Plug-In Two-Stage Normal Density Estimation Under MISE Loss: Unknown Variance

Abstract

Consider independent observations X 1, X 2,… having a common normal probability density function with − ∞ < x < ∞ and unknown variance σ2 ( > 0). We propose to estimate f(x;σ2) by a plug-in maximum likelihood (ML) two-stage estimator under the mean integrated squared error (MISE) loss function. Our goal is to make the associated risk not to exceed a preassigned positive number c, referred to as the risk-bound. Because no fixed-sample-size methodology would handle this estimation problem, we design an appropriate two-stage estimation methodology that is shown to satisfy the asymptotic (i) first-order efficiency property (Theorem 2.1), (ii) first-order risk-efficiency property (Theorem 2.2), as well as (iii) second-order efficiency property (Theorem 2.3). The performances of the proposed methodology for small, moderate, and large sample sizes are examined with the help of simulations. We have noticed some limited robustness of the proposed methodology under mixture-normal population densities, in which case the asymptotic second-order efficiency property (Theorem 3.1) is shown. Illustrations are included with real data and analysis. Overall, we feel that the proposed two-stage plug-in density estimation methodology performs remarkably well.