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

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

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 $A \prec B$. 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 $u$, ${\rm E}[u(A)] \le {\rm E}[u(B)]$.
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 $A \overset{d}{=} B$ (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 $A \le B$ 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 $\mathcal G$. $\mathcal G$ 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.

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

See also

• Stochastic dominance

References

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