Comparing Location Estimators for Exponential Mixtures Under Pitman's Measure of Closeness

Abstract

Carpenter and Hebert [Carpenter, M., Hebert, J. L. (1997). Estimating guaranteed lifetimes of systems in a random environment. Comm. Statistics Theory Methods 26:309–316] consider competing estimators for the minimum and maximum location parameters from joint samples of an exponential mixture. They develop bias-corrected estimators for these extrema and show that the new estimators dominate the maximum likelihood estimators in terms of absolute bias and mean squared error. In this article, we compare these estimators in terms of Pitman's measure of closeness. We present a general result that facilitates the calculation of the Pitman probabilities when the moment-generating function for the mixing distribution is known. We also provide sufficient conditions under which the estimators proposed by Carpenter and Hebert dominate the maximum likelihood estimators in terms of Pitman's measure of closeness, and we provide sufficient conditions under which the bias-corrected estimators are inadmissible in this measure.

Key Words:

• Moment generating function
• Gamma distribution
• Inverse Gaussian distribution

Acknowledgments