The S‐Distribution A Tool for Approximation and Classification of Univariate, Unimodal Probability Distributions

Abstract

AbstractIn many statistical applications a data set needs to be evaluated but there is no solid information about which probability distribution might be most appropriate. Typical solutions to this problems are: to make assumptions that facilitate mathematical treatment; to use a family of distribution functions that contains all relevant distributions as special cases; or, to employ nonparametric methods. All three solutions have disadvantages since assumptions are usually difficult to justify, families of distributions contain too many parameters to be of practical use, and nonparametric methods make it difficult to characterize data in a succinct quantitative form. The S‐distribution introduced here is a compromise between the conflicting goals of simplicity in analysis and generality in scope. It is characterized by four parameters, one of which reflects its location, the second one its spread, and the remaining two its shape; transformation to a standard form reduces the number of free parameters to two. Cumulatives and densities are computed numerically in fractions of seconds, key features like quantiles and moments are easily obtained, and results can be presented in terms of parameter values. The S‐distribution rather accurately models different distribution functions, including central and noncentral distributions, and thus competes in flexibility with some distribution families. As an approximation, the S‐distribution provides a graphical method for demonstrating relationships between distributions, such as the relationships between central F, χ2 and χ−2 or central and noncentral t, χ‐1, and normal.