- Stochastic ordering
In

statistics , a**stochastic order**quantifies the concept of onerandom variable being "bigger" than another. These are usuallypartial order s, 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

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

where $P(cdot)$ denotes the probability of an event.This is sometimes denoted $A\; preceq\; B$ or $A\; le\_\{st\}\; B$. If additionally $P(A>x)\; <\; P(B>x)$ 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 $u$, $E\; [u(A)]\; le\; E\; [u(B)]$.

#If $u$ is non-decreasing and $Apreceq\; B$ then $u(A)\; preceq\; u(B)$

#If $u:mathbb\{R\}^nmapstomathbb\{R\}$ 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 $u(A\_1,dots,A\_n)\; preceq\; u(B\_1,dots,B\_n)$ and in particular $sum\_\{i=1\}^n\; A\_i\; preceq\; sum\_\{i=1\}^n\; B\_i$ Moreover, the $i$thorder statistic s satisfy $A\_\{(i)\}\; preceq\; B\_\{(i)\}$.

#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 $P(A>u|C=c)le\; P(B>u|C=c)$ for all $u$ and $c$, then $Apreceq\; B$**Other properties**If $Apreceq\; B$ and $E\; [A]\; =E\; [B]$ then $A=B$ in distribution.

**tochastic dominance**Stochastic dominance [*http://www.mcgill.ca/files/economics/stochasticdominance.pdf*] is a stochastic ordering used indecision theory . Several "orders" of stochastic dominance are defined.

*Zeroth order stochastic dominance consists of simple inequality: $A\; preceq\_\{(0)\}\; B$ 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 thedistribution 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:$r(t)\; =\; frac\{d\}\{dt\}(-log(1-F(t)))\; =\; frac\{f(t)\}\{1-F(t)\}$.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 $X\; le\_\{hr\}Y$) if:$r(t)ge\; q(t)$ for all $tge\; 0$,or equivalently if:$frac\{1-F(t)\}\{1-G(t)\}$ 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 $u$, $E\; [u(A)]\; <\; E\; [u(B)]$.

**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

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