Estimation of the geometric survival distribution

Abstract

This paper discusses both point and interval estimation of the survivor function S/sub 0/=Pr{X/spl ges/x/sub 0/} for the geometric distribution. When the number of devices n/spl ges/50, the performance of the maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) of S/sub 0/ are essentially equivalent with respect to the relative mean-square-error (RMSE) to S/sub 0/. However, when the failure probability per time unit p/spl ges/0.50, and n/spl les/30, the UMVUE is preferable to the MLE with respect to the RMSE. For interval estimation of S/sub 0/ 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/spl les/0.2 and n/spl ges/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"/spl ges/"desired confidence", this estimator is probably preferable to the other asymptotic ones when p/spl ges/0.50, and n/spl les/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.