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Regret Indices and Capability Quantification

psychological perception; EDFs are quite easily and meaningfully drawn as if their cell-width were zero! FigurÄ™ 8 redisplays the information in the histograms of FigurÄ™ 7 as CC curves, thus giving the maximum level of detail on these distributions. NotÄ™ that, in order to use the same scaling for all four of these CC curves, the index rangÄ™ 0 < I < 8 is used in FigurÄ™ 8.

Another advantage of EDFs over histograms is that there are established methods that placing confidence limits on EDFs, such as CC curves, as shown in FigurÄ™ 9. For example, Nair and Freeny (1993) point out that asymptotic confidence bands for nonparametric EDFs are of the form

CC(I) ± S • VCC(I) • [1 - CC(I)] / N    [16]

where S is a (significance point) constant that depends upon the desired confidence level, (1-a), and N is the total number of regret observations defining the CC curve.

Exact confidence intervals that are valid pointwise (for a given index, I) can be constructed from the fact that N*CC(I) follows a binomial distribution. When a normal approximation is used, as in [16], the S constant becomes the (1 - a / 2) Ä…uantile of the z (standard normal) distribution.

Simultaneous (rather than pointwise) confidence bands of the form given in [16] can be constructed from the variance-weighted Kolmogorov-Smimov statistic, but the rangÄ™ of CC values to be covered [L < CC(I) < U] will then have a strong influence on S. In fact, S approaches infinity in the limit as L approaches 0 and U approaches 1. Stephens (1986) provides tables of asymptotic S values (large N), a few of which are given here in Table 2.

Nair and Freeny (1993) recommend the choice L = 0.05 and U = 0.95 as yielding a relatively wide CC rangÄ™ without producing excessively wide confidence limits. Thus FigurÄ™ 9 uses S = 2.91 to display asymptotic, 95%

Table 2. S Values for Variance-Weighted CC Bounds a = Significance Level

0.01    0.05    0.10

CC RangÄ™ Covered
L = 0.01, U = 0.99	3.81	3.31	3.08
L = 0.05, U = 0.95	3.68	3.16	2.91
L = 0.10, U = 0.90	3.59	3.06	2.79