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

Abstract

SummaryIn 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. I t 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, xz and x-'. or central and noncentral t, I-', and normal.

Biochemical Systems Theory: A Review

“…Recognizing that cumulative distribution functions always grow monotonically from 0 to 1 and very much resemble growth functions, Voit proposed a single S-system equation as a good approximation of cumulative distribution functions, calling it the S-distribution [731]. It turned out that this S-distribution has interesting features.…”

“…For instance, the initial value is directly related to the median of the distribution, the rate constant, which must be the same for both terms in the S-system equation, is related to the variance, and the kinetic orders determine the shape of the distribution. Indeed, the two kinetic orders were used as a shape classi�cation system for continuous as well as discrete distribution functions [731][732][733]. e same distribution was subsequently used in survival analysis and risk assessment [28,625,[734][735][736][737][738], and as a tool for random number generation and quantile analysis [739,740], as well as for inference [741,742].…”

“…Parameters of the distribution were estimated with leastsquares and maximum likelihood methods [731,740,746].…”

Scalability Properties of the S-Distribution

“…A few years ago, the S-distribution was introduced as a tool for classifying data and distributions (Voit 1992, Voit andYu, 1994). It has been shown that this distribution is capable of representing various degrees of skewness and that it is structurally rich enough to approximate most densities and mass functions rather well.…”

A method constructing density functions: The case of a generalized Rayleigh variable

“…Let us mention also the so-called "suprasystem of probability distributions" proposed by Savageau [42] which is a set of simultaneous ordinary differential equations, and the S-system of distributions presented by Voit [53] and defined as a four parameter ordinary differential equations which "appears to be a good candidate for representing and analyzing failure data" ([54, p. 596]). A generalization of the BurrHatke and S-system of distributions has been recently done by the present author and posted on Gnedenko forum [52].…”