Stochastic ordering – Wikipedia

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{displaystyle A}

may be neither stochastically greater than, less than nor equal to another random variable

B{displaystyle B}

. Many different orders exist, which have different applications.

Usual stochastic order[edit]

A real random variable

A{displaystyle A}

is less than a random variable

B{displaystyle B}

in the “usual stochastic order” if

Pr(A>x)≤Pr(B>x) for all x∈(−∞,∞),{displaystyle Pr(A>x)leq Pr(B>x){text{ for all }}xin (-infty ,infty ),}

Pr(⋅){displaystyle Pr(cdot )}

denotes the probability of an event. This is sometimes denoted A⪯B{displaystyle Apreceq B}

or A≤stB{displaystyle Aleq _{st}B}

. If additionally Pr(A>x)<Pr(B>x){displaystyle Pr(A>x)x)}

x{displaystyle x}

, then A{displaystyle A}

is stochastically strictly less than B{displaystyle B}

, sometimes denoted A≺B{displaystyle Aprec B}

. 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. 
   A⪯B{displaystyle Apreceq B}
   if and only if for all non-decreasing functions u{displaystyle u}
   , E[u(A)]≤E[u(B)]{displaystyle {rm {E}}[u(A)]leq {rm {E}}[u(B)]}
   .
2. If
   u{displaystyle u}
   is non-decreasing and A⪯B{displaystyle Apreceq B}
   then u(A)⪯u(B){displaystyle u(A)preceq u(B)}
   
3. If
   u:Rn→R{displaystyle u:mathbb {R} ^{n}to mathbb {R} }
   is increasing in each variable and Ai{displaystyle A_{i}}
   and Bi{displaystyle B_{i}}
   are independent sets of random variables with Ai⪯Bi{displaystyle A_{i}preceq B_{i}}
   for each i{displaystyle i}
   , then u(A1,…,An)⪯u(B1,…,Bn){displaystyle u(A_{1},dots ,A_{n})preceq u(B_{1},dots ,B_{n})}
   and in particular ∑i=1nAi⪯∑i=1nBi{displaystyle sum _{i=1}^{n}A_{i}preceq sum _{i=1}^{n}B_{i}}
   Moreover, the i{displaystyle i}
   th order statistics satisfy A(i)⪯B(i){displaystyle A_{(i)}preceq B_{(i)}}
   .
4. If two sequences of random variables
   Ai{displaystyle A_{i}}
   and Bi{displaystyle B_{i}}
   , with Ai⪯Bi{displaystyle A_{i}preceq B_{i}}
   for all i{displaystyle i}
   each converge in distribution, then their limits satisfy A⪯B{displaystyle Apreceq B}
   .
5. If
   A{displaystyle A}
   , B{displaystyle B}
   and C{displaystyle C}
   are random variables such that ∑cPr(C=c)=1{displaystyle sum _{c}Pr(C=c)=1}
   and Pr(A>u|C=c)≤Pr(B>u|C=c){displaystyle Pr(A>u|C=c)leq Pr(B>u|C=c)}
   u{displaystyle u}
   and c{displaystyle c}
   such that Pr(C=c)>0{displaystyle Pr(C=c)>0}
   A⪯B{displaystyle Apreceq B}
   .

Other properties[edit]

If

A⪯B{displaystyle Apreceq B}

and

E[A]=E[B]{displaystyle {rm {E}}[A]={rm {E}}[B]}

then

A=dB{displaystyle Amathrel {overset {d}{=}} B}

(the random variables are equal in distribution).

Stochastic dominance[edit]

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

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

A≺(i)B⟹A≺(i+1)B{displaystyle Aprec _{(i)}Bimplies Aprec _{(i+1)}B}

.

Multivariate stochastic order[edit]

An

Rd{displaystyle mathbb {R} ^{d}}

-valued random variable

A{displaystyle A}

is less than an

Rd{displaystyle mathbb {R} ^{d}}

-valued random variable

B{displaystyle B}

in the “usual stochastic order” if

E[f(A)]≤E[f(B)] for all bounded, increasing functions f:Rd⟶R{displaystyle {rm {E}}[f(A)]leq {rm {E}}[f(B)]{text{ for all bounded, increasing functions }}fcolon 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{displaystyle A}

is said to be smaller than

B{displaystyle B}

in upper orthant order if

Pr(A>x)≤Pr(B>x) for all x∈Rd{displaystyle Pr(A>mathbf {x} )leq Pr(B>mathbf {x} ){text{ for all }}mathbf {x} in mathbb {R} ^{d}}

A{displaystyle A}

is smaller than B{displaystyle B}

in lower orthant order if^[2]

Pr(A≤x)≤Pr(B≤x) for all x∈Rd{displaystyle Pr(Aleq mathbf {x} )leq Pr(Bleq 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{displaystyle A}

is smaller than

B{displaystyle B}

if and only if

E[f(A)]≤E[f(B)]{displaystyle {rm {E}}[f(A)]leq {rm {E}}[f(B)]}

for all

f:Rd⟶R{displaystyle fcolon mathbb {R} ^{d}longrightarrow mathbb {R} }

in a class of functions

G{displaystyle {mathcal {G}}}

.^[3]

G{displaystyle {mathcal {G}}}

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

A⪰ddB{displaystyle Asucceq _{dd}B}

, means that every possible outcome of

A{displaystyle A}

is at least as good as every possible outcome of

B{displaystyle B}

: for all x<y,

Pr[A=x]⋅Pr[B=y]=0{displaystyle Pr[A=x]cdot Pr[B=y]=0}

. In other words:

Pr[A≥B]=1{displaystyle Pr[Ageq B]=1}

. For example,

0.6∗30+0.4∗20⪰dd0.5∗20+0.5∗10{displaystyle 0.6*30+0.4*20succeq _{dd}0.5*20+0.5*10}

.

