# Stochastic ordering

In probability theory and 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

$\Pr(A>x)\leq \Pr(B>x){\text{ for all }}x\in (-\infty ,\infty ),$

where $\Pr(\cdot )$  denotes the probability of an event. This is sometimes denoted $A\preceq B$  or $A\leq _{st}B$ . If additionally $\Pr(A>x)<\Pr(B>x)$  for some $x$ , then $A$  is stochastically strictly less than $B$ , sometimes denoted $A\prec B$ . In decision theory, under this circumstance B is said to be first-order stochastically dominant over A.

### 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.

1. $A\preceq B$  if and only if for all non-decreasing functions $u$ , ${\rm {E}}[u(A)]\leq {\rm {E}}[u(B)]$ .
2. If $u$  is non-decreasing and $A\preceq B$  then $u(A)\preceq u(B)$
3. If $u:\mathbb {R} ^{n}\to \mathbb {R}$  is an increasing function[clarification needed] 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 $i$ th order statistics satisfy $A_{(i)}\preceq B_{(i)}$ .
4. 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 $A\preceq B$ .
5. If $A$ , $B$  and $C$  are random variables such that $\sum _{c}\Pr(C=c)=1$  and $\Pr(A>u|C=c)\leq \Pr(B>u|C=c)$  for all $u$  and $c$  such that $\Pr(C=c)>0$ , then $A\preceq B$ .

## Other properties

If $A\preceq B$  and ${\rm {E}}[A]={\rm {E}}[B]$  then $A{\overset {d}{=}}B$  (the random variables are equal in distribution).

## Stochastic dominance

Stochastic dominance is a stochastic ordering used in decision theory. Several "orders" of stochastic dominance are defined.

• Zeroth order stochastic dominance consists of simple inequality: $A\preceq _{(0)}B$  if $A\leq B$  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

An $\mathbb {R} ^{d}$ -valued random variable $A$  is less than an $\mathbb {R} ^{d}$ -valued random variable $B$  in the "usual stochastic order" if

${\rm {E}}[f(A)]\leq {\rm {E}}[f(B)]{\text{ for all bounded, increasing functions }}f:\mathbb {R} ^{d}\longrightarrow \mathbb {R}$

Other types of multivariate stochastic orders exist. For instance the upper and lower orthant order which are similar to the usual one-dimensional stochastic order. $A$  is said to be smaller than $B$  in upper orthant order if

$\Pr(A>\mathbf {x} )\leq \Pr(B>\mathbf {x} ){\text{ for all }}\mathbf {x} \in \mathbb {R} ^{d}$

and $A$  is smaller than $B$  in lower orthant order if

$\Pr(A\leq \mathbf {x} )\geq \Pr(B\leq \mathbf {x} ){\text{ for all }}\mathbf {x} \in \mathbb {R} ^{d}$

All three order types also have integral representations, that is for a particular order $A$  is smaller than $B$  if and only if ${\rm {E}}[f(A)]\leq {\rm {E}}[f(B)]$  for all $f:\mathbb {R} ^{d}\longrightarrow \mathbb {R}$  in a class of functions ${\mathcal {G}}$ . ${\mathcal {G}}$  is then called generator of the respective 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 $X\leq _{hr}Y$ ) if

$r(t)\geq q(t)$  for all $t\geq 0$ ,

or equivalently if

${\frac {1-F(t)}{1-G(t)}}$  is decreasing in $t$ .

### Likelihood ratio order

Let $X$  and $Y$  two continuous (or discrete) random variables with densities (or discrete densities) $f\left(t\right)$  and $g\left(t\right)$ , respectively, so that ${\frac {g\left(t\right)}{f\left(t\right)}}$  increases in $t$  over the union of the supports of $X$  and $Y$ ; in this case, $X$  is smaller than $Y$  in the likelihood ratio order ($X\leq _{lr}Y$ ).

### 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.[citation needed]

#### Convex order

Convex order is a special kind of variability order. Under the convex ordering, $A$  is less than $B$  if and only if for all convex $u$ , ${\rm {E}}[u(A)]\leq {\rm {E}}[u(B)]$ .

### Laplace transform order

Laplace transform order compares both size and variability of two random variables. Similar to convex order, Laplace transform order is established by comparing the expectation of a function of the random variable where the function is from a special class: $u(x)=-\exp(-\alpha x)$ . This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with $\alpha$  a positive real number.

### Realizable monotonicity

Considering a family of probability distributions $({P}_{\alpha })_{\alpha \in F}$  on partially ordered space $(E,\preceq )$  indexed with $\alpha \in F$  (where $(F,\preceq )$  is another partially ordered space, the concept of complete or realizable monotonicity may be defined. It means, there exists a family of random variables $(X_{\alpha })_{\alpha }$  on the same probability space, such that the distribution of $X_{\alpha }$  is ${P}_{\alpha }$  and $X_{\alpha }\preceq X_{\beta }$  almost surely whenever $\alpha \preceq \beta$ . It means the existence of a monotone coupling. See