Estimation of the mean vector in a singular multivariate normal distribution

Abstract

This paper addresses the problem of estimating the mean vector of a singular multivariate normal distribution with an unknown singular covariance matrix. The maximum likelihood estimator is shown to be minimax relative to a quadratic loss weighted by the Moore-Penrose inverse of the covariance matrix. An unbiased risk estimator relative to the weighted quadratic loss is provided for a Baranchik type class of shrinkage estimators. Based on the unbiased risk estimator, a sufficient condition for the minimaxity is expressed not only as a differential inequality, but also as an integral inequality. Also, generalized Bayes minimax estimators are established by using an interesting structure of singular multivariate normal distribution.

Improved second order estimation in the singular multivariate normal model

“…An application of the Tsukuma and Kubokawa (2015) technique developed in Subsection 2.1 allows in essence to reduce the dimension from p to r. Since r < min(n, p), this in effect turns the problem into a classical setting where the sample size is greater than the dimension, and allows for classical proof techniques to be applied.…”

“…Before presenting the proofs of the statements from Section 2, we explain the techniques employed by Tsukuma and Kubokawa (2015) to work around the singularity of the covariates in the model. Define the sample mean and covariance matrix to bē…”

“…The decision-theoretic results are described in Section 2. For each of the three problems, we construct an appropriate unbiased estimator of the risk (URE) using Stein's and Haff's lemmas (Stein, 1986;Haff, 1979b;Tsukuma and Kubokawa, 2015). We then consider the class of estimator given by constant multiples of a naive estimator, and minimize an upper bound on the difference in risk to obtain estimators that dominate the naive estimator.…”

“…Recently, Tsukuma and Kubokawa (2015) investigated the problem of estimating the mean vector of a multivariate normal distribution when the unknown covariance matrix is singular. By deriving an unbiased risk estimator for the quadratic loss, they were able to give sufficient conditions for an estimator to dominate the maximum likelihood estimator.…”

On shrinkage estimators improving the James-Stein estimator under balanced loss function

“…Recent studies, in the context of shrinkage estimation, include Selahattin et al(2011), Amin et al(2020), Yuzba et al(2020). Tsukuma and Kubukawa(2015) address the problem of estimating the mean vector of a singular multivariate normal distribution with an unknown singular covariance matrix. Xie et al(2016) introduced a class of semi-parametric/parametric shrinkage estimators and established their asymptotic optimality properties.…”