on Tools & Techniques

What are the covariances of Kaplan-Meier reliability estimates at different ages? I need them for the variance of actuarial demand forecasts and for confidence bands on reliability. I thought cohort reliability estimate variances and covariances in the previous article were a good idea. How good? Not as good as bootstrap and jackknife resampling alternatives!

The Kaplan-Meier reliability function estimator uses right-censored and grouped time-to-failure counts in periodic cohorts (rows in table 1). The Nelson-Aalen cumulative failure rate function estimators are theoretically independent [Aalen, Nelson], but not for some examples. The Kaplan-Meier reliability and actuarial failure rate function estimates at different ages are dependent, so their covariances matter to actuarial forecasts and confidence bands on reliability.

The well-known variance of the Kaplan-Meier reliability function estimator [Greenwoood, Wikipedia] can drastically under-or over-estimate variance. The covariances of the Kaplan-Meier reliability pairs at different ages are ignored or neglected. Variance errors and covariance neglect bias the variance of actuarial demand forecasts. Imagine what errors and neglect do to confidence bands on reliability functions.

My first task at Apple Computer was to recommend the warranty duration for the Apple II computer. Apple didn’t have a warranty! So, I looked at competitors’ warranties and recommended the same, one year. I wish I had known Apple’s computers’ and service parts’ reliabilities before that recommendation; I would have used actuarial forecasts of warranty returns to compare alternative warranties. Apple’s hardware warranty is still one year. Is that equitable to Apple, its customers, and society?

Imagine observing inputs and outputs of a self-service system, without individual service times. How would you estimate the distribution of service time without following individuals from input to output? The maximum likelihood estimator for an M/G/Infinity self-service-time distribution function from ships and returns counts works for nonstationary arrival process M(t)/G/Infinity self-service systems, under a condition. A constant or linearly increasing arrival (ships) rate satisfies the condition. If you identify outputs by failure mode then you could estimate reliability by failure mode or quantify reliability growth, without life data. [Read more…]

Nonparametric, age-specific field reliability estimates helped deal with a Customer’s bad experience using a Hewlett-Packard part in the Customer’s product: 110 part failures out of 3001 shipped in the first five months. Comparison of HP population vs. Customer reliability estimates showed the Customer’s infant mortality was not typical. Using population ships and failures or returns data eliminated sample uncertainty from the HP population field reliability estimate.

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.