Plug-In Sequential Normal Density Estimation Under Unknown Variance: Mise Loss

Abstract

Consider independent observations X1, X2,… having a common normal probability density function [Formula: see text] and unknown variance σ2(> 0). We propose estimating f( x; σ2) with a purely sequential methodology 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. Since no fixed-sample-size methodology would be able to handle this estimation problem, Mukhopadhyay and Pepe (2009) first gave a two-stage sampling method. We show that our purely sequential density estimation methodology satisfies asymptotic (i) first-order efficiency property (Theorem 2.1) and (ii) first-order risk-efficiency property (Theorem 2.2) just like the Mukhopadhyay and Pepe two-stage methodology did. But, the present purely sequential density estimation methodology has better second-order efficiency property (Theorems 2.3 and 3.1) than those associated with the Mukhopadhyay and Pepe two-stage methodology. Small, moderate, and large sample performances are investigated with the help of simulations. Illustrations are included with the help of a real dataset.