Optimal, nearly optimal and robust procedures for the exponential distribution with relative linear exponential loss

Abstract

In this paper, the problem of sequentially estimating the mean of the exponential distribution with relative linear exponential loss and fixed cost for each observation is considered within the Bayesian framework. An optimal procedure with a deterministic stopping rule is derived. Since the corresponding value of the optimal deterministic stopping rule cannot be obtained directly, an approximate optimal deterministic stopping rule and an asymptotically pointwise optimal rule are proposed. In addition, we propose a robust procedure with a deterministic stopping rule, which does not depend on the parameters of the prior distribution. All of the proposed procedures are shown to be asymptotically optimal. Some numerical studies are conducted to investigate the performances of the proposed procedures. A real data set is provided to illustrate the use of the proposed procedures.

Keywords:

• asymptotically optimal
• asymptotically pointwise optimal
• deterministic stopping rule
• optimal procedure
• relative LINEX loss
• robust

AMS 2010 Subject Classifications:

• Primary: 62L12
• Secondary: 62C10

Acknowledgements

This research is partially supported by the National Science Council of ROC. The authors thank the referees and editor for constructive suggestions leading to an improved presentation of the paper.