00386 Te1730accb779852cc586c0c62f5e4d

390

Obenchain

necessary, EN rescaling is still recommended because the estimated gamma scalę parameter will then be approximately 1, at least when all of the available data are used in the analysis.

Unfortunately, the maximum likelihood equations of Wilk, Gnanadesikan, and Huyett (1962a, 1962b) depend on the geometrie mean (n-th root of the product of n values) as well as on the arithmetic mean of the chosen regret order statistics. Drastic rounding of X values, as in the fill-volumes data, would then have to be avoided because even a single observed regret value of zero implies a zero geometrie mean. Thus the following morę "robust” approach to gamma Q-Q plotting, based upon simple approximate methods, is recommended for generał EE estimation from order-statistics (and is implemented in the software described in the Appendix.)

For gamma Q-Q probability plotting, Chambers, CIeveland, Kleiner and Tukey (1983), Chapter 6, recommend use of the Wilson-Hilferty (1931) normal approximation for the cube root of gamma variables and the Hastings (1955) rational function approximation for normal quantiles. Then it is straight-forward to estimate the slope, bfc, of a zero-intercept regression of the k smallest observed EN regrets onto their corresponding approximate gamma EE ąuantiles. At iteration I + 1, the trial value for EE(i+i) could be, say, one third of the way from EEi to [EEi times the i-th bfc estimate.] Iterative Fitting would then halt when either (1) the i+l-th zero-intercept regression fails to show inereased multiple correlation between the k smallest observed EN(i+i) regrets and their fitted gamma EE(j+i) ąuantiles, or (2) bfc converges to 1.0.

For logistic regret with target at 75 ml and half-with 15 ml on the fill-volumes data, Poissonization using rounded values for the sample moments of Table 1 yields an initial EE estimate of (0.20)^/0.0225 = 1.778 and a Kolmogorov-Smimov lack-of-fit statistic of 0.211 that is not significant at the 5% level. But if EE is to be estimated using only the 10 smallest regrets (out of 31), then the above iteration converges in 8 steps to EE = 1.647 and b₁₀ =0.991, where the multiple correlation between the 10 smallest observed EN regrets and their fitted gamma EE quantiles has inereased from 0.913 to 0.919. The corresponding Q-Q plot is given in Figurę 12.

Composite Regret Indices

The Poissonization arguments leading to relationship [21] also provides the following additional insight: Eąuation [8] is equivalent to