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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.^[1]

Definition

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

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

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:^[1]

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

Properties

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

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

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:^[1]

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

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.^[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).

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 $u$ :^[1]

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

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

References

^ ^a ^b ^c ^d ^e ^f Muñoz-Pérez, J.; Sánchez-Gómez, A. (1990). “A characterization of the distribution function: The dispersion function”. Statistics & Probability Letters. 10 (3): 235–239. doi:10.1016/0167-7152(90)90080-Q.
^ Bickel, P.J.; Lehmann, E.L. (1976). “Descriptive statistics for non-parametric models. III. Dispersion”. Annals of Statistics. 4 (6): 1139–1158. doi:10.1214/aos/1176343650.
^ Shaked, Moshe; Shanthikumar, J. George (2007). Stochastic Orders. Springer Series in Statistics. New York: Springer. ISBN 978-0-387-32915-4.

[rdp-we-cite_note-Munoz1990-1] ^ ^a ^b ^c ^d ^e ^f Muñoz-Pérez, J.; Sánchez-Gómez, A. (1990). “A characterization of the distribution function: The dispersion function”. Statistics & Probability Letters. 10 (3): 235–239. doi:10.1016/0167-7152(90)90080-Q.

[rdp-we-cite_note-2] Bickel, P.J.; Lehmann, E.L. (1976). “Descriptive statistics for non-parametric models. III. Dispersion”. Annals of Statistics. 4 (6): 1139–1158. doi:10.1214/aos/1176343650.

[rdp-we-cite_note-Shaked2007-3] Shaked, Moshe; Shanthikumar, J. George (2007). Stochastic Orders. Springer Series in Statistics. New York: Springer. ISBN 978-0-387-32915-4.

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