Bayes and robust Bayes predictions in a subfamily of scale parameters under a precautionary loss function

ABSTRACT

This paper deals with Bayes, robust Bayes, and minimax predictions in a subfamily of scale parameters under an asymmetric precautionary loss function. In Bayesian statistical inference, the goal is to obtain optimal rules under a specified loss function and an explicit prior distribution over the parameter space. However, in practice, we are not able to specify the prior totally or when a problem must be solved by two statisticians, they may agree on the choice of the prior but not the values of the hyperparameters. A common approach to the prior uncertainty in Bayesian analysis is to choose a class of prior distributions and compute some functional quantity. This is known as Robust Bayesian analysis which provides a way to consider the prior knowledge in terms of a class of priors Γ for global prevention against bad choices of hyperparameters. Under a scale invariant precautionary loss function, we deal with robust Bayes predictions of Y based on X. We carried out a simulation study and a real data analysis to illustrate the practical utility of the prediction procedure.

KEYWORDS:

• Bayes prediction
• Gamma distribution
• Minimax prediction
• Precautionary loss function
• Robust Bayes prediction

MATHEMATICS SUBJECT CLASSIFICATION:

• 62C10
• 62F15
• 62F35

Acknowledgments

The authors are grateful to the Editor in Chief and two anonymous referees for making valuable comments and suggestions on an earlier version of this article which improved our paper.

Funding

Ahmad Parsian’s research is supported by a grant of the Research Council of the University of Tehran.