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Obenchain

an empirical histogram or cumulative distribution function. In fact, this sort of smoothing can even provide good methods for estimating both ER and VR. Of course, because regret distributions that are highly skewed and concentrated near zero are desirable, it usually would make no sense to attempt to smooth an observed regret distribution with a normal (or any other symmetric and doubly unbounded) distribution.

There are a number of relatively straight-forward ways to fit a Poisson or standard gamma distribution to a sample of values, nonę of which are negative. For example, once your data have been EN transformed, the maximum likelihood estimate of EE (for both Poisson and gamma distributions) is simply the sample mean.

Smoothing Regrets with a Discrete Poisson Distribution EN rescaling of observed regrets freąuently allows them to be well approximated by a discrete Poisson distribution. I have used this tactic in practical applications for morę than eight years, and I have enjoyed great success in all cases except those where the regret distribution either had multiple modes or was much morę discrete than an integer-valued Poisson distribution. We will see in the environmental monitoring example presented below that a simple EN Poisson approximation can work very well in actual practice.

To smooth regrets with a Poisson approximation, we start by computing the mean and variance of regret from historical X characteristic values and by picking convenient, rounded numerical values for ER and VR that are close to these sample estimates. This yields initial values for equivalent nonconformances, EN(X) = ER • R(X)/VR, and for their mean, EE = (ER)2/VR. These observed EN values will usually not be integers, so we have the option at this point of rounding each observed EN to its nearest fuli integer. Ideally, the rounded and un-rounded EN values will have approximately the same mean value, which is now our "trial" value for EE.

Next, a P-P probability plot [Wilk and Gnanadesikan (1968) or Chambers, Cleveland, Kleiner and Tukey (1983)] as well as a Kolmogorov-Smimov statistic [say, Conover (1980), page 462] can be used to judge the lack-of-fit of the entire EN cumulative distribution function to a Poisson at the trial intensity, EE. When lack-of-fit is statistically significant, it can usually be reduced by making smali, trial-and-error changes in the numerical values for ER and/or VR and then recomputing the implied ENs. On the other hand, Poisson lack-of-fit can be significant simply because EE is smali (say, EE < 1), as explained next.