Improved confidence set estimators of a multivariate normal mean and generalizations

Abstract

LetX be the observed vector of thep-variate (p≧3) normal distribution with mean θ and covariance matrix equal to the identity matrix. Denotey +=max{0,y} for any real numbery. We consider the confidence set estimator of θ of the formC δa,φ={θ:|θ−δa,φ(X)}≦c}, whereδ a,φ=[1−aφ({X})/{X}2]+X is the positive part of the Baranchik (1970,Ann. Math. Statist.,41, 642–645) estimator. We provide conditions on ϕ(•) anda which guarantee thatC δa.φ has higher coverage probability than the usual one, {θ:|θ−X|≦c}. This dominance result will be shown to hold for spherically symmetric distributions, which include the normal distribution,t-distribution and double exponential distribution. The latter result generalizes that of Hwang and Chen (1983,Technical Report, Dept. of Math., Cornell University).