Progress in Field Reliability?

Dear Larry

Thank you for your data request for breast implant data and apologies for the delay in responding. The data available is:

• The number of women receiving implants, by year, by major manufacturer
• Number of Explants: All Manufacturers (inc. Others and Unknown Brands)

My colleagues have been copied into this email to show your request has been actioned. I hope this is helpful. [Read more…]

Email from www.smartcorp.com advertised how to forecast inventory requirements using time-series analyses: single and double exponential smoothing, linear and simple moving average, and Winters models. SmartCorp compares alternative times-series forecasts in a “tournament” that picks the best forecast. Charles Smart says forecasting, “…particularly for low-demand items like service and spare parts — is especially difficult to predict with any accuracy.”

Time series forecasts also quantify variance. Excel’s time-series FORECAST() functions do exponential smoothing, account for seasonality and trend, and “pointwise” confidence intervals. Pointwise means only one confidence interval is valid at a time; not a confidence band on several forecasts!

[Read more…]

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.

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!

ANSI-IES TM-21 standard method may predict negative L70 LED lives. (L70 is the age at which LED lumens output has deteriorated to less than 70% of initial lumens.) Philips-Lumileds deserves credit for publishing the data that inspired an alternative L70 reliability estimation method based on geometric Brownian motion of stock prices in the Black-Scholes-Merton options price model. This gives the inverse Gauss distribution of L70 for LEDs. 

“Aircraft LRUs test NFF (No-Failure-Found) approximately 50% of the time” {Anderson] Wabash Magnetics claimed returned crankshaft position sensors had 89-90% NTF (No-Trouble-Found), Uniphase had 20%, Apple computer had 50% [George].

How is failure testing done on the Space Station? Could FTA (Fault Tree Analysis) be used in reverse to detect multiple failures given symptoms? That’s what NASA was programming in the 1990s. I proposed that the ratios P[part failure]/(part test time) be used to optimally sequence tests. Those ratios work if there are multiple failures, as long as failure rates are constant and failure times are statistically independent. 

Shotgun repair is trying to fix a system problem by replacing parts until the problem goes away. It is often done without regard to parts’ age-specific reliability information. Should you test before replacement? Which test(s) should you do? In which order? How long? Which part should you replace next if the test gave no indication of what’s wrong? What if test indication is imperfect or the fault is intermittent? What if there are more than one part failure?

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.

COVID-19 Case Fatality Rate (CFR) is easy to estimate: CFR=deaths/cases. Regression forecasts of COVID-19 cases and deaths are easy but complicated by variants and nonlinearity. Epidemiologists use SIR models (Susceptible, Infectious, and “Removed”) to estimate Ro. These are baseball statistics. Reliability people need age-specific reliability and failure-rate function estimates, by failure mode, to diagnose problems, recommend spares, plan maintenance, do risk analyses, etc. Markov models use actuarial transition rates.