On Exact and Asymptotic Properties of Two-Stage and Sequential Estimation of the Normal Mean Under LINEX Loss

Abstract

In this article, we consider the bounded risk point estimation problem for the mean μ in a N(μ, σ²) distribution under a LINEX loss function. We have proposed both two-stage and sequential procedures with a goal that the associated risk functions approximately fall under a preassigned risk-bound ω (>0). Our two-stage and sequential procedures and the associated terminal estimators for μ are different from those of Takada (2006) and Chattopadhyay et al. (2005), respectively. We begin by presenting interesting asymptotic properties of the two-stage point estimator for μ followed by the exact distribution of the two-stage stopping time along with some exact properties of the terminal estimator for μ. We also develop the exact distribution of the sequential stopping time along with some exact properties of the terminal estimator for μ. In either case, the exact expressions are illustrated with numerical computations.

Keywords:

• Asymptotic risk bound
• Bounded risk problem
• Exact properties
• LINEX loss
• Normal mean
• Sequential procedure
• Two-stage procedure

Mathematics Subject Classification:

• Primary 62L10
• Secondary 62L12

Acknowledgment

The referee enthusiastically shared important thoughts and input that ultimately led us to correct some mishaps in earlier versions. We remain indebted to this referee for helpful remarks.