00376 …094574a3f70db509441cdc0f13c558

380

Obenchain

the left extreme). My personal Computer capability software (see Appendix) displays CC(I) over the rangÄ™ 0 < I < 4. (QMP Trend Charts have the regret index on their vertical axis and display the rangÄ™ 0 < I < 5, with zero regret at the top of the plot.]

Scott Vander Wiel (1993) has pointed out an elegant use for the well known eÄ…uality

co    ao

E(X) = J x dF(x) = J(1 - F(x)]dx

o    o

that applies to nonnegative random variables, X > 0. When applied to CC curves for regret indices, this eÄ…uality becomes

ao

1 = J[l-CC(I)]dI

o

which means that the total area above a properly normalized CC curve is one. In tum, this implies that two different CC curves with the same expected value must cross at least once. After all, if one CC curve starts out higher and increases morÄ™ Ä…uickly (for smali index values) than does a second CC curve with the same expected value, the first curve must also have a much longer right-hand (large index value) taiÅ‚ than does that second CC curve. Again, the key property here is simply that the total area above both CC curves is the same!

NOTES In practical applications, the area above a CC curve may not be exactly one simply because the numerical value of ER has been rounded to only a very smali number of decimal places. On the other hand, the area above an empirical CC curve might be much larger than one when a (smali) value of ER is used to represent a "stretch" goal for regret not yet within current capability.

EDFs Rather Than Histograms Most people find it easier to visualize a stochastic distribution from a histogram plot than from its Empirical (cumulative) Distribution Function, EDF, plot. This is why we used histograms in FigurÄ™ 7 to depict the performance index distributions corresponding to Ä…uadratic, inverted normal, absolute value and logistic regret for the fill-volume data. But histograms have the distinct disadvantage that a viewer's perception of a distribution can be strongly affected by the histogram creator's subjective choice for cell-width. EDF plots place no corresponding impediment to