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
In decision theory, under this circumstance, B is said to be first-order stochastically dominant over A.
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.
(the random variables are equal in distribution).
Stochastic dominance relations are a family of stochastic orderings used in decision theory:[1] There also exist higher-order notions of stochastic dominance.
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 then called generator of the respective order.
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.
They are exemplified on random variables over the finite support {30,20,10}.
yields the worse one is at least as large as the probability the other way around: for all x Stochastic dominance (already mentioned above), denoted Upward-lexicographic dominance is defined analogously based on the probability to return the worst outcomes. The hazard rate of a non-negative random variable with absolutely continuous distribution function in the hazard rate order (denoted as two continuous (or discrete) random variables with densities (or discrete densities) 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 is a special kind of variability 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: This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with is another partially ordered space, the concept of complete or realizable monotonicity may be defined. It means, there exists a family of random variables It means the existence of a monotone coupling.