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(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 familyF={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.

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

Consider the two-sided truncation distrbution families written in the form $$\f(x,\theta )dx = \omega ({\theta _1},{\theta _2})h(x){I_{{\theta _1},{\theta _2}]}}(x)dx,\theta = ({\theta _1},{\theta _2}).\]$$ $$\T(x) = ({t_1}(x),{t_2}(x)) = (\min ({x_1}, \cdots {x_m}),\max ({x_1}, \cdots {x_m}))\]$$ is a sufficient statistic and its marginal density is denoted by $\f(t)d{\mu ^T}\]$. The prior distribution of $\\theta \]$ belongs to the family $$\F = \{ G:{\int {\int {\left\| \theta \right\|} } ^2}dG(\theta ) < \infty \} \]$$ In this paper, the author constructs the empirical Bayes estimator (EBE) of $\\theta \]$, $\{\phi _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 $\{\phi _n}(t)\]$ is an asymptotically optimal EBE of $\\theta \]$.