| Summary: | The four-parameter kappa distribution (Hosking 1994) was analyzed with respect to the various possible shapes of the probability density function. The general form for the cumulative distribution function when both h and k are non-zero is:
F(x) = { 1 - h [ 1 - k ( x - ξ ) / α ]<sup>1/k</sup> }<sup>1/h</sup>.
The parameters h and k work together to define the function's shape, ξ affects location, and α is the scale parameter. A method of selecting parameters to minimize the Kolmogorov-Smirnov test statistic, D, was developed. The technique was first described for the logistic distribution, which is the special case of the kappa distribution with k = 0 and h = -1. Then the more general case, k = 0 and h ≠ 0, was further explored as a possibility for expanding the optimization technique. The optimization method was shown to provide the parameters h, ξ, and α such that the Kolmogorov-Smirnov test statistic, D, was minimized. This optimization was applied to several example data sets and found to produce distributions that fit the empirical data much better than the normal or lognormal distribution functions. The results have potential applications in describing the distributions of many types of real data, including, but not limited to, weather, hydrologic and other environmental data. Matching empirical data to an invertible probability distribution makes it convenient to simulate random data that follow closely the characteristics of the natural data. Preliminary inquiry suggested that the technique might be expanded to allow non-zero values of k. This would improve the shape flexibility slightly and produce slightly better fits to empirical data.
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