Stochastic ordering

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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 may be neither stochastically greater than, less than nor equal to another random variable . Many different orders exist, which have different applications.

Usual stochastic order[edit]

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

where denotes the probability of an event. This is sometimes denoted or . If additionally for some , then is stochastically strictly less than , sometimes denoted . In decision theory, under this circumstance B is said to be first-order stochastically dominant over A.


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. if and only if for all non-decreasing functions , .
  2. If is non-decreasing and then
  3. If is an increasing function[clarification needed] and and are independent sets of random variables with for each , then and in particular Moreover, the th order statistics satisfy .
  4. If two sequences of random variables and , with for all each converge in distribution, then their limits satisfy .
  5. If , and are random variables such that and for all and such that , then .

Other properties[edit]

If and then (the random variables are equal in distribution).

Stochastic dominance[edit]

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: if 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[edit]

An -valued random variable is less than an -valued random variable 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. is said to be smaller than in upper orthant order if

and is smaller than in lower orthant order if

All three order types also have integral representations, that is for a particular order is smaller than if and only if for all in a class of functions .[2] is then called generator of the respective order.

Other stochastic orders[edit]

Hazard rate order[edit]

The hazard rate of a non-negative random variable with absolutely continuous distribution function and density function is defined as

Given two non-negative variables and with absolutely continuous distribution and , and with hazard rate functions and , respectively, is said to be smaller than in the hazard rate order (denoted as ) if

for all ,

or equivalently if

is decreasing in .

Likelihood ratio order[edit]

Let and two continuous (or discrete) random variables with densities (or discrete densities) and , respectively, so that increases in over the union of the supports of and ; in this case, is smaller than in the likelihood ratio order ().

Mean residual life order[edit]

Variability orders[edit]

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[edit]

Convex order is a special kind of variability order. Under the convex ordering, is less than if and only if for all convex , .

Laplace transform order[edit]

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: . This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with a positive real number.

Realizable monotonicity[edit]

Considering a family of probability distributions on partially ordered space indexed with (where is another partially ordered space, the concept of complete or realizable monotonicity may be defined. It means, there exists a family of random variables on the same probability space, such that the distribution of is and almost surely whenever . It means the existence of a monotone coupling. See[3]

See also[edit]


  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.
  1. ^
  2. ^ Alfred Müller, Dietrich Stoyan: Comparison methods for stochastic models and risks. Wiley, Chichester 2002, ISBN 0-471-49446-1, S. 2.
  3. ^ Stochastic Monotonicity and Realizable Monotonicity James Allen Fill and Motoya Machida, The Annals of Probability, Vol. 29, No. 2 (Apr., 2001), pp. 938-978, Published by: Institute of Mathematical Statistics, Stable URL: