# Stochastic ordering

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{{Short description|Type of random variable ordering}}
{{Citations|date=May 2024}}

In [probability theory](/source/probability_theory) and [statistics](/source/statistics), a '''stochastic order''' quantifies the concept of one [random variable](/source/random_variable) being "bigger" than another.  These are usually [partial order](/source/partial_order)s, so that one random variable <math>A</math> may be neither stochastically greater than, less than, nor equal to another random variable <math>B</math>.  Many different orders exist, which have different applications.

==Usual stochastic order==
A real random variable <math>A</math> is less than a random variable <math>B</math> in the "usual stochastic order" if

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

where <math>\Pr(\cdot)</math> denotes the probability of an event. This is sometimes denoted <math>A \preceq B</math> or <math>A \le_\mathrm{st} B</math>.

If additionally <math>\Pr(A>x) < \Pr(B>x)</math> for some <math>x</math>, then <math>A</math> is stochastically strictly less than <math>B</math>, sometimes denoted <math>A \prec B</math>. In [decision theory](/source/decision_theory), under this circumstance, {{mvar|B}} is said to be [first-order stochastically dominant](/source/stochastic_dominance) 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.
#<math>A\preceq B</math> if and only if for all non-decreasing functions <math>u</math>, <math>\operatorname E[u(A)] \le \operatorname E[u(B)]</math>.
#If <math>u</math> is non-decreasing and <math>A\preceq B</math> then <math>u(A) \preceq u(B)</math>
#If <math>u:\mathbb{R}^n\to\mathbb{R}</math> is increasing in each variable and <math>A_i</math> and <math>B_i</math> are independent sets of random variables with <math>A_i \preceq B_i</math> for each <math>i</math>, then <math>u(A_1,\dots,A_n) \preceq u(B_1,\dots,B_n)</math> and in particular <math>\sum_{i=1}^n A_i \preceq \sum_{i=1}^n B_i</math> Moreover, the <math>i</math>th [order statistic](/source/order_statistic)s satisfy <math>A_{(i)} \preceq B_{(i)}</math>.
#If two sequences of random variables <math>A_i</math> and <math>B_i</math>, with <math>A_i \preceq B_i</math> for all <math>i</math> each [converge in distribution](/source/convergence_in_distribution),  then their limits satisfy <math>A \preceq B</math>.
#If <math>A</math>, <math>B</math> and <math>C</math> are random variables such that <math>\sum_c\Pr(C=c)=1</math> and <math>\Pr(A>u\mid C=c)\le \Pr(B>u\mid C=c)</math> for all <math>u</math> and <math>c</math> such that <math>\Pr(C=c)>0</math>, then <math>A\preceq B</math>.

===Other properties===
If <math>A\preceq B</math> and <math>\operatorname E[A]=\operatorname E[B]</math> then <math> A \mathrel{\overset{d}{=}} B</math> (the random variables are equal in distribution).

==Stochastic dominance==
[Stochastic dominance](/source/Stochastic_dominance) relations are a family of stochastic orderings used in [decision theory](/source/decision_theory):<ref>{{cite book |last=Perrakis |first1=Stylianos |title=Stochastic Dominance Option Pricing |publisher=Palgrave Macmillan, Cham |year=2019 |isbn=978-3-030-11589-0 |doi=10.1007/978-3-030-11590-6_1 }}</ref>
* Zeroth-order stochastic dominance: <math>A \prec_{(0)} B</math> if and only if <math>A \le B</math> for all realizations of these random variables and <math>A < B</math> for at least one realization.
* First-order stochastic dominance: <math>A \prec_{(1)} B</math> if and only if <math>\Pr(A>x) \le \Pr(B>x)</math> for all <math>x</math> and there exists <math>x</math> such that <math>\Pr(A>x) < \Pr(B>x)</math>.
* Second-order stochastic dominance: <math>A \prec_{(2)} B</math> if and only if <math>\int_{-\infty}^x [\Pr(B>t) - \Pr(A>t)] \, dt \geq 0</math> for all <math>x</math>, with strict inequality at some <math>x</math>.

