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Regret Indices and Capability Quantification

restrict attention only to variation within rational subsets.) On the other hand, when a rather complete characterization of "current" capability is our objective, we might wish to retain almost all available data. The practical difficulty associated with too much data cleaning before setting our ER (and VR) standard(s) is that we could end-up with base-lines that actually are well beyond the current capability of our process. The corresponding psychological disadvantage of setting a futuristic ("stretch") goal for regret is that workers might think that further improvement would be unnecessary if that stated goal were ever actually achieved.

When setting standards by estimating ER and VR, it will freÄ…uently be best to retain almost all available data not directly traceable to "assignable causes." This is especially true, of course, when using a "robust" regret function that is relatively insensitive to large deviations of X from T (like logistic or inverted normal with a relatively smali half-width). The resulting estimates of ER and VR are then unÄ…uestionably "realistic" and within the current capability of our process. Quality improvement programs can then focus not only on continual reduction in both ER and VR but also on continua! improvement in the shape of the entire regret distribution!

Another sound strategy, illustrated below, would be to find out whether ER and VR would be quite different when estimated using only the smaller order-statistics of the available regrets than when estimated from all of your data.

Illustration that ER and VR Can Depend Upon Process Centering In actual practice, we will almost always estimate ER and VR directly from observed R(X) values rather than either from observed X values or from their theoretical distribution. But we can illustrate an important point by considering the following exact calculations.

When regret follows Taguchi's Ä…uadratic form with K=1 in equation [1] and the X characteristic has mean p and variance o², it is easily shown that

ER = ct² + (p-T)²    [9]

This ER expresses the well-known relationship "mean-squared-error equals variance plus sÄ…uared bias." Under the additional assumption that the X characteristic has a normal distribution, the regret variance is

[10]

VR = 2ct² [a² + 2(p - T)² ]