Sequential Estimation Results for a Two-Parameter Exponential Family of Distributions

Abstract

We consider the problem of sequentially estimating one parameter in a class of two-parameter exponential family of distributions. We assume a weighted squared error loss with a fixed cost of estimation error. The stopping rule, based on the maximum likelihood estimate of the nuisance parameter, is shown to be independent of the terminal estimate. The asymptotic normality of the stopping variable is established and approximations to its mean and to the regret associated with it are also provided. The general results are exemplified by the normal, gamma and the inverse Gaussian densities.