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Regret Indices and Capability Quantification

Motivation for Forming Composite Indices When EE Is Smali It is straight-forward to represent the approximating Poisson distribution on the same scalÄ™ as the observed regret indices. One simply applies the appropriate inverse transformation:

(fitted I) = (theoretical EN distribution) / EE    [21]

In other words, our Poisson approximation with unit cell-widths corresponds to a discrete regret index distribution with cell-widths of VR/ER^ = 1/EE . Because these regret index cell-widths will be veiy wide whenever EE is smali, there is very little hope of getting a good Poisson approximation in cases where EE is smali.

In the following sense, smali EE cases are merely examples of the usual distinction between statistical significance and practical importance. Once composite indices are formed (as explained below in Equation [23]) to combine results over multiple measurements, over time, and/or over related processes, equivalent expectancy will accumulate to, say, EE > 2. The Poisson step-size corresponding to this composite regret index will then shrink to 0.5 or less, and all lack-of-fit in the approximating Poisson distribution may simply vanish!

Lack-of-Fit Only in the Right-Hand TaiÅ‚ Poisson lack-of-fit that occurs only in the extreme right-hand taiÅ‚ of an EN sample is easily tolerated. The essential feature of a "successful" smoothing for a regret index distribution is that it provides a good representation over the rangÄ™ corresponding to relatively good performance, I < 1. In my experience at least, processes in doubtful States of statistical control tend to be unstable primarily in their right-hand regret taiÅ‚.

Visual examination of a Poisson P-P probability plot will reveal whether the Kolmogorov-Smimov statistic is detecting lack-of-fit primarily in the right-hand taiÅ‚. And significant lack-of-fit may simply be ignored in this case. On the other hand, a morÄ™ systematic approach is provided for this sort of situation by the methods for fitting gamma distributions using order-statistics, as described next.

Smoothing Regrets with a Continuous Gamma Distribution Wilk, Gnanadesikan, and Huyett (1962a, 1962b) describe Q-Q probability plotting methods for fitting gamma distributions that depend only upon a subjectiyely chosen subset of the smallest regret index order-statistics. These Q-Q methods estimate the scalÄ™ parameter as well as the shape parameter of gamma distributions. Thus, while preliminary EN rescaling of regret is not really