statistics, a stochastic order quantifies the concept of one random variablebeing "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
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 .
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.
# if and only if for all non-decreasing functions , .
#If is non-decreasing and then
#If is an increasing function and and are independent sets of random variables with for each , then and in particular Moreover, the th
order statistics satisfy .
#If two sequences of random variables and , with for all each converge in distribution, then their limits satisfy .
#If , and are random variables such that for all and , then
If and then in distribution.
Stochastic dominance[http://www.mcgill.ca/files/economics/stochasticdominance.pdf] 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
*Lower order stochastic dominance implies higher order stochastic dominance.
Multivariate stochastic order
Other stochastic orders
Hazard rate order
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
Mean residual life order
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.
Under the convex ordering, is less than if and only if for all convex , .
#M. Shaked and J. G. Shanthikumar, "Stochastic Orders and their Applications", Associated Press, 1994.
#E. L. Lehmann. Ordered families of distributions. "The Annals of Mathematical Statistics", 26:399-419, 1955.
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