Mean Shrinkage Estimation for High-Dimensional Diagonal Natural Exponential Families.

Shrinkage estimators have been studied widely in statistics and have profound impact in many applications. In this paper, we study simultaneous estimation of the mean parameters of random observations from a diagonal multivariate natural exponential family. More broadly, we study distributions for which the diagonal entries of the covariance matrix are certain quadratic functions of the mean parameter. We propose two classes of semi-parametric shrinkage estimators for the mean vector and construct unbiased estimators of the corresponding risk. Further, we establish the asymptotic consistency and convergence rates for these shrinkage estimators under squared error loss as both $n$, the sample size, and $p$, the dimension, tend to infinity. Finally, we consider the diagonal multivariate natural exponential families, which have been classified as consisting of the normal, Poisson, gamma, multinomial, negative multinomial, and hybrid classes of distributions. We deduce consistency of our estimators in the case of the normal, gamma, and negative multinomial distributions if $p n^{-1/3}\log^{4/3}{n} \rightarrow 0$ as $n,p \rightarrow \infty$, and for Poisson and multinomial distributions if $pn^{-1/2} \rightarrow 0$ as $n,p \rightarrow \infty$.