ON ESTIMATION FOLLOWING SELECTION FROM NONREGULAR DISTRIBUTIONS

Abstract

Independent random samples are drawn from k (≥ 2) populations, having probability density functions belonging to a general truncation parameter family. The populations associated with the smallest and the largest truncation parameters are called the lower extreme population (LEP) and the upper extreme population (UEP), respectively. For the goal of selecting the LEP (UEP), we consider the natural selection rule, which selects the population corresponding to the smallest (largest) of k maximum likelihood estimates as the LEP (UEP), and study the problem of estimating the truncation parameter of the selected population. We unify some of the existing results, available in the literature for specific distributions, by deriving the uniformly minimum variance unbiased estimator (UMVUE) for the truncation parameter of the selected population. The conditional unbiasedness of the UMVUE is also checked. The cases of the left and the right truncation parameter families are dealt with separately. Finally, we consider an application to the Pareto probability model, where the performances of the UMVUE and three other natural estimators are compared with each other, under the mean squared error criterion.

Keywords:

• Conditionally unbiased
• Left truncation parameter family
• Lower extreme population
• Maximum likelihood estimator
• Mean squared error
• Natural selection rule
• Right truncation parameter family
• UMVU estimator
• Upper extreme population

ACKNOWLEDGMENTS

The research of the first author was supported by a fellowship awarded by the research council of the Katholieke University Leuven (Belgium), while the research of the second author was supported by the F.W.O.—Flanders Research Grant No. 1.5593.98N and Project GOA/98/06 of Research Fund K.U. Leuven. Authors are also thankful to the associate editor for suggestions.