ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETERS OF TWO-SIDED TRUNCATION DISTRIBUTION FAMILIES

Consider the two-sided truncation distrbution families written in the formf（x,θ）dx=w（θ1, θ2）h（x）I[θ1,θ2]（x）dx, where θ=（θ1,θ2）.T（x）=（t1（x）, t2（x））=（min（x1,…,xm）, max（x1, …,xm））is a sufficient statistic and its marginal density is denoted by f（t）dμT. The prior distribution of θ belongs to the familyF={G:∫‖θ‖2dG（θ）<∞}.In this paper, the author constructs the empirical Bayes estimator （EBE） of θ, φn （t）, by using the kernel estimation of f（t）. Under a quite general assumption imposed upon f（t） and h（x）, it is shown that φn（t） is an asymptotically optimal EBE of θ.