Distributions of Sequential and Two-Stage Stopping Times for Fixed-Width Confidence Intervals in Bernoulli Trials: Application in Reliability

Abstract

We develop the exact distribution of the stopping variable of a sequential procedure that was originally given by Robbins and Siegmund (1974). The stopping variable was designed for estimating the log-odds in a sequence of Bernoulli trials. Using our exact distribution of the stopping variable, we give explicit formulas for the expected value and mean squared error for the estimator of the odds at stopping. An alternative two-stage procedure is then given and some of its important characteristics are exactly evaluated. It is shown that if the probability of success p is not too small or too large, the two-stage procedure is nearly as efficient as the purely sequential procedure. The results of this paper are then applied for designing an appropriate stopping time in a reliability experiment for estimating the ratio of the mean time between failures of two independent systems with exponential lifetimes.

Keywords:

• Bernoulli trials
• Exponential distributions
• Fixed-width confidence intervals
• Log-odds
• Ratio of MTBFs
• Sequential sampling
• Two-stage sampling

Subject Classifications:

• 62L12
• 62N05

ACKNOWLEDGMENTS

The authors are grateful for the comments of two referees and of the Associate Editor, which led to substantial improvement of the presentation.

Notes

Recommended by Linda J. Young