Characterization of a Family of Distributions by the Independence of Size and Shape Variables

Abstract

Let $X_1, \cdots, X_n$ be $n \geqslant 2$ positive random variables and $G(\mathbf{X})$ a positive variable satisfying $G(a\mathbf{X}) = aG(\mathbf{X})$ for all $a > 0$. Then $G$ is a size variable, and $\mathbf{X}/G$ is a shape vector. If $X_1, \cdots, X_n$ are independent, then the independence of shape and the size variable $G(\mathbf{X})$ characterizes (i) the lognormal distribution if $G(\mathbf{X}) = \Pi X^{1/n}_i$, (ii) the generalized gamma distribution if $G(\mathbf{X}) = (\sum X^b_i)^{1/b}$, (iii) the Pareto distribution or its discrete analogue if $G(\mathbf{X}) = \min(\mathbf{X})$, and (iv) the power-function distribution or its discrete analogue if $G(\mathbf{X}) = \max(\mathbf{X})$. It is shown here that if $X_1, \cdots, X_n$ have piecewise continuous density functions and $G$ is a continuous function then these four size variables are effectively the only ones for which such independence properties are attainable. A connection with the theory of sufficient statistics for a scale parameter is also considered.