There also exist higher-order notions of stochastic dominance. With the definitions above, we have <math>A \prec_{(i)} B \implies A \prec_{(i+1)} B</math>.

==Multivariate stochastic order==
An <math>\mathbb R^d</math>-valued random variable <math>A</math> is less than an <math>\mathbb R^d</math>-valued random variable <math>B</math> in the "usual stochastic order" if

:<math>\operatorname E[f(A)] \le \operatorname E[f(B)]\text{ for all bounded, increasing functions } f\colon\mathbb R^d\longrightarrow\mathbb R </math>

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. <math>A</math> is said to be smaller than <math>B</math> in upper orthant order if

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

and <math>A</math> is smaller than <math>B</math> in lower orthant order if<ref>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</ref>

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

All three order types also have integral representations, that is for a particular order <math>A</math> is smaller than <math>B</math> if and only if <math>\operatorname E[f(A)] \le \operatorname E[f(B)] </math> for all <math>f\colon\mathbb R^d\longrightarrow \mathbb R</math> in a class of functions <math>\mathcal G</math>.<ref>Alfred Müller, Dietrich Stoyan: ''Comparison methods for stochastic models and risks.'' Wiley, Chichester 2002, {{ISBN|0-471-49446-1}}, S. 2.</ref> <math>\mathcal G</math> is then called generator of the respective order.

== Other dominance orders ==
The following stochastic orders are useful in the theory of [random social choice](/source/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.<ref name=":0">{{Cite book|last=Felix Brandt|chapter-url=https://books.google.com/books?id=0qY8DwAAQBAJ&dq=multiwinner++voting+a+new+challenge&pg=PA27|title=Trends in Computational Social Choice|date=2017-10-26|publisher=Lulu.com|isbn=978-1-326-91209-3|editor-last=Endriss|editor-first=Ulle|language=en|chapter=Roling the Dice: Recent Results in Probabilistic Social Choice}}</ref> 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 <math>A\succeq_\mathrm{dd} B</math>, means that every possible outcome of <math>A</math> is at least as good as every possible outcome of <math>B</math>:  for all ''x''&nbsp;<&nbsp;''y'',  <math>\Pr[A=x]\cdot \Pr[B=y] = 0</math>. In other words: <math>\Pr[A\geq B] = 1</math>. For example, <math>0.6 \times 30 + 0.4 \times 20 \succeq_\mathrm{dd} 0.5 \times 20 + 0.5 \times 10</math>.

'''Bilinear dominance''', denoted <math>A\succeq_\mathrm{bd} B</math>, means that, for every possible outcome, the probability that <math>A</math> yields the better one and <math>B</math> yields the worse one is at least as large as the probability the other way around: for all x<y, <math>\Pr[A=x]\cdot \Pr[B=y] \leq \Pr[A=y]\cdot \Pr[B=x]</math> For example, <math>0.5 \times 30 + 0.5 \times 20 \succeq_\mathrm{bd} 0.33 \times 30 + 0.33 \times 20 + 0.34 \times 10</math>.

'''Stochastic dominance''' (already mentioned above), denoted <math>A\succeq_\mathrm{sd} B</math>, means that, for every possible outcome ''x'', the probability that <math>A</math> yields at least ''x'' is at least as large as the probability that <math>B</math> yields at least ''x'': for all x, <math>\Pr[A\geq x]\geq  \Pr[B\geq x]</math>. For example, <math>0.5 \times 30 + 0.5 \times 10 \succeq_\mathrm{sd} 0.5 \times 20 + 0.5\times 10</math>.

