Applications of Sequentially Estimating the Mean in a Normal Distribution Having Equal Mean and Variance

Abstract

We suppose that we have at our disposal a sequence of independent observations from a N(θ, θ) distribution where θ(>0) is an unknown parameter. Such a model will make good sense to approximate a Poisson(θ) distribution, especially when θ(>0) is moderately large. The Fisher-information contained in the sample mean, the sample variance, and the MLE are included (Appendix A). The derivation of the UMVUE of θ and some remarks regarding computational complexities in numerically evaluating its expression, even for small fixed-sample-size n, are also included (Appendix B). Assuming the availability of a lower bound θ _L (>0) for θ, both purely sequential and two-stage bounded risk methodologies are developed for estimating θ. We have considered the analogs of fixed-sample-size MLE, sample mean, sample median, sample variance, and the UMVUE. First-order asymptotic properties of both purely sequential as well as two-stage sample sizes and the associated risk-bound for an analog of the MLE have been found. Extensive investigations based on computer-simulations have been carried out and the previously stated estimators of θ are compared with one another. The effect of the ratio θ _L /θ on the pilot sample size is critically examined. Overall, we have found that an analog of the fixed-sample-size MLE performs most satisfactorily. In the end, both proposed methodologies are successfully implemented to investigate how these work in two practical situations with the help of real datasets. They show that these methodologies stay fairly robust under mild departures from normality.

Key Words:

• Poisson mean
• Normal distribution
• Preassigned risk-bound
• MLE
• UMVUE
• Fisher-information
• Sequential sampling
• Two-stage sampling
• Simulations
• Real data examples
• Gestation time
• Typographical error

AMS 2000 Subject Classifications:

• 62L05
• 62L12

Acknowledgments

The two referees and the associate editor gave encouraging remarks. Some of these comments required us to rethink some of the practical issues. We thank the referees and the associated editor for their thoughtful comments.

Notes

Recommended by Tumulesh K. S. Solanky