摘要
Consider the two-sided truncation distrbution families written in the formf(x,θ)dx=w(θ<sub>1</sub>, θ<sub>2</sub>)h(x)I<sub>[θ<sub>1</sub>,θ<sub>2</sub>]</sub>(x)dx, where θ=(θ<sub>1</sub>,θ<sub>2</sub>).T(x)=(t<sub>1</sub>(x), t<sub>2</sub>(x))=(min(x<sub>1</sub>,…,x<sub>m</sub>), max(x<sub>1</sub>, …,x<sub>m</sub>))is a sufficient statistic and its marginal density is denoted by f(t)dμ<sup>T</sup>. The prior distribution of θ belongs to the familyF={G:∫‖θ‖<sup>2</sup>dG(θ)【∞}.In this paper, the author constructs the empirical Bayes estimator (EBE) of θ, φ<sub>n</sub> (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 φ<sub>n</sub>(t) is an asymptotically optimal EBE of θ.
Consider the two-sided truncation distrbution families written in the form f(x,θ)dx=w(θ_1, θ_2)h(x)I_([θ_1,θ_2])(x)dx, where θ=(θ_1,θ_2). T(x)=(t_1(x), t_2(x))=(min(x_1,…,x_m), max(x_1, …,x_m)) is a sufficient statistic and its marginal density is denoted by f(t)dμ~T. The prior distribution of θ belongs to the family F={G:∫‖θ‖~2dG(θ)<∞}. In this paper, the author constructs the empirical Bayes estimator (EBE) of θ, φ_n (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 φ_n(t) is an asymptotically optimal EBE of θ.
基金
Projects supported by the Science Fund of the Chinese Academy of Sciences.