A general theory of purely sequential minimum risk point estimation (MRPE) of a function of the mean in a normal distribution

Abstract

A purely sequential minimum risk point estimation (MRPE) methodology with associated stopping time N is designed to come up with a useful estimation strategy. We work under an appropriately formulated weighted squared error loss (SEL) due to estimation of $g(\mu ),$ a function of μ, with $g({\overline{X}}_N)$ plus linear cost of sampling from a $N(\mu ,{\sigma }^2)$ population having both parameters unknown. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, $\to 0$) results is laid out including the first-order and second-order efficiency properties. Then, accurate sequential risk calculations are launched, which are then followed by two main results: (i) Theorem 4.1 shows an asymptotic risk efficiency property, and (ii) Theorem 5.1 shows an asymptotic second-order regret expansion associated with the proposed purely sequential MRPE strategy assuming suitable conditions on g(.). We also provide a bias-corrected version of the terminal estimator, $g({\overline{X}}_N).$ We follow up with a number of interesting illustrations where Theorems 4.1–5.1 are readily exploited to conclude an asymptotic risk efficiency property and second-order regret expansion, respectively. A number of other interesting illustrations are highlighted where it is possible to verify the conclusions from Theorems 4.1–5.1 more directly with less stringent assumptions on the pilot sample size.

Keywords:

• Asymptotic efficiency
• asymptotic risk efficiency
• bias correction
• first-order properties
• illustrations
• linear cost
• minimum risk
• optimal fixed sample size
• regret
• regret expansion
• risk efficiency
• second-order properties
• squared error loss (SEL)
• uniform integrability
• weighted SEL

Subject Classifications:

• 62L12
• 62L10
• 62L05

Acknowledgments

A preliminary version of this work was presented as an invited paper at the 7th International Workshop in Sequential Methodlogies (IWSM) hosted at Binghamton University, New York, June 18–21, 2019. The Associate Editor and reviewers shared helpful comments with us. We heartily thank them.