Accurate estimation with one order statistic

Estimating parameters from certain survival distributions is shown to suffer little loss of accuracy in the presence of left censoring. The variance of maximum likelihood estimates (MLE) in the presence of type II right-censoring is almost un-degraded if there also is heavy left-censoring when estimating certain parameters. In fact, if only a single data point, the rth recorded failure time, is available, the MLE estimates using the one data point are similar in variance to the estimates using all r failure points for all but the most extreme values of r. Analytic results are presented for the case of the exponential and Rayleigh distributions, to include the exact distributions of the estimators for the parameters. Simulated results are also presented for the gamma distribution. Implications in life test design and cost savings are explained as a result. Also computational considerations for finding analytic results as well as simulated results in a computer algebra system are discussed.