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, 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 orderEdit

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.

CharacterizationsEdit

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 propertiesEdit

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

Stochastic dominanceEdit

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 orderEdit

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 ordersEdit

Hazard rate orderEdit

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 orderEdit

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 orderEdit

Variability ordersEdit

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 orderEdit

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 orderEdit

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 monotonicityEdit

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 alsoEdit

ReferencesEdit

  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. ^ https://www.mcgill.ca/files/economics/stochasticdominance.pdf
  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: https://www.jstor.org/stable/2691998