Dispersion function

Short description: Statistical characterization of distribution functions

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 $X$ be a real-valued random variable with a finite expectation ($X\in {{ℒ}}_1$). The dispersion function $D_X(u)$ is defined as the absolute moment of order $r=1$ of the random variable $X$ with respect to $u$:

$$D_X(u)=E|X-u|,\mathrm{\forall }u\in \mathbb{ℝ}$$

Characterization of the distribution

The dispersion function uniquely determines the cumulative distribution function (CDF) of $X$. If $C_F$ is the set of continuity points of $F_X$, the distribution function can be recovered via the derivative of the dispersion function:

$$F_X(u)=\frac{1}{2}({D_X}^{\prime }(u)+1)$$

Properties

The dispersion function has the following properties:

• Convexity: $D_X$ is a convex function on $\mathbb{ℝ}$.
• Differentiability: It is differentiable, and its derivative ${D_X}^{\prime }$ has at most a countable number of discontinuity points.
• Asymptotic behavior of the derivative: The limits of the derivative are $\underset{x\to \mathrm{\infty }}{\lim }{D_X}^{\prime }(x)=1$ and $\underset{x\to -\mathrm{\infty }}{\lim }{D_X}^{\prime }(x)=-1$.
• Mean relationship: The limits involving the mean $EX$ are given by $\underset{x\to \mathrm{\infty }}{\lim }[D_X(x)-x]=-EX$ and $\underset{x\to -\mathrm{\infty }}{\lim }[D_X(x)+x]=EX$.

Relation to Variance

For a random variable with finite variance ${\sigma }^2$, the $L_1$-distance between its dispersion function and the dispersion function of the degenerate random variable at its mean ($D_{EX}(u)=|EX-u|$) is exactly the variance:

$$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}|D_X(u)-D_{EX}(u)|du={\sigma }^2$$

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 $L_p$-distance between the quantile function $Q_X$ and the quantile function of a degenerate variable at $u$:

$$D_X(u,p)={(\underset{0}{\overset{1}{\int }}[Q_X(t)-u{]}^{p}d\mathrm{\Lambda }(t))}^{1/p}$$

where $\mathrm{\Lambda }$ is a probability distribution on $(0,1)$ and $p$ is any positive number.

See also

• Statistical dispersion
• Stochastic ordering
• Quantile function
• Lp space