'''Pairwise-comparison dominance''', denoted <math>A\succeq_\mathrm{pc} B</math>, means that the probability that that <math>A</math> yields a better outcome than <math>B</math> is larger than the other way around: <math>\Pr[A\geq B]\geq\Pr[B\geq A]</math>. For example, <math>0.67 \times 30 + 0.33 \times 10 \succeq_\mathrm{pc} 1.0 \times 20</math>.

'''Downward-lexicographic dominance,''' denoted <math>A\succeq_\mathrm{dl} B</math>, means that <math>A</math> has a larger probability than <math>B</math> of returning the best outcome, or both <math>A</math> and <math>B</math> have the same probability to return the best outcome but  <math>A</math> has a larger probability than <math>B</math> 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](/source/lexicographic_dominance).

==Other stochastic orders==

===Hazard rate order===
The ''[hazard rate](/source/hazard_rate)'' of a non-negative random variable <math>X</math> with absolutely continuous distribution function <math>F</math> and density function <math>f</math> is defined as
:<math>r(t) = \frac{d}{dt}(-\log(1-F(t))) = \frac{f(t)}{1-F(t)}.</math>

Given two non-negative variables <math>X</math> and <math>Y</math> with absolutely continuous distribution <math>F</math> and <math>G</math>, and with hazard rate functions <math>r</math> and <math>q</math>, respectively, <math>X</math> is said to be smaller than <math>Y</math> in the hazard rate order  (denoted as <math>X \preceq_\mathrm{hr}Y</math>) if
:<math>r(t)\ge q(t)</math> for all <math>t\ge 0</math>,
or equivalently if
:<math>\frac{1-F(t)}{1-G(t)}</math> is decreasing in <math>t</math>.

===Likelihood ratio order===
Let <math>X</math> and <math>Y</math> two continuous (or discrete) random variables with densities (or discrete densities) <math>f(t)</math> and <math>g(t)</math>, respectively, so that <math>\frac{g(t)}{f(t)}</math> increases in <math>t</math> over the union of the supports of <math>X</math> and <math>Y</math>; in this case, <math>X</math> is smaller than <math>Y</math> in the ''likelihood ratio order'' (<math>X \preceq_\mathrm{lr} Y</math>).

===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](/source/variance), but more fully by a range of stochastic orders.{{citation needed|date=February 2012}}

====Convex order====
Convex order is a special kind of variability order. Under the convex ordering, <math>A</math> is less than <math>B</math> if and only if for all convex <math>u</math>, <math>\operatorname E[u(A)] \leq \operatorname E[u(B)]</math>.

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

===Realizable monotonicity===
Considering a family of probability distributions  <math> ({P}_\alpha)_{\alpha \in F} </math> on partially ordered space <math> (E,\preceq) </math>
indexed with <math> \alpha \in F </math> (where <math> (F,\preceq) </math> is another partially ordered space, the concept of complete or realizable monotonicity may be defined. It means, there exists a family of random variables <math> (X_\alpha)_\alpha </math> on the same probability space, such that the distribution of <math> X_\alpha </math> is <math> {P}_\alpha </math> and <math> X_\alpha \preceq X_\beta </math> almost surely whenever <math> \alpha \preceq \beta </math>. It means the existence of a monotone [coupling](/source/coupling_(probability)).<ref>{{Cite journal |last=Fill |first=James Allen |last2=Machida |first2=Motoya |date=2001 |title=Stochastic Monotonicity and Realizable Monotonicity |url=https://www.jstor.org/stable/2691998 |journal=The Annals of Probability |volume=29 |issue=2 |pages=938–978 |issn=0091-1798}}</ref>

==See also==
*[Stochastic dominance](/source/Stochastic_dominance)
*[Stochastic](/source/Stochastic)

==References==
{{reflist}}

==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'', 26:399&ndash;419, 1955.

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Adapted from the Wikipedia article [Stochastic ordering](https://en.wikipedia.org/wiki/Stochastic_ordering) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Stochastic_ordering?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.