Stochastic ordering

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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 order[edit]

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.

Characterizations[edit]

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.

  1. if and only if for all non-decreasing functions , .
  2. If is non-decreasing and then
  3. If is increasing in each variable 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 properties[edit]

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

Stochastic dominance[edit]

Stochastic dominance relations are a family of stochastic orderings used in decision theory:[1]

  • Zeroth-order stochastic dominance: if and only if for all realizations of these random variables and for at least one realization.
  • First-order stochastic dominance: if and only if for all and there exists such that .
  • Second-order stochastic dominance: if and only if for all , with strict inequality at some .

There also exist higher-order notions of stochastic dominance. With the definitions above, we have .

Multivariate stochastic order[edit]

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[2]

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 .[3] is then called generator of the respective order.

Other dominance orders[edit]

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.[4] 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, denoted , means that every possible outcome of is at least as good as every possible outcome of : for all x<y, . In other words: . For example, .

Bilinear dominance, denoted , means that, for every possible outcome, the probability that yields the better one and yields the worse one is at least as large as the probability the other way around: for all x<y, For example, .

Stochastic dominance (already mentioned above), denoted , means that, for every possible outcome x, the probability that yields at least x is at least as large as the probability that yields at least x: for all x, . For example, .

Pairwise-comparison dominance, denoted , means that the probability that that yields a better outcome than is larger than the other way around: . For example, .

Other stochastic orders[edit]

Hazard rate order[edit]

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 order[edit]

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 ().

Variability orders[edit]

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[edit]

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 order[edit]

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 monotonicity[edit]

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[5]

See also[edit]

References[edit]

  1. ^ Perrakis, Stylianos (2019). Stochastic Dominance Option Pricing. Palgrave Macmillan, Cham. doi:10.1007/978-3-030-11590-6_1. ISBN 978-3-030-11589-0.
  2. ^ Definition 2.3 in Thibaut Lux, Antonin Papapantoleon: "Improved Fréchet-Hoeffding bounds for d-copulas and applications in model-free finance." Annals of Applied Probability 27, 3633-3671, 2017
  3. ^ Alfred Müller, Dietrich Stoyan: Comparison methods for stochastic models and risks. Wiley, Chichester 2002, ISBN 0-471-49446-1, S. 2.
  4. ^ Felix Brandt (2017-10-26). "Roling the Dice: Recent Results in Probabilistic Social Choice". In Endriss, Ulle (ed.). Trends in Computational Social Choice. Lulu.com. ISBN 978-1-326-91209-3.
  5. ^ 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

Bibliography[edit]

  • 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.