The selection of a specific statistical distribution as a model for describing the population behavior of a given variable is seldom a simple problem. One strategy consists in testing different distributions (normal, lognormal, Weibull, etc.), and selecting the one providing the best fit to the observed data and being the most parsimonious. Alternatively, one can make a choice based on theoretical arguments and simply fit the corresponding parameters to the observed data. In either case, different distributions can give similar results and provide almost equivalent models for a given data set. Model selection can be more complicated when the goal is to describe a trend in the distribution of a given variable. In those cases, changes in shape and skewness are difficult to represent by a single distributional form. As an alternative to the use of complicated families of distributions as models for data, the S-distribution [E. O. Voit, Biom. J. 34, No. 7, 855–878 (1992; Zbl 0775.62027)] provides a highly flexible mathematical form in which the density is defined as a function of the cumulative. S-distributions can accurately approximate many known continuous and unimodal distributions, preserving the well known limit relationships between them. Besides representing well-known distributions, S-distributions provide an infinity of new possibilities that do not correspond with known classical distributions. Although the utility and performance of this general form has been clearly proved in different applications, its definition as a differential equation is a potential drawback for some problems.
In this paper we obtain an analytical solution for the quantile equation that highly simplifies the use of S-distributions. We show the utility of this solution in different applications. After classifying the different qualitative behaviors of the S-distribution in parameter space, we show how to obtain different S-distributions that accomplish specific constraints. One of the most interesting cases is the possibility of obtaining distributions that acomplish \(P(X\leq Xc) = 0\). Then, we demonstrate that the quantile solution facilitates the use of S-distributions in Monte-Carlo experiments through the generation of random samples. Finally, we show how to fit an S-distribution to actual data, so that the resulting distribution can be used as a statistical model for them.