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SLfSL19P.'s INZQUALITY VIA THECENTRAL LIMIT THEORZK

FRED W. UF

TECHNICAL REPORT NO. 378

AUGUST 5, 1986

PREPARED UNDER CONTRACT

1000144 6-K-0156 (11-042-267)

FOR THE 0FFCE OF NAVAL RESEARCH

Reproduction in Whole or In Part is Permittedfor any purpose of the United States Goverment

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DEPARTMENT OF STATISTICS___ ~~STANFORD UNIVERSITY \ E'r

CL. STANFORD, CALIFORNIA S ELECTE

,P3wA ~AUG 2 5M6

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SLEPIAN'S INEQUALITY VIA THE

CENTRAL LIMIT THEOREM

BY

FRED W. HUFFER

TECHNICAL REPORT NO. 378

AUGUST 5, 1986

Prepared Under Contract

NOO014-86-K-0156 (NR-042-267)

For the Office of Naval Research

Herbert Solomon, Project Director

Reproduction in Whole or in Part is Permittedfor any purpose of the United States Government

Approved for public release; distribution unlimited.

DTICELECTEAUG 25 1986

DEPARTMENT OF STATISTICS -U--

STANFORD UNIVERSITY

STANFORD, CALIFORNIA B

IT0

'- 12

SLEPIANIS INEQUALITY VIA THE

CENTRAL LIMIT THEOREM

By

Fred W. Huffer

We give a proof of Slepian's (1962) inequality which does not

rely on Plackett's identity or geometric arguments. The proof uses a

partial ordering of distributions which is preserved under convolutions

and scale transformations. Slepian's inequality may be formulated in .

terms of such a partial ordering. The properties of this partial

ordering allow us to obtain results for the multivariate normal distri-

bution by using the central limit theorem.

Tchen (1980) also noted the preservation under convolution

property and from this obtained Slepian's inequality in the bivariate

case. Further information and references concerning partial orderings

of probability distributions may be found in Eaton (1982) or Chapter I

of Stoyan (1983).

Let F be a collection of bounded continuous real-valued functions

defined on R k . Suppose that F is invariant under both translations and

scaling, that is, for any bcRk, c > 0 and f F, the function g

defined by g(x) - f(cx+b) also belongs to F. Define X << Y if X and i

Y are random vectors satisfying Ef(X) < Ef(Y) for all f c F.

Our object will be to prove inequalities concerning multivariate %

normal distributions. Let Z and A be any k xk covariance matrices.

Let X and Y be k-dimensional normal random vectors with = EY* -0,

%

a n~r ,enr., %. ~-rO .. *n . a a a -' 1 tn AMa aMan n p

coy - , coy - A. We wish to determine if X << * in which case

we also say that Z <<A. The following result can sometimes be used to

make this determination.

Proposition 1: Let X * and Y be as given above. Suppose the random

vectors X and Y satisfy EX - EY - 0, cov X - E, coy Y = A. If

X<<Y, then X <<Y*.

In our application X and Y will be chosen to have simple

discrete distributions so that the relationship X<< Y is easy to

verify.

Proof: First note that

(a) If U,V,W are independent with U <<V, then U+W<<V+W .

This follows by conditioning on the value of W and using the translation

invariance of F. Let X1,X2,X3.... be i.i.d. copies of X and

Y1 ,Y2,Y3,... be i.i.d. copies of Y. Since X<< Y, using (a) twice in

succession gives XI+X2 << Y1 +X2 << Y1 +Y2. By induction we obtain

S.+ Xn<< YY2 +'''+ Yn. By the scale invariance of F,

(b) U << V implies cU <<cV for all c > 0.

Thus n-/2 (X I.++ xn) << n-1/2(Y+Y2+.--+Yn). Now let n and use

the Central Limit Theorem to obtain X*<< Y

The next proposition is useful in extending the ordering << to a

broader class of covariance matrices.

2

14.

Proposition 2: Suppose A1 and A2 are any kx k covariance matrices

satisfying A1 < < A2. Define r - A2-A1 . Choose t > 0. If E and

z+tr are both nonsingular covariance matrices, then E << +tr.

Proof: Let 41,t2903 denote kx k covariance matrices. In terms of

covariance matrices (a) and (b) become:

(c) 01 << 2 implies 0143 << 02+43

(d) 1<<2 implies co 1 " c 0 2 for all c > 0 .

These are used implicitly in the following argument. Choose A small

enough so that both (E-cA1 ) and (E+tr-cA1 ) are positive definite

whenever 0 < c < A. By taking convex combinations of these matrices we

find that (E+sr-cA1 ) is positive definite (and therefore a covariance

matrix) when 0 < s < t and 0 < c < A. Now

z+sr - (E+sr-cAI ) + cA << (r+sr-Al) + cA 2 u E+ (s+e)r

Here we have used A A<< Thus z+sr<< E+(s+c)r for 0 < s < t and

0 < c < A. Since A does not depend on s, it is clear that

E<<E+tr as desired.

