Wikipedia / Stochastic ordering
Wikipedia / 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. Many different orders exist, which have different applications.
Usual stochastic order
in the "usual stochastic order" if
.
. In decision theory, under this circumstance, is said to be first-order stochastically dominant over A.
Characterizations
The following rules describe situations when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist.
.
th order statistics.
each converge in distribution.
.
Other properties
(the random variables are equal in distribution).
Stochastic dominance
Stochastic dominance relations are a family of stochastic orderings used in decision theory:
for at least one realization.
.
.
.
Multivariate stochastic order
in the "usual stochastic order" if
in upper orthant order if
in lower orthant order if
. is then called generator of the respective order.
Other dominance orders
The following stochastic orders are useful in the theory of random social choice. They are used to compare the outcomes of random social choice functions, in order to check them for efficiency or other desirable criteria. The dominance orders below are ordered from the most conservative to the least conservative. They are exemplified on random variables over the finite support {30,20,10}.
Deterministic dominance: for all x<y.
Bilinear dominance.
Stochastic dominance, means that, for every possible outcome x yields at least x yields at least x.
Pairwise-comparison dominance.
Downward-lexicographic dominance, of returning the second-best best outcome, etc. Upward-lexicographic dominance is defined analogously based on the probability to return the worst outcomes. See lexicographic dominance.
Other stochastic orders
Hazard rate order
The hazard rate is defined as
) if
,
or equivalently if
.
Likelihood ratio order
in the likelihood ratio 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
.
Laplace transform order
Laplace transform a positive real number.
Realizable monotonicity
. It means the existence of a monotone coupling.
See also
References
Bibliography
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, 26:399419, 1955.