The Kaplan-Meier reliability estimator errs on Fred’s bicycle ships and failure data! The Kaplan-Meier estimate was computed from Fred’s bicycles’ grouped failure data in the body of a “Nevada” table. It disagrees with the reliability estimate from ships cohorts and monthly failures (without knowing which cohort the failures came from). It disagrees with least squares nonparametric reliability estimates. All but the Kaplan-Meier estimate agree! Which would you prefer?

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The (age-specific or actuarial) force of mortality drives the demand for spares, service parts, and most products. The actuarial demand forecast is Σd(t‑s)*n(s), where d(t-s) is (age-specific) actuarial demand rate and n(s) is the installed base of age s, s=0,1,2,…,t. Ulpian, 220 AD, made actuarial forecasts of pension costs for Roman Legionnaires. (Imagine computing actuarial demand forecasts with Roman numerals.) Actuarial demand rates are functions of reliability. What if reliability changes? We Need Statistical Reliability Control (SRC).

Actuarial demand forecasts require updating as installed base and field reliability data accumulates. Actuarial failure rate function, a(t), is related to reliability function, R(t), by a(t) = (R(t)-R(t-1))/R(t-1), t=1,2,… If products or parts are renewable or repairable, then actuarial demand rate function, d(t), depends on the number of prior renewals or repairs by age t [George, Sept. 2021].

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Do you want easy demand forecasts or do you want to learn and use the reliabilities of service parts and make demand forecasts and distribution estimates, without sample uncertainty? Would you like to do something about service parts’ reliability? Would you like demand forecast distributions so you could set inventory policies to meet fill rate or service level requirements? Without sample uncertainty? Without life data? Don’t believe people who write that it can’t be done!

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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.

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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!

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Which of these six failure rate functions do your products and their service parts have? You don’t know? You don’t have field reliability lifetime data by product name or part serial number? That’s OK. Lifetime data are not required to estimate and classify failure-rate functions, including attrition and retirement. GAAP requires statistically sufficient field reliability data to classify failure rate functions for RCM.

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Suppose installed base or cohorts in successive periods have different reliabilities due to nonstationarity? What does that do to forecasts, estimates, reliability predictions, diagnostics, spares stock levels, maintenance plans, etc.? Assuming stationarity is equivalent to assuming all installed base, cohorts, or ships have the same reliability functions. At what cost? Assuming a constant failure rate is equivalent to assuming everything has exponentially distributed time to failure or constant failure rate. At what cost?

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I needed multivariate fragility functions for seismic risk analysis of nuclear power plants. I didn’t have any test data, so Lawrence Livermore Lab paid “experts” for their opinions! I set up the questionnaires, asked for percentiles, salted the sample to check for bias, asked for percentiles of conditional fragility functions to estimate correlations, and fixed pairwise correlations to make legitimate multivariate correlation matrixes. Subjective percentiles provide more distribution information than parameter or distribution assumptions, RPNs, ABCD, high-medium-low, or RCM risk classifications.

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