Sequential Fixed-Width Confidence Interval Procedures for the Mean Under Multiple Boundary Crossings

Abstract

The Anscombe-Ray-Chow-Robbins purely sequential procedure provided a breakthrough for constructing a fixed-width (= 2d) confidence interval for a normal population mean μ with a confidence coefficient 1 − α when the standard deviation σ was unknown. They proposed a purely sequential procedure of the following kind:

with the associated confidence interval for μ.

But, a purely sequential procedure does not meet the exact consistency property. While Simons (1968) proved the existence of a universal fixed non negative integer r such that for all fixed μ, σ, α, and d, the magnitude of r has remained unknown.

We introduce a new methodology along with appropriate truncation in which the number of additional observations required beyond Q ₁ is determined by the sequential sampling process itself. Interesting properties and performances of this methodology under successive crossings are explored theoretically and compared with those of the existing procedures by large-scale simulations.

Keywords:

• Consistency
• Exact consistency
• First-order efficiency
• Repeated boundary crossing
• Second-order efficiency
• Simulations
• Truncation

Subject Classifications:

• 62L12
• 62L10
• 60G40

ACKNOWLEDGMENTS

A preliminary version of this article was presented by one of the authors (Sankha) in an invited paper session in the 3rd International Workshop in Sequential Methodologies (IWSM) held in Stanford, California, during June 14–16, 2011. The enthusiastic comments made by the participants from that conference were highly appreciated. Thoughtful comments made by an Associate Editor and the referees were very helpful and we express our sincere gratitude to these colleagues.

Notes

Recommended by T. K. S. Solanky