Stochastic ordering

Usual stochastic order

A real random variable A is less than a random variable B in the "usual stochastic order" if

where Pr ( ⋅ ) denotes the probability of an event. This is sometimes denoted A ⪯ B or A ≤ s t B . If additionally Pr ( A > x ) < Pr ( B > x ) for some x , then A is stochastically strictly less than B , sometimes denoted A ≺ 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 ⪯ B if and only if for all non-decreasing functions u , E [ u ( A ) ] ≤ E [ u ( B ) ] .
2. If u is non-decreasing and A ⪯ B then u ( A ) ⪯ u ( B )
3. If u : R n → R is an increasing function and A i and B i are independent sets of random variables with A i ⪯ B i for each i , then u ( A 1 , … , A n ) ⪯ u ( B 1 , … , B n ) and in particular ∑ i = 1 n A i ⪯ ∑ i = 1 n B i Moreover, the i th order statistics satisfy A ( i ) ⪯ B ( i ) .
4. If two sequences of random variables A i and B i , with A i ⪯ B i for all i each converge in distribution, then their limits satisfy A ⪯ B .
5. If A , B and C are random variables such that ∑ c Pr ( C = c ) = 1 and Pr ( A > u | C = c ) ≤ Pr ( B > u | C = c ) for all u and c such that Pr ( C = c ) > 0 , then A ⪯ B .

Other properties

If A ⪯ B and E [ A ] = E [ B ] then A = 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.

Multivariate stochastic order

An R d -valued random variable A is less than an R d -valued random variable B in the "usual stochastic order" if

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

and A is smaller than B in lower orthant order if

All three order types also have integral representations, that is for a particular order A is smaller than B if and only if E [ f ( A ) ] ≤ E [ f ( B ) ] for all f : R d ⟶ R in a class of functions G . G is then called generator of the respective order.

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

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 ≤ h r Y ) if

or equivalently if

Likelihood ratio order

Let X and Y two continuous (or discrete) random variables with densities (or discrete densities) f ( t ) and g ( t ) , respectively, so that g ( t ) f ( t ) 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 ≤ l r 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 , E [ u ( A ) ] ≤ 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 ⁡ ( − α x ) . This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with α a positive real number.

Realizable monotonicity

Considering a family of probability distributions ( P α ) α ∈ F on partially ordered space ( E , ⪯ ) indexed with α ∈ F (where ( F , ⪯ ) 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 α ) α on the same probability space, such that the distribution of X α is P α and X α ⪯ X β almost surely whenever α ⪯ β . It means the existence of a monotone coupling. See