> 0,
1
lim p( Max X. ;;:: e nP }
n-.cc
l~i~n ~
Proof:
For every e
>
1
P
0, 13( e n )~ as n-.cc.
=0
.
Consequently there exists an
an n (e, 0) such that
o
1
P
13( e n ) ~ 0 e P for n ;;:: n (e 0)
o '
Also from (3.1) ,
Hence by (3.3) and (3.4) ,
1
n
pC Max X. ;;:: e nP} ~ ~ PlX.
l~i~n ~
i=l
~
1
~ ~ 13(e nP)/e P n = e- P 13(e nP ) ~ 0
i=l
n
1
-
-
for n ;;:: n (e, 0) •
o
Lenuna 3.2
For every e
>
0, there exists a positive k
e
which is finite and
independent of ..§ such that for some n ,
o
uniformly in
Proof:
~
•
By Markov inequality, the left hand side of
(3.6)
is bounded above
1
1
'
Nothing that ~ (x) = ELF (x)!T } and using
k~ ElSUp n~1 *n (x) - F(x, ..§)
n
n-n
x
Jensen's inequality on conditional expectation
I} .
1
1
E[Sup n2 \'n(X) - F(x, ~)I} ~ E[Sup n2 \F (X) -.F(x, ~)I}.
n
x
x
From Dvoretsky, Kiefer and Wolfowitz (1956, p646) we have
1
p[Sup n2 (F(x, 8) - F (x)) > r} ~ c el
x
n
for all r
~
° where cl
does not depend on n or 8.
2r2
From this result the
lemma follows.
Lemma 3.3
Proof:
For every sequence [s } of positive s
n
n
>
° with lim s
n
= 0,
The proof follows easily from the tightness of the Rao-Blackwell
empirical process.
(Bhattacharyya-Sen (1977)).
Let ern(x)
= er(Xl ,
X '
2
••• , X ) be the er-field generated by Xl' X2 ' '.', X and er(T ) be the
n
n
~
er-field generated by T. Also let d = er(T
T
••• ) and therefore d
-n
n
-n' -n+l'
n
is a monotone non-increasing er-field.
Lemma 3.4
Proof:
[Sup x(l-, (x)), d} is a reverse submartingale process.
n
n
Since [X , n ~ I} are iid random variables and {T } is complete
n
-n
and sufficient, it follows from Bahadur (1954, p443) that it is transitive.
Hence by Wijsman Theorem (Zacks 1971, p84) erne!) and er(T + ) are condi~
n l
tionally independent given er(T ).
-n
'n+l(X)
= E[Fn(x)!er(!n+l)} = E[E{Fn(x)\er(!n'
= E[, n (x)ler(T-n+l)} = E[, n (x)\d+
n l}
Hence [, (x),
n
!n+l)}ler(!n+l)
a.s.
d}
is a reverse martingale process. From this the result
n
follows immediately.
Under assumptions (VI) for every e > 0,
Theorem 3.1
=0
lim p[Sup Ix(l- *n(x)) - x(l-F(x, ..§))\ > e}
n-P
x
Proof:
Sup IX(l-
~O
~ max[
1lr
n
(x) - x(l- F(x, ~)) I
X
Sup
OS:~X
n,n
n,n
Iw (x) - F(x, ~I,
n
Sup
X>Xn,n x(l- F(x, ..§))}
.1.
.1.
By Lemma 3.1 X
= 0 (n 2 ) and n 2 Sup 1* (x) - F(x, 8)1 is bound in
n,n
p
OS:~
n
probability by Lemma 3.2. Moreover E8(~) <co implies lim x(l-F(x, 8))
.p
and hence
Sup
x:':X
-
.
x (1 - F (x, 8)) ... 0 as X
-
:x-tco
=0
-
-P as n-P.
n,n
n,n
Corollary 1
Under the conditions of Theorem 3.1,
x
8)).
almost surely to Sup x(l - F(x,
Sup x(l-
1\1
n
(x)) converges
x
Proof:
Since
Isup x(l-w (x) - Sup x(l-F(x, 8))1
x
n
-
(3.12 )
p
~ Sup lx(l- "'n(x) - x(l-F(x, ..§))I ... 0,
x
and by Lemma 3.4, Sup x(l- * (x)) is a nonnegative reverse martingale which
n
x
converges almost surely to a random variable, the Corollary follows.
Corollary 2
-
If x
no
almost surely to x
Proof:
(1 - 111 (x
))
'n no
=
Sup x(l x
(e).
1lr
'n
(x)), then x
no
converges
0-
Since x(l- * (x)) converges a.s. to the continous function
n
x(l-F(x, e)) uniformly in x and x (8) is unique by (2.2), hence x
-
0 -
converges a.s. to x (8).
