on Tools & Techniques

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!

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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…]