ESTIMATION OF THE MEAN OF A e1-EXPONENTIAL MULTIVARIATE DISTRIBUTION

Abstract

We consider the estimation of the mean Ö of a £i-exponential multivariate distribution with density i ^ j^exp (-J!^), where ||i,||, = ZLi lf<l is the fi-norm in IP' and a > 0 is a scale parameter. We investigate the general class of estimators, X + g{X), and show they dominate the minimum risk equivariant estimator X under the usual quadratic loss ||i5(A') -where llfll = (EiLi I/.-)'^^ is the Eudidean norm in K'. When er = 1, the main domination condition is obtained through the partial differential inequality 4 divff + 2X • S^g + ||s|p < 0. We also consider the case where u is unknown. Furthermore we also treat the general case where the sampling distribution is a Scale mixture of £i-exponential distributions.

Estimation under ℓ1-Symmetry

“…Furthermore his method is very specific to this type of estimators and cannot be generalized. In [13], we give a general condition for improvement of d over X. It is expressed in terms of a differential inequality involving the function g, that is 2(2 div g+ X."…”

“…In the case where the distribution is specified as the a 1 -exponential distribution, Fourdrinier and Lemaire [13] show that a Stein-type-like identity is available for a weakly differentiable function g, that is,…”

“…However it is worth noting that (see [13]), when we specify the distribution as the a 1 -exponential distribution given by (1) (8) and (10) Condition (8) is an assumption on the sampling distribution and it is tractable for numerous distributions. Thus we first give three examples of a 1 -symmetric distributions with their corresponding second invert moment.…”

“…The regularity and, particularly, the finiteness of the risk are satisfied (see [13] for a full account). Straightforward calculations show that…”

“…Because this name is also used for the spherical density [16]) we will refer to (1) as the a 1 -exponential distribution (see [13]). Osiewalski and Steel [18] consider the general class of multivariate a q -spherical distributions (for 0 < q < .).…”