# Stochastic ordering

Stochastic ordering

In statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable $A$ may be neither stochastically greater than, less than nor equal to another random variable $B$. Many different orders exist, which have different applications.

Usual stochastic order

A real random variable $A$ is less than a random variable $B$ in the "usual stochastic order" if

:$P\left(A>x\right) le P\left(B>x\right) ext\left\{ for all \right\}x in \left(-infty,infty\right),$

where $P\left(cdot\right)$ denotes the probability of an event.This is sometimes denoted $A preceq B$ or $A le_\left\{st\right\} B$. If additionally $P\left(A>x\right) < P\left(B>x\right)$ for some $x$, then $A$ is stochastically strictly less than $B$, sometimes denoted $A prec B$.

Characterizations

The following rules describe cases when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist.

#$Apreceq B$ if and only if for all non-decreasing functions $u$, $E \left[u\left(A\right)\right] le E \left[u\left(B\right)\right]$.
#If $u$ is non-decreasing and $Apreceq B$ then $u\left(A\right) preceq u\left(B\right)$
#If $u:mathbb\left\{R\right\}^nmapstomathbb\left\{R\right\}$ is an increasing function and $A_i$ and $B_i$ are independent sets of random variables with $A_i preceq B_i$ for each $i$, then $u\left(A_1,dots,A_n\right) preceq u\left(B_1,dots,B_n\right)$ and in particular $sum_\left\{i=1\right\}^n A_i preceq sum_\left\{i=1\right\}^n B_i$ Moreover, the $i$th order statistics satisfy $A_\left\{\left(i\right)\right\} preceq B_\left\{\left(i\right)\right\}$.
#If two sequences of random variables $A_i$ and $B_i$, with $A_i preceq B_i$ for all $i$ each converge in distribution, then their limits satisfy $A preceq B$.
#If $A$, $B$ and $C$ are random variables such that $P\left(A>u|C=c\right)le P\left(B>u|C=c\right)$ for all $u$ and $c$, then $Apreceq B$

Other properties

If $Apreceq B$ and $E \left[A\right] =E \left[B\right]$ then $A=B$ in distribution.

tochastic dominance

Stochastic dominance [http://www.mcgill.ca/files/economics/stochasticdominance.pdf] is a stochastic ordering used in decision theory. Several "orders" of stochastic dominance are defined.
*Zeroth order stochastic dominance consists of simple inequality: $A preceq_\left\{\left(0\right)\right\} B$ if $A le B$ for all states of nature.
*First order stochastic dominance is equivalent to the usual stochastic order above.
*Higher order stochastic dominance is defined in terms of integrals of the distribution function.
*Lower order stochastic dominance implies higher order stochastic dominance.

Multivariate stochastic order

Other stochastic orders

Hazard rate order

The "hazard rate" of a non-negative random variable $X$ with absolutely continuous distribution function $F$ and density function $f$ is defined as:$r\left(t\right) = frac\left\{d\right\}\left\{dt\right\}\left(-log\left(1-F\left(t\right)\right)\right) = frac\left\{f\left(t\right)\right\}\left\{1-F\left(t\right)\right\}$.

Given two non-negative variables $X$ and $Y$with absolutely continuous distribution $F$ and $G$, and with hazard rate functions$r$ and $q$, respectively,$X$ is said to be smaller than $Y$ in the hazard rate order (denoted as $X le_\left\{hr\right\}Y$) if:$r\left(t\right)ge q\left(t\right)$ for all $tge 0$,or equivalently if:$frac\left\{1-F\left(t\right)\right\}\left\{1-G\left(t\right)\right\}$ is decreasing in $t$.

Likelihood ratio order

Mean residual life order

Variability orders

If two variables have the same mean, they can still be compared by how "spread out" their distributions are. This is captured to a limited extent by the variance, but more fully by a range of stochastic orders.

Convex order

Under the convex ordering, $A$ is less than $B$ if and only if for all convex $u$, $E \left[u\left(A\right)\right] < E \left[u\left(B\right)\right]$.

References

#M. Shaked and J. G. Shanthikumar, "Stochastic Orders and their Applications", Associated Press, 1994.
#E. L. Lehmann. Ordered families of distributions. "The Annals of Mathematical Statistics", 26:399-419, 1955.

ee also

*Stochastic dominance

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Stochastic dominance — is a form of stochastic ordering. The term is used in decision theory to refer to situations where one lottery (a probability distribution over outcomes) can be ranked as superior to another. It is based on preferences regarding outcomes (e.g.,… …   Wikipedia

• List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

• Mean-preserving spread — In probability and statistics, a mean preserving spread (MPS)[1] is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A s probability density function… …   Wikipedia

• List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

• Gewöhnliche stochastische Ordnung — Stochastische Ordnungen sind Ordnungsrelationen für Zufallsvariablen. Sie verallgemeinern das Konzept von größer und kleiner auf zufällige Größen und dienen zum Beispiel dem Vergleich von Risiken in der Versicherungswirtschaft. Die Theorie der… …   Deutsch Wikipedia

• Stochastische Ordnung — Stochastische Ordnungen sind Ordnungsrelationen für Zufallsvariablen. Sie verallgemeinern das Konzept von größer und kleiner auf zufällige Größen und dienen zum Beispiel dem Vergleich von Risiken in der Versicherungswirtschaft. Die Theorie der… …   Deutsch Wikipedia

• Mann–Whitney U — In statistics, the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW) or Wilcoxon rank sum test) is a non parametric statistical hypothesis test for assessing whether one of two samples of independent observations tends to have… …   Wikipedia

• Jack C. Hayya — is professor emeritus of management science at the Pennsylvania State University.Education*B.S., Civil Enginering, University of Illinois at Champaign Urbana, 1952 *M.S., Management, California State University, Northridge, 1961 [Hayya, Jack C.… …   Wikipedia

• Filtration (mathematics) — In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj. The concept… …   Wikipedia

• CMA-ES — stands for Covariance Matrix Adaptation Evolution Strategy. Evolution strategies (ES) are stochastic, derivative free methods for numerical optimization of non linear or non convex continuous optimization problems. They belong to the class of… …   Wikipedia