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We study 17 different statistical distributions for sizes obtained {}from theclassical and recent literature to describe a relevant variable in the socialsciences and Economics, namely the firms' sales distribution in six countriesover an ample period. We find that the best results are obtained with mixturesof lognormal (LN), loglogistic (LL), and log Student's $t$ (LSt) distributions.The single lognormal, in turn, is strongly not selected. We then find that thewhole firm size distribution is better described by a mixture, and there existsubgroups of firms. Depending on the method of measurement, the best fittingdistribution cannot be defined by a single one, but as a mixture of at leastthree distributions or even four or five. We assess a full sample analysis, anin-sample and out-of-sample analysis, and a doubly truncated sample analysis.We also provide the formulation of the preferred models as solutions of theFokker--Planck or forward Kolmogorov equation.

Composite distributions in the social sciences: A comparative empirical  study of firms' sales distribution for France, Germany, Italy, Japan, South  Korea, and Spain