Sequential Point Estimation of the Location Parameter in the Location-Scale Family of Non-Regular Distributions

Abstract

In this paper, we consider sequential estimation of the location parameter based on the midrange in the presence of an unknown scale parameter when the underlying distribution has a bounded support. The estimation is done under squared loss plus cost of sampling. Stopping rules based on the range are proposed and are shown to be asymptotically efficient. The risks of the sequential procedures are compared with the Robbins sequential estimation procedure based on the sample mean. The former are shown to be asymptotically more efficient than the latter in the sense of the sample size when the density function changes sharply at the end points of the support. Koike (2007) observed a similar asymptotic superiority of the sequential estimation procedure based on the midrange in the sequential interval estimation procedure under the same condition.

Keywords:

• Extreme value
• Non-regular case
• Robbin' procedure
• Sequential point estimation

Subject Classifications:

• 62L12
• 62F10

ACKNOWLEDGMENTS

The author would like to thank Professor Masafumi Akahira of the University of Tsukuba for his valuable comments throughout the preparation of the paper. The author also thanks the Editor and the referee for helpful comments, which have led to improvements to the original version of the manuscript. This research was partly supported by the Grant-in-Aid for Scientific Research (C) 17540101, Japan Society for the Promotion of Science.

Notes

Recommended by T. N. Sriram