# 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 ${\displaystyle A}$ may be neither stochastically greater than, less than nor equal to another random variable ${\displaystyle B}$. Many different orders exist, which have different applications.

## Usual stochastic order

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

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

where ${\displaystyle \Pr(\cdot )}$  denotes the probability of an event. This is sometimes denoted ${\displaystyle A\preceq B}$  or ${\displaystyle A\leq _{st}B}$ . If additionally ${\displaystyle \Pr(A>x)<\Pr(B>x)}$  for some ${\displaystyle x}$ , then ${\displaystyle A}$  is stochastically strictly less than ${\displaystyle B}$ , sometimes denoted ${\displaystyle A\prec B}$ . 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. ${\displaystyle A\preceq B}$  if and only if for all non-decreasing functions ${\displaystyle u}$ , ${\displaystyle {\rm {E}}[u(A)]\leq {\rm {E}}[u(B)]}$ .
2. If ${\displaystyle u}$  is non-decreasing and ${\displaystyle A\preceq B}$  then ${\displaystyle u(A)\preceq u(B)}$
3. If ${\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} }$  is an increasing function[clarification needed] and ${\displaystyle A_{i}}$  and ${\displaystyle B_{i}}$  are independent sets of random variables with ${\displaystyle A_{i}\preceq B_{i}}$  for each ${\displaystyle i}$ , then ${\displaystyle u(A_{1},\dots ,A_{n})\preceq u(B_{1},\dots ,B_{n})}$  and in particular ${\displaystyle \sum _{i=1}^{n}A_{i}\preceq \sum _{i=1}^{n}B_{i}}$  Moreover, the ${\displaystyle i}$ th order statistics satisfy ${\displaystyle A_{(i)}\preceq B_{(i)}}$ .
4. If two sequences of random variables ${\displaystyle A_{i}}$  and ${\displaystyle B_{i}}$ , with ${\displaystyle A_{i}\preceq B_{i}}$  for all ${\displaystyle i}$  each converge in distribution, then their limits satisfy ${\displaystyle A\preceq B}$ .
5. If ${\displaystyle A}$ , ${\displaystyle B}$  and ${\displaystyle C}$  are random variables such that ${\displaystyle \sum _{c}\Pr(C=c)=1}$  and ${\displaystyle \Pr(A>u|C=c)\leq \Pr(B>u|C=c)}$  for all ${\displaystyle u}$  and ${\displaystyle c}$  such that ${\displaystyle \Pr(C=c)>0}$ , then ${\displaystyle A\preceq B}$ .

## Other properties

If ${\displaystyle A\preceq B}$  and ${\displaystyle {\rm {E}}[A]={\rm {E}}[B]}$  then ${\displaystyle A{\overset {d}{=}}B}$  (the random variables are equal 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: ${\displaystyle A\preceq _{(0)}B}$  if ${\displaystyle A\leq B}$  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 ${\displaystyle \mathbb {R} ^{d}}$ -valued random variable ${\displaystyle A}$  is less than an ${\displaystyle \mathbb {R} ^{d}}$ -valued random variable ${\displaystyle B}$  in the "usual stochastic order" if

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

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. ${\displaystyle A}$  is said to be smaller than ${\displaystyle B}$  in upper orthant order if

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

and ${\displaystyle A}$  is smaller than ${\displaystyle B}$  in lower orthant order if

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

All three order types also have integral representations, that is for a particular order ${\displaystyle A}$  is smaller than ${\displaystyle B}$  if and only if ${\displaystyle {\rm {E}}[f(A)]\leq {\rm {E}}[f(B)]}$  for all ${\displaystyle f:\mathbb {R} ^{d}\longrightarrow \mathbb {R} }$  in a class of functions ${\displaystyle {\mathcal {G}}}$ .[2] ${\displaystyle {\mathcal {G}}}$  is then called generator of the respective order.

## Other stochastic orders

### Hazard rate order

The hazard rate of a non-negative random variable ${\displaystyle X}$  with absolutely continuous distribution function ${\displaystyle F}$  and density function ${\displaystyle f}$  is defined as

${\displaystyle r(t)={\frac {d}{dt}}(-\log(1-F(t)))={\frac {f(t)}{1-F(t)}}.}$

Given two non-negative variables ${\displaystyle X}$  and ${\displaystyle Y}$  with absolutely continuous distribution ${\displaystyle F}$  and ${\displaystyle G}$ , and with hazard rate functions ${\displaystyle r}$  and ${\displaystyle q}$ , respectively, ${\displaystyle X}$  is said to be smaller than ${\displaystyle Y}$  in the hazard rate order (denoted as ${\displaystyle X\leq _{hr}Y}$ ) if

${\displaystyle r(t)\geq q(t)}$  for all ${\displaystyle t\geq 0}$ ,

or equivalently if

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

### Likelihood ratio order

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

### 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, ${\displaystyle A}$  is less than ${\displaystyle B}$  if and only if for all convex ${\displaystyle u}$ , ${\displaystyle {\rm {E}}[u(A)]\leq {\rm {E}}[u(B)]}$ .

### 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: ${\displaystyle u(x)=-\exp(-\alpha x)}$ . This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with ${\displaystyle \alpha }$  a positive real number.

### Realizable monotonicity

Considering a family of probability distributions ${\displaystyle ({P}_{\alpha })_{\alpha \in F}}$  on partially ordered space ${\displaystyle (E,\preceq )}$  indexed with ${\displaystyle \alpha \in F}$  (where ${\displaystyle (F,\preceq )}$  is another partially ordered space, the concept of complete or realizable monotonicity may be defined. It means, there exists a family of random variables ${\displaystyle (X_{\alpha })_{\alpha }}$  on the same probability space, such that the distribution of ${\displaystyle X_{\alpha }}$  is ${\displaystyle {P}_{\alpha }}$  and ${\displaystyle X_{\alpha }\preceq X_{\beta }}$  almost surely whenever ${\displaystyle \alpha \preceq \beta }$ . It means the existence of a monotone coupling. See[3]