Rupert Miller said, “Surprisingly, no efficiency comparison of the sample distribution function with the mles (maximum likelihood estimators) appears to have been reported in the literature.” (Statistical “efficiency” measures how close an estimator’s sample variance is to its Cramer-Rao lower bound.) In “What Price Kaplan-Meier?” Miller compares the nonparametric Kaplan-Meier reliability estimator with mles for exponential, Weibull, and gamma distributions.

This report compares the bias, efficiency, and robustness of the Kaplan-Meier reliability estimator from grouped failure counts (grouped life data) with the nonparametric maximum likelihood reliability estimator from ships (periodic sales, installed base, cohorts, etc.) and returns (periodic complaints, failures, repairs, replacement, spares sales, etc.) counts, estimator vs. estimator and population vs. sample.

Dick Mensing said, “Larry, you can’t give an estimate without some measure of its uncertainty!” For seismic risk analysis of nuclear power plants, we had plenty of multivariate earthquake stress data but paltry strength-at-failure data on safety-system components. So we surveyed “experts” for their opinions on strengths-at-failures distribution parameters and for the correlations between pairs of components’ strengths at failures. 

If you make estimates from population field reliability data, do the estimates have uncertainty? If all the data were population lifetimes or ages-at-failures, estimates would have no sample uncertainty, perhaps measurement error. Estimates from population field reliability data have uncertainty because typically some population members haven’t failed. If field reliability data are from renewal or replacement processes, some replacements haven’t failed and earlier renewal or replacement counts may be unknown. Regardless, estimates from population data are better than estimates from a sample, even if the population data is ships and returns counts!