Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function

Abstract

For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax.

Keywords:

• Bayes estimator
• Bounded parameter space
• Discrete distributions
• Least favorable prior
• LINEX loss function
• Minimax estimation

Mathematics Subject Classification:

• Primary 62C20, 62F30
• Secondary 62C10, 62F10

Acknowledgments

The authors are grateful to the Editor, two anonymous referees, and Professor Ahmad Parsian of University of Tehran for making helpful comments and suggestions on an earlier version of this article.