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 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[edit]

A real random variable A is less than a random variable B in the "usual stochastic order" if

\Pr(A>x) \le \Pr(B>x)\text{ for all }x \in (-\infty,\infty),

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

Characterizations[edit]

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. A\preceq B if and only if for all non-decreasing functions u, {\rm E}[u(A)] \le {\rm E}[u(B)].
  2. If u is non-decreasing and A\preceq B then u(A) \preceq u(B)
  3. If u:\mathbb{R}^n\to\mathbb{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)}.
  4. 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.
  5. If A, B and C are random variables such that \sum_c\Pr(C=c)=1 and \Pr(A>u|C=c)\le \Pr(B>u|C=c) for all u and c such that \Pr(C=c)>0, then A\preceq B.

Other properties[edit]

If A\preceq B and {\rm E}[A]={\rm E}[B] then A=B in distribution.

Stochastic dominance[edit]

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: 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[edit]

An \mathbb R^d-valued random variable A is less than an \mathbb R^d-valued random variable B in the "usual stochastic order" if

{\rm E}[f(A)] \le {\rm E}[f(B)]\text{ for all bounded, increasing functions } f:\mathbb R^d\longrightarrow\mathbb R

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. A is said to be smaller than B in upper orthant order if

\Pr(A>\mathbf x) \le \Pr(B>\mathbf x)\text{ for all } \mathbf x \in \mathbb R^d

and A is smaller than B in lower orthant order if

\Pr(A\le\mathbf x) \ge \Pr(B\le\mathbf x)\text{ for all } \mathbf x \in \mathbb R^d

All three order types also have integral representations, that is for a particular order A is smaller than B if and only if {\rm E}[f(A)] \le {\rm E}[f(B)] for all f:\mathbb R^d\longrightarrow \mathbb R in a class of functions \mathcal G.[2] \mathcal G is then called generator of the respective order.

Other stochastic orders[edit]

Hazard rate order[edit]

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 Y with absolutely continuous distribution F and G, and with hazard rate functions r 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 t\ge 0,

or equivalently if

\frac{1-F(t)}{1-G(t)} is decreasing in t.

Likelihood ratio order[edit]

Let X and Y two continuous (or discrete) random variables with densities (or discrete densities) f \left( t \right) and g \left( t \right), respectively, so that \frac{g \left( t \right)}{f \left( t \right)} increases in t over the union of the supports of X and Y; in this case, X is smaller than Y in the likelihood ratio order (X \le _{lr} Y).

Mean residual life order[edit]

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, A is less than B if and only if for all convex u, {\rm E}[u(A)] \leq {\rm E}[u(B)].

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: u(x) = -\exp(-\alpha x). This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with  \alpha a positive real number.

Realizable monotonicity[edit]

Considering a family of probability distributions  ({P}_{\alpha})_{\alpha \in F} on partially ordered space  (E,\preceq) indexed with  \alpha \in F (where  (F,\preceq) is another partially ordered space, the concept of complete or realizable monotonicity may be defined. It means, there exists a family of random variables  (X_\alpha)_{\alpha} on the same probability space, such that the distribution of  X_\alpha is  {P}_\alpha and  X_\alpha \preceq X_\beta almost surely whenever  \alpha \preceq \beta . It means the existence of a monotone coupling. See[3]

See also[edit]

References[edit]

  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. ^ http://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: http://www.jstor.org/stable/2691998