Estimation of the geometric survival distribution

This paper discusses both point and interval estimation of the survivor function S₀=Pr{X⩾x₀} for the geometric distribution. When the number of devices n⩾50, the performance of the maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) of S₀ are essentially equivalent with respect to the relative mean-square-error (RMSE) to S₀. However, when the failure probability per time unit p⩾0.50, and n⩽30, the UMVUE is preferable to the MLE with respect to the RMSE. For interval estimation of S₀ with no censoring, 4 asymptotic interval-estimators are derived from large-sample theory, and one from the exact distribution of the negative binomial. When p⩽0.2 and n⩾30, all 5 interval-estimators perform reasonably well with respect to coverage probability. Since using the interval estimator derived from the exact distribution can assure “coverage probability”⩾“desired confidence”, this estimator is probably preferable to the other asymptotic ones when p⩾0.50, and n⩽10. Finally, consider right-censoring, in which the failure-time that occurs after a fixed follow-up time period, is censored. We extend the interval estimator using the asymptotic properties of the MLE to account for right censoring. Monte Carlo simulation is used to evaluate the performance of this interval estimator; the censoring effect on efficiency is discussed for a variety of situations