For x - (x1,x2,...,x k ) and y ' (y1 ,Y2 ,...,yk) define

rVy= (x1 vYl'x 2 vY 2,...,xkvYk) and xAy - (X1AYlX2AY2,...,XkAYk)

where v and A denote the maximum and minimum respectively. A function

f defined on R k is called L-superadditive if f(x)+f(y) < f(xVy)+f(xAy)

3.

for all x and y. This condition was introduced by Lorentz (1953) who

also showed that when f has continuous second partial derivatives, f

is L-superadditive if and only if

a2f(x) > 0 for all x and all i xj

See Marshall and Olkin (1979) for further information on L-superadditivity.

Proposition 3: Let F be the class of bounded, continuous, L-superadditive

k ~functions on R . Suppose E - (a) and H - (wi) are kx k non- *

ii ii

singular covariance matrices. If ai = ii for all i and aij < 7ij

for all i #J, then Z<< .

This result is very similar to Proposition 1 of Joag-dev, Perlman

and Pitt (1983) and it easily implies Slepian's inequality as given in

Slepian (1962). The argument needed to obtain Slepian's inequality is

basically the same as that in Corollary 1 of Joag-dev, et al. V

Proof: Let ,,0,,0 be k-dimensional vectors defined by

, a -1, 0 for i~p or q,

p q ,ai

op -0-, e i = O for ip or q

p -q -l, *i O for i p or qSq

Define the random vectors X and Y by

4

. .&

1 1P{X-ct} - P{X-0} - ' P{Y-e} - P{Y-0} 2

Since av 8 - 8 and aA 1 - 0, it is clear that X<<Y. Now applying

Proposition 1 leads to a corresponding ordering between normal random

vectors with covariance matrices

coy X Sp and coy Y - T

where the entries of Spq and T are given by Sp q = Spq = 1,pp qa

-1and 5 q 0 otherwise, T T~P T ~p Pij pp qq pq qp

and Tp = 0 otherwise. Since p and q are arbitrary, we have shownii

Spq <<Tp for all p # q. Now we can use (c) and (d) to deduce that

A <<A 2 where

A1 - . bpqpq A - bp1pq 2 pqpq p,qp<q p<q

and bpq are arbitrary nonnegative numbers. Choose bij (Wij -ij)/2

for all i< J. Now using Proposition 2 with r A-A n -E completes21

the proof.

".o For

b - 1

":T-:--

,,-.

References

Eaton, M.L. (1982). A review of selected topics in multivariate prob-

ability inequalities. Ann. Statistics, 10, 11-43.

Joag-dev, K., Perlman, M.D., and Pitt, L.D. (1983). Association of "-.

normal random variables and Slepian's inequality. Ann. Probability,

11, 451-455.

Lorentz, G.G. (1953). An inequality for rearrangements. Amer. Math.

Monthly, 60, 176-179.

Marshall, A.W., and Olkin, I. (1979). Inequalities: Theory of Majorization

and Its Applications. Academic Press, New York.4.°

Slepian, D. (1962). The one-sided barrier problem for Gaussian Noise.

Bell System Tech. J., 41, 463-501.

Stoyan, D., and (edited with revisions by) Daley, D.J. (1983). Comparison

Methods for Queues and Other Stochastic Models. John Wiley and

Sons, New York.

Tchen, A.H. (1980). Inequalities for distributions with given marginals. '.

Ann. Probability, 8, 814-827.

5?."9

6%

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Slepian's Inequality Via The TECHNICAL REPORTCentral Limit Theorem __ _ _ _

6. PERFORMING ORO. REPORT NUNBER

7. AUTHOR(*) V. CONTRACT OR GRANT NUMBZEfa)

Fred W. Huffer N00014-86-K-0156

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Department of Statistics AREA & WORK UNIT NUMBERS

Stanford University NR-042-267Stanford, CA 94305

O. CONTROLLINGOFFICE NAME AND ADDRESS 12. REPORT DATE

Office of Naval Research August 5, 1986Statistics & Probability Program Code 1111. 13. NUMBEROF PAGES ,

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APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (Conuinue on r#,vet, side II necoeear and Identif/y by block number)

Slepian's inequality, partial orderings of distributions,L-superadditive functions, multivariate normal distribution.

20. A§STRACT (Continue ,anrevere.* sde It necee. md Identity by block number)

We define a partial ordering of distributions which is preserved underconvolutions and scale transformations. Some properties of this partialordering are developed and then used to give a new argument for Slepian's(1962) inequality.

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