Draft:Dispersion function

Let ${\displaystyle X}$ be a real-valued random variable with a finite expectation (${\displaystyle X\in {\mathcal {L}}_{1}}$). The dispersion function ${\displaystyle D_{X}(u)}$ is defined as the absolute moment of order ${\displaystyle r=1}$ of the random variable ${\displaystyle X}$ with respect to ${\displaystyle u}$:^[1]

$${\displaystyle D_{X}(u)=E|X-u|,\quad \forall u\in \mathbb {R} }$$

The dispersion function uniquely determines the cumulative distribution function (CDF) of ${\displaystyle X}$. If ${\displaystyle C_{F}}$ is the set of continuity points of ${\displaystyle F_{X}}$, the distribution function can be recovered via the derivative of the dispersion function:^[1]

$${\displaystyle F_{X}(u)={\frac {1}{2}}(D_{X}'(u)+1)}$$

The dispersion function has the following properties:^[1]

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

For a random variable with finite variance ${\displaystyle \sigma ^{2}}$, the ${\displaystyle L_{1}}$-distance between its dispersion function and the dispersion function of the degenerate random variable at its mean (${\displaystyle D_{EX}(u)=|EX-u|}$) is exactly the variance:^[1]

$${\displaystyle \int _{-\infty }^{+\infty }|D_{X}(u)-D_{EX}(u)|du=\sigma ^{2}}$$

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.^[2] According to Shaked and Shanthikumar,^[3] 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).

A generalized dispersion function of order p is defined as the ${\displaystyle L_{p}}$-distance between the quantile function ${\displaystyle Q_{X}}$ and the quantile function of a degenerate variable at ${\displaystyle u}$:^[1]

$${\displaystyle D_{X}(u,p)=\left(\int _{0}^{1}[Q_{X}(t)-u]^{p}d\Lambda (t)\right)^{1/p}}$$

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

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