A counterexample to a conjecture on order statistics

Statistics & Probability North-Holland

Letters

21 January

19 (1994) 129-130

A counterexample on order statistics Luning

Received October Revised February

Keywords:

to a conjecture

Li

Indiana University, Bloomington,

Abstract: functions

1994

IN, USA

1992 1993

We give a counterexample of two order statistics. Order

statistics;

to a conjucture

inequalities;

in Ma (1992) and find bounds

ample for a conjecture that

X,, be the i.i.d. r.v.‘s with distriLet xi, x,,..., bution F, define Xo,, Xc2,, . . . , Xcn, as order statistics of Xi, X,, . . . , X,,, such that Xo, < Xc2, < .** -
O.

cov(u)

in Ma (1992b) which says

=cov(u(xci,),

U(X~j,))n*n~o

(4)

i.e., the covariance matrix of the vector nonnegative elements.

u has

(1)

func-

w7)1* (2)

U(X@))) = 4[Cov(u(X),

Theorem 1. Let XC1, and XC2) denote the order statistics from a sample of size 2, and let g( ) and h( > be nondecreasing functions, then Cov(g(X& G

where X has distribution [Cov(u(X),

nondecreasing

2. A theorem h(x(j,))

Ma (1992a) shows that for any real-valued tion u(. 1 when IZ= 2, Cov(~(X&

between

counterexample

1. Introduction

cov(g(x~i~)~

for the covariance

F(X))]‘<&

F, and

+ COVM?2,)J

i[Var( g( X))Var(

h( X))]

vu,))

“2.

The inequality becomes equality exactly if

Var(u(X)).

(3)

Equality (3) holds exactly if u(x) = c(F(x) - i) for some constant c. In this paper, we derive some bounds for Cov(g(Xo,), h(X,,,)) when n = 2. Also, for general n, we will give a counterex-

g(x)

=clh(x)

=c2(F(x)

-5)

for some constants c1 and c2. Proof. By (21, we have

Cov( g(Xo,) Correspondence to: Luning Li, Department Indiana University, Bloomington, IN, USA. 0167-7152/94/$07.00 0 1994 - Elsevier SSDI 0167-7152(93)E0092-8

h&2,))

- h(Xo,),

gPG2,)

- wG2,))

of Mathematics,

Science

=4[Cov(g(X)

B.V. All rights reserved

-h(X),

F(X))]*,

129

Volume

19, Number

2

STATISTICS

Gh)) = 4(Cov(g(X), ww)2, C4vd~ G%,))

& PROBABILITY

(5)

P(X,r, E [a,

F(X))Y,

(6)

= 8 Cov(g(X),

+ Cov(g(X& F(X))

hG%,))

Cov(h(X),

F(X))

+ Cov(g(X&

+?r,))

using (3), we get Cov(g(XoJ7

WG,))

< $[Var(g(X))

Hence

0

3. A counterexample The following is a counterexample to (4), i.e., to the conjecture of Ma (1992b). Let u(x) = Ita,bl(~), n = 3, let X be a random variable with distribution F, let F(a) = i, F(b) = f, then

bl X(3,E [a, q =P(X, E [a, bl, x2 E [a, bl, x, E [a, bl) =P(XE [a, b]) 1 =-8.

130

=P(+,,

u&31)) E [a,

bl,

X(3) E [a,

bl)

-P(+,, E [a, bl)P(X,,, E [a, bl) +_ [2!?]‘=-.g$
Var(h(X))]“*

which completes the proof.

P(X(1, E [a,

bl) = $i

P(X(,, E [a, bl) = %

Cov(u(XoJ7 W&J)

1994

and

plugging (5) and (6) to the first equation, we have that Cov(g(Xo,),

27 january

Similarly,

COVE

=4(Cov(h(X),

LETTERS

Acknowledgment

The author wishes to thank Professor B.D. Flury and Dr. B.E. Neuenschwander for their suggestions and comments.

References Esary, J.D., F. Proschan and D.W. Walkap (19671, Association of random variables, with applications, Ann. Math. Statist. 38, 1466-1474. Ma, C. (1992a), Variance bound of function of order statistics, Statist. Probab. Lett. 13 (11, 25-27. Ma, C. (1992b), Moments of functions of statistics, Statist. Probab. Lett. 15 (3), 57-62.