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The Kaplan-Meier estimator is the maximum likelihood, nonparametric reliability estimator for censored, grouped lifetime data. It’s traditional. It’s in statistical software. Greenwood’s variance formula is well known. Could Kaplan-Meier be improved: smaller variance, better actuarial forecasts, seasonality, separate cohort variability from reliability? Could you estimate reliability without life data and preserve privacy?
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I estimated actuarial failure rates, made actuarial forecasts, and recommended stock levels for automotive aftermarket stores. I wondered how to account for seasonality in their sales? Time series forecasts account for seasonality but not for age, the force of mortality accounted for by actuarial forecasts. I finally figured out how to seasonally adjust actuarial forecasts. It’s the same method, David Cox’ “Proportional Hazards” model, used to make “Semi-Parametric” estimates and “Credible Reliability Predictions”.
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Lifetime data is nice to have, but lifetime data is not necessary! Generally Accepted Accounting Principles require statistically sufficient data to estimate nonparametric reliability and failure rate functions. Some work is required!
ISO 14224 “Petroleum, Petrochemical and Natural Gas Industries—Collection and Exchange of Reliability and Maintenance Data for Equipment” requires lifetime data to estimate exponential or Weibull reliability functions! Sales or ships and returns or failure counts are statistically sufficient to make nonparametric estimates of reliability and failure rate functions, without unwarranted distribution assumptions or lifetime data!
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Fred asked me to explain why use nonparametric statistics? The answer is reality. Reality trumps opinion, mathematical convenience, and tradition. Reality is more interesting, but quantifying reality takes work, especially if you track lifetimes. Using field reliability reality provides credibility and could reduce uncertainty due to tradition and unwarranted, unverified assumptions.
Data is inherently nonparametric. Cardinal numbers are used for period counts: cohorts, cases, failures, etc. Accounting data is numerical; it is derived from data or from dollars required by GAAP (Generally Accepted Accounting Principles); e.g., revenue = price*(products sold), service cost = (Cost per service)*(Number of services), or numbers of spare parts sold. Why not do nonparametric reliability estimation, with or without lifetime data?
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Time series forecasts are easy to make and data are available. They’re like driving while looking in the rear-view mirror. A survey listed 31 forecasting software programs: none actuarial [Yurkewicz]. Actuarial failure forecasts are less biased and are more precise than time series failure forecasts, because actuarial failure forecasts use age-specific failure rates. How much better?
The example in this article shows the 5% to 95% time series confidence interval width is 44.78 vs. the nonparametric actuarial Kaplan-Meier actuarial forecast width of 12.63, from grouped failure data, and actuarial forecast width of 15.45, from ships and returns counts.
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AFRs are periodic ratios of failure counts divided by installed base. Have you seen meeting rooms wallpapered with AFR charts (Annualized Failure Rate)? Have you sat through debates about the wiggles in AFR charts? Fred Schenkelberg wondered if reliability could be estimated from AFRs and their input data? How about age-specific reliability and actuarial failure rate functions? Actuarial forecasts? MTBFs? Wonder no more!
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The Kaplan-Meier reliability estimator is for dead-forever products or parts, given individual lifetime data or a “Nevada” table of periodic ships cohorts and their grouped failure counts. This estimator presumes that ships cohorts are NOT random. Production, sales, installed base, and cohort case counts are random! What does that do to Kaplan-Meier reliability estimates? What is the nonparametric reliability function estimator if ships cohorts are random?
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The New-Products manager asked me, “Your actuarial failure rate estimates (from vehicle registrations, bills-of-materials, and automotive aftermarket store sales) are for dead-forever parts with at most one failure. What if auto parts could be renewed or replaced more than once?” Chagrined, I wrote a spreadsheet program to estimate actuarial rates for renewal processes, without life data. But what is the corresponding estimator from grouped, cohort renewal counts like the Kaplan-Meier estimator for grouped, cohort failure counts?
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Fred wrote, “I would like to suggest that you continue writing articles – make them more tutorial in nature as if teaching someone the stats from scratch that you use in your articles. Instead of loads of references to papers and procedures, explain the concepts and math involved.” OK, I’ll try.
Nonparametric field reliability estimators require no unwarranted distribution assumptions and they preserve all information in data. Here’s how to compute them, without life data, while preserving all relevant information in ships and returns counts.
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Azmat Siddiqi suggested a certification in reliability statistics in 2022. Azmat believes in knowing and using the reliability statistical information in test, installed base, failures, and service data. Thanks Azmat.
I propose Certification in Reliability Statistics to recognize statistics knowledge, work experience, and applications. Certification in Reliability Statistics should provide assurance to employers, contractors, and collaborators that reliability statistics are estimated and used to the best extent with available data, including uncertainty quantification, with or without life data.
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The LinkedIn ASQ RRD group published this question from a reliability manager. Replies included:
Generations of products have similar field reliability functions because they are designed, processed, shipped, sold, and used in similar environments by similar customers. Replacement parts have similar reliability functions depending on replacement number: 1st, 2nd,….
Biostatisticians use David Cox’ proportional hazard (PH) survival function models to quantify effects of treatment or risk factors. Proportional hazard models could describe product’s failure modes, parts’ reliabilities in successive replacements, or products’ reliabilities in successive generations. [Read more…]
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].
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!
My wife says I am wasting my time trying to change reliability statistics, so I polled the www.linkedin.com Reliability Leadership…, ASQRRD, IEEE Reliability, “Biostatistics, and No MTBF groups. The polls claimed that “Life data, censored or not, is required to estimate MTBF, reliability function, failure rate function, or survivor function. TRUE? FALSE? or DON’T KNOW.” I am grateful for the responses.
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