# Stochastic ordering

(Redirected from Stochastic order)

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

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

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

### 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 $u$, ${\rm E}[u(A)] \le {\rm E}[u(B)]$.
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 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 $\Pr(A>u|C=c)\le \Pr(B>u|C=c)$ 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 $A=B$ in distribution.

## Stochastic dominance

Stochastic dominance[1] 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 $A \le B$ 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

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

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

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

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

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

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

All three order types also have integral representations, that is for a particular order $A$ is smaller than $B$ if and only if ${\rm E}[f(A)] \le {\rm E}[f(B)]$ for all $f:\mathbb R^d\longrightarrow \mathbb R$ in a class of functions $\mathcal G$.[2] $\mathcal G$ 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 $X \le_{hr}Y$) if

$r(t)\ge q(t)$ for all $t\ge 0$,

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 ($X \le _{lr} Y$).

### 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[3]