Robust Bayesian prediction under a general linear-exponential posterior risk function and its application in finite population

ABSTRACT

We consider robust Bayesian prediction of a function of unobserved data based on observed data under an asymmetric loss function. Under a general linear-exponential posterior risk function, the posterior regret gamma-minimax (PRGM), conditional gamma-minimax (CGM), and most stable (MS) predictors are obtained when the prior distribution belongs to a general class of prior distributions. We use this general form to find the PRGM, CGM, and MS predictors of a general linear combination of the finite population values under LINEX loss function on the basis of two classes of priors in a normal model. Also, under the general ε-contamination class of prior distributions, the PRGM predictor of a general linear combination of the finite population values is obtained. Finally, we provide a real-life example to predict a finite population mean and compare the estimated risk and risk bias of the obtained predictors under the LINEX loss function by a simulation study.

KEYWORDS:

• Conditional gamma-minimax
• finite population
• linear-exponential loss
• posterior regret gamma-minimax
• prediction
• robust Bayesian analysis
• the most stable.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62D05
• 62F10
• 62F15

Acknowledgments

The authors are grateful to the anonymous referees for making helpful comments and suggestions on an earlier version of this article which resulted in this improved version. The authors would like to thank Dr. Mohammad Jafari Jozani from the University of Manitoba for useful comments and his help regarding the Application in Section 5.