In probability theory and statistics, the dispersion function is a functional that characterizes a probability distribution by measuring the expected absolute deviation of a random variable from any given point. It was introduced by J. Muñoz-Pérez and A. Sánchez-Gómez in 1990 as a tool for studying statistical dispersion and inducing a partial ordering of distributions.
Definition Let be a real-valued random variable with a finite expectation (). The dispersion function is defined as the absolute moment of order of the random variable with respect to :
Characterization of the distribution The dispersion function uniquely determines the cumulative distribution function (CDF) of . If is the set of continuity points of , the distribution function can be recovered via the derivative of the dispersion function:
Properties The dispersion function has the following properties: Convexity: is a convex function on . Differentiability: It is differentiable, and its derivative has at most a countable number of discontinuity points. Asymptotic behavior of the derivative: The limits of the derivative are and . Mean relationship: The limits involving the mean are given by and .
Relation to Variance For a random variable with finite variance , the -distance between its dispersion function and the dispersion function of the degenerate random variable at its mean () is exactly the variance:
Dispersive Ordering In the study of stochastic orders, the dispersion function provides a necessary and sufficient condition for the dispersive ordering. This concept builds upon earlier work by Bickel and Lehmann regarding descriptive statistics for non-parametric models. According to Shaked and Shanthikumar, this characterization allows for the comparison of distributions even when they have the same finite support, such as comparing a continuous uniform distribution to a triangular distribution (Simpson's distribution).
Generalizations A generalized dispersion function of order p is defined as the -distance between the quantile function and the quantile function of a degenerate variable at :
where is a probability distribution on and is any positive number.
See also Statistical dispersion Stochastic ordering Quantile function Lp space
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