0-
no
Corollary 3 Under the conditions of Theorem 3.1 and Corollary 2,
as
,(x )- F(x (8))'" O.
n no
0 Proof: 1\1 (x ) - F(x (8)) = F(x ) - F(x (8)) + 1\1 (x ) - F(x ).
'no
0 no
0 - n no
no
as
By Corollary 2 and continuity of F, F(x
)- F(x (8))'" 0 and also
no
0as
1lf (x ) - F(x ) ... O. Hence the result.
n no
no
Theorem 3.2
Under the same conditions as in Theorem 3.1,
lim E[Sup x(l-,~ (x))}k ~ (x (8))k (l-F(x (8), 8)k
n-.co
x
'n
0 . 0 Proof:
for k=l, 2,
This follows directly from the fact [Sup x(l-1lf (x))}k, k=l, 2, •••
x
n
are reverse submartingale (Chung 1974, p238) sequences.
4. ASYMPTOTIC NORMALITY
1
n2 ( [Sup x(l - 1\r n (x)} h (8)) converges in law to a normal distribution with mean 0 and variance
Theorem 4.1
Under the conditions stated in Section 2
0-
(x (8))2 hex (8), x (8), 8).
0-
Proof:
0-
0-
-
Let us choose a o(~) satisfying (2.3) and (2.4).
Then using (2.4)
and (2.9), it is easy to show
max[Sup x(l-1lf n (x); Ix-x (.§.)\ > o(~)}
o
x
"S;
ho(~) -
n
(4.1)
where n > 0, in probability as n-.co.
With the definition k
I
n
*
2
(13) =
I (13)
n
(~l)
in (2.4) we let 13=1/2k
Lx: Ix - x
0
2
and put
(§) I < n-13 log n}
= [x: Ix - x0(8)
\
-
< 0(8)
but x ~ I (13)}.
'n
Following the same lines of argument as in Sen-Bhattacharyya (1976), we
can show
(4.2)
(4.4)
n~
in probability as
where 0 < d < c •
2
Finally, for x e I (~),
n
lsup x(l- 1\1 (x»
n
- h (6) - x (6)[1\1 (x (6»
00n 0-
Since x (6)[1\1 (x (6») - F(x (6), 6)} is
0-
n
0-
0-
-
- F(x (6), 6)}
0--
I
_.1.
= 0 (n 2)
p
(4.5)
_.1.
0p (n 2), it follows from. (4.1),
(4.4) and (4.5)
*
Sup x(l- n (x»
in probability as
Combining
= [Sup x(l-1\1 (x»; xeI (~)}
n
n
(4.6)
n~.
(4.5), 4.6) and assumptions (IV), (V) of Section 2 Theorem 4.1
is established.
5.
CONCLUDING REMARKS
An alternative method of proving Theorem
Mapping Theorem (Billingsley
4.1 may be given using Continous
1968) as was done by Phoenix and Taylor (1973).
We can also use the method used in this paper to establish asymptotic properties of a general class of nonnegative statistics expressible as an extremum
of a functional of Rao-Blackwell empirical process.
Acknowledgement:
I am indebted to Dr. P. K. Sen of the University of North Carolina at
Chapel Hill for helpful discussions.
REFERENCES
Bahadur, R. R. (1954). Sufficiency and statistical decision function.
Ann. Math. Stat. 25:423-462.
Bhattacharyya, B. B. and Sen, P. K. (1977). Weak convergence of the
Rao-Blackwell estimator of a distribution function. Annals of
Probability 5:500-510.
[3J
Billingsley, P. (1968).
Wiley, New York.
[4J
Chung, Kai Lai (1974).
Press, New York.
Convergence of Probability Measures.
A Course in Probability Theory.
John
Academic
Daniels, H. E. (1945). The statistical theory of the strength of a
bundle of threads. Proc. Roy. Soc. A. 183:405-425.
[6J
Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic
minimax character of the sample distribution function and
of classical multinormal estimator. Ann. Math. Stat. 27:642-669.
[7J
Phoenix, S. I. and Taylor, H. M. (1973). The asymptotic strength
distribution of a fiber bundle. Advances in Applied Probability.
5:200-216.
[8J
Sen, P. K. Bhattacharyya, B. B. and Suh, M. W. (1973). Limiting
behaviour of the extremum of a certain sample function. Ann.
Statistics. 1:297-311.
[9J
Sen, P. K. and Bhattacharyya, B. B. (1976). Asymptotic normality of
the extremum of a sample function. Z. Wahrscheinlichkeitstheorie
Verw Gebiete 14:113-118.
[lOJ
Suh, M. W. Bhattacharyya, B. B. and Grandage, A. (1970). On the
distribution and moments of the strength of a bundle of filaments.
Journal of Applied Probability 7(3):712-720.
[llJ
Zacks, S. (1971).
New York.
The Theory of Statistical Inference.
John Wiley.