Asymptotic Normality of Sequential Stopping Times with Applications: Confidence Intervals for an Exponential Mean

Abstract

In sequential methodologies, finally accrued data customarily look like $\{N_{\nu },X_1,...,X_{N_{\nu }}\}$ where $N\equiv N_{\nu }$ is the total number of observations collected through termination. Under mild regulatory conditions, a standardized version of $N_{\nu }$ follows an asymptotic normal distribution (Ghosh–Mukhopadhyay theorem) which we highlight with a number of illustrations from the recent literature for completeness. Then, we emphasize the role of such asymptotic normality results along with second-order approximations for stopping times in the construction of sequential fixed-width confidence intervals for the mean in an exponential distribution. Two kinds of confidence intervals are developed: (a) one centred at the randomly stopped sample mean ${\bar{X}}_{N_{\nu }}$ and (b) the two other centred at appropriate constructs using the stopping variable $N_{\nu }$ alone. Ample comparisons among all three proposed methodologies are summarized via simulations. We emphasize our finding that the two fixed-width confidence intervals centred at appropriate constructs using the stopping variable $N_{\nu }$ alone perform as well or better than the customary one centred at the randomly stopped sample mean.