Plug-In Two-Stage and Sequential Normal Density Estimation Under MISE Loss: Both Mean and Variance are Unknown

Abstract

Consider independent observations having a common normal probability density function with x ∈ R, unknown mean μ(∈R), and unknown variance σ2(∈R +). We propose estimating f(x; μ, σ2) with both two-stage and purely sequential methodologies 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. No fixed-sample-size methodology would handle this estimation problem. We show that both density estimation methodologies satisfy an asymptotic (a) first-order efficiency property and a (b) first-order risk-efficiency property. Interestingly, purely sequential density estimation methodology has a better second-order efficiency property than that associated with two-stage methodology. Some robustness issues have been addressed. Small, moderate, and large sample performances are examined with the help of simulations. Illustrations are given with real data sets.