Expectation bounds on linear estimators from dependent samples

Abstract

Let X1,X2,...,Xn be a sample of arbitrary, possibly dependent, random variables, with possibly different marginal distributions, and denote by X1:n≤X2:n≤≤Xn:n the corresponding order statistics. Using the notation μi=EXi and σi 2=VarXi, i=1,2,...,n (assumed finite), it is proved that for any real constants λ1,λ2,...,λ n,∑i=1nλi(EX i:n-μ̄)≤∑i=1n(ci-λ̄) 21/2∑i=1n((μi-μ̄) 2+σi 2)-nVarX̄1/2,where μ̄=n-1∑i=1 nμi, λ̄=n-1∑i=1 nλ i, X̄=n-1∑i=1nXi and (c1,c2,...,cn)′ is the l2-projection of the vector (λ1,λ2,...,λn)′ onto the convex cone of componentwise nondecreasing vectors in Rn (in particular, ci=λi for all i if and only if λi is nondecreasing in i). A similar lower bound is also given. The bound is sharp when the X's are exchangeable

moreover, it provides an improvement over the known bounds given by (Arnold and Groeneveld, 1979 Ann. Statist. 7, 220-223, Aven, 1985 J. Appl. Probab. 22, 723-728 and Lefèvre, 1986 Stochastic Anal. Appl. 4, 351-356). © 2001 Elsevier Science B.V.