Density Estimation: Normal with Unknown Variance
by
William Pepe
University of Connecticut
Coauthors: Nitis Mukhopadhyay

Consider independent observations having a common normal probability density function with zero mean and unknown variance. We propose estimating the normal density function sequentially under the mean integrated squared error (MISE) loss function. Our goal is to make the associated risk not to exceed a pre-assigned positive number c, referred to as the risk-bound. Since no fixed-sample-size methodology would be able to handle this estimation problem, we design an appropriate two-stage density estimation methodology that is shown to satisfy the asymptotic (i) first-order efficiency property, (ii) first-order risk-efficiency property, as well as (iii) second-order efficiency property. The performances of the proposed methodology for small, moderate, and large sample sizes are also investigated with the help of large-scale simulations. With the help of simulations, we have seen a kind of robustness of the proposed methodology under mixture-normal population densities. Illustrations are included with the help of real data and analysis. Overall, we can report that the proposed two-stage density estimation methodology performs remarkably well.

Date received: April 22, 2007