Stochastic ordering

In 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

:P(A>x) le P(B>x) ext{ for all }x in (-infty,infty),

where P(cdot) denotes the probability of an event.This is sometimes denoted A preceq B or A le_{st} B. If additionally P(A>x) < P(B>x) for some x, then A is stochastically strictly less than B, sometimes denoted A prec B.

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.

#Apreceq B if and only if for all non-decreasing functions u, E [u(A)] le E [u(B)] .
#If u is non-decreasing and Apreceq B then u(A) preceq u(B)
#If u:mathbb{R}^nmapstomathbb{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 ith order statistics satisfy A_{(i)} preceq B_{(i)}.
#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.
#If A, B and C are random variables such that P(A>u|C=c)le P(B>u|C=c) for all u and c, then Apreceq B

Other properties

If Apreceq B and E [A] =E [B] then A=B in distribution.

tochastic dominance

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: 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

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 Ywith absolutely continuous distribution F and G, and with hazard rate functionsr 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 tge 0,or equivalently if:frac{1-F(t)}{1-G(t)} is decreasing in t.

Likelihood ratio order

Mean residual life order

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

Under the convex ordering, A is less than B if and only if for all convex u, E [u(A)] < E [u(B)] .

References

#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.

ee also

*Stochastic dominance


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