Point estimation based on confidence intervals: exponential families

Abstract

Point estimators are usually judged in terms of their centering and spread properties, the most common measures being expectation and mean squared error. These measures, however, do not agree with the requirement of equivariance under reparameterization. In exponential families, when the parameter of interest is a scalar linear function of the natural parameter, an optimal equivariant estimator exists and is given by the zero-level optimal confidence interval. This optimal procedure is obtained by conditioning on the sufficient statistic for the nuisance parameter. However, the required conditional distribution is usually difficult to compute exactly. Quite accurate approximations are provided by recently developed higher-order asymptotic methods, in particular, by the modified directed likelihood introduced by Barndorff-Nielsen (1986). In this paper, an approximation for the optimal conditional estimator is obtained, using the modified directed likelihood as an estimating function. A simple explicit version of the approximate optimal conditional estimator is also derived. Simulation results with many nuisance parameters confirm that the approximate conditional estimators improve remarkably on the maximum likelihood estimator.

Keywords:

• Conditional cumulants
• conditional inference
• confidence intervals
• estimating equations
• exponential families
• many nuisance parameters
• median unbiased estimators
• modified directed likelihood
• point estimation

^∗Corresponding Author

^∗Corresponding Author

Notes

^∗Corresponding Author