Sequential interval estimation procedures for the mean exponential survival time and reliability function

The problems of confidence interval estimation are considered (a) for estimating the mean of a one-parameter exponential distribution and (b) for estimating the reliability function associated with the one-parameter exponential distribution. For the estimation problem (a), the confidence interval of ‘preassigned width and coverage probability’ is considered. For the estimation problem (b), the confidence interval of ‘fixed-ratio width and preassigned coverage probability’ is proposed. The failure of the fixed sample size procedures to deal with these estimation problems is established and sequential procedures are proposed to deal with them. The proposed sequential procedures are proved to be ‘asymptotically efficient and consistent’ in the Chow-Robbins [Chow and Robbins, Ann. Math. Statist. 36,457–462 (1965)] sense. Asymptotic distributions of the stopping times are derived and second-order approximations are obtained for the average sample numbers associated with them.