Bilinear dominance, denoted

A⪰bdB{displaystyle Asucceq _{bd}B}

, means that, for every possible outcome, the probability that

A{displaystyle A}

yields the better one and

B{displaystyle B}

yields the worse one is at least as large as the probability the other way around: for all x

Pr[A=x]⋅Pr[B=y]≤Pr[A=y]⋅Pr[B=x]{displaystyle Pr[A=x]cdot Pr[B=y]leq Pr[A=y]cdot Pr[B=x]}

For example,

0.5∗30+0.5∗20⪰bd0.33∗30+0.33∗20+0.34∗10{displaystyle 0.5*30+0.5*20succeq _{bd}0.33*30+0.33*20+0.34*10}

.

Stochastic dominance (already mentioned above), denoted

A⪰sdB{displaystyle Asucceq _{sd}B}

, means that, for every possible outcome x, the probability that

A{displaystyle A}

yields at least x is at least as large as the probability that

B{displaystyle B}

yields at least x: for all x,

Pr[A≥x]≥Pr[B≥x]{displaystyle Pr[Ageq x]geq Pr[Bgeq x]}

. For example,

0.5∗30+0.5∗10⪰sd0.5∗20+0.5∗10{displaystyle 0.5*30+0.5*10succeq _{sd}0.5*20+0.5*10}

.

Pairwise-comparison dominance, denoted

A⪰pcB{displaystyle Asucceq _{pc}B}

, means that the probability that that

A{displaystyle A}

yields a better outcome than

B{displaystyle B}

is larger than the other way around:

Pr[A≥B]≥Pr[B≥A]{displaystyle Pr[Ageq B]geq Pr[Bgeq A]}

. For example,

0.67∗30+0.33∗10⪰pc1.0∗20{displaystyle 0.67*30+0.33*10succeq _{pc}1.0*20}

.

Downward-lexicographic dominance, denoted

A⪰dlB{displaystyle Asucceq _{dl}B}

, means that

A{displaystyle A}

has a larger probability than

B{displaystyle B}

of returning the best outcome, or both

A{displaystyle A}

and

B{displaystyle B}

have the same probability to return the best outcome but

A{displaystyle A}

has a larger probability than

B{displaystyle B}

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.

Other stochastic orders[edit]

Hazard rate order[edit]

The hazard rate of a non-negative random variable

X{displaystyle X}

with absolutely continuous distribution function

F{displaystyle F}

and density function

f{displaystyle f}

is defined as

r(t)=ddt(−log⁡(1−F(t)))=f(t)1−F(t).{displaystyle r(t)={frac {d}{dt}}(-log(1-F(t)))={frac {f(t)}{1-F(t)}}.}

Given two non-negative variables

X{displaystyle X}

and

Y{displaystyle Y}

with absolutely continuous distribution

F{displaystyle F}

and

G{displaystyle G}

, and with hazard rate functions

r{displaystyle r}

and

q{displaystyle q}

, respectively,

X{displaystyle X}

is said to be smaller than

Y{displaystyle Y}

in the hazard rate order (denoted as

X⪯hrY{displaystyle Xpreceq _{hr}Y}

) if

r(t)≥q(t){displaystyle r(t)geq q(t)}

for all t≥0{displaystyle tgeq 0}

,

or equivalently if

1−F(t)1−G(t){displaystyle {frac {1-F(t)}{1-G(t)}}}

is decreasing in t{displaystyle t}

.

Likelihood ratio order[edit]

Let

X{displaystyle X}

and

Y{displaystyle Y}

two continuous (or discrete) random variables with densities (or discrete densities)

f(t){displaystyle fleft(tright)}

and

g(t){displaystyle gleft(tright)}

, respectively, so that

g(t)f(t){displaystyle {frac {gleft(tright)}{fleft(tright)}}}

increases in

t{displaystyle t}

over the union of the supports of

X{displaystyle X}

and

Y{displaystyle Y}

; in this case,

X{displaystyle X}

is smaller than

Y{displaystyle Y}

in the likelihood ratio order (

X⪯lrY{displaystyle Xpreceq _{lr}Y}

).

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{displaystyle A}

is less than

B{displaystyle B}

if and only if for all convex

u{displaystyle u}

,

E[u(A)]≤E[u(B)]{displaystyle {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⁡(−αx){displaystyle 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

α{displaystyle alpha }

a positive real number.

Realizable monotonicity[edit]

Considering a family of probability distributions

(Pα)α∈F{displaystyle ({P}_{alpha })_{alpha in F}}

on partially ordered space

(E,⪯){displaystyle (E,preceq )}

indexed with

α∈F{displaystyle alpha in F}

(where

(F,⪯){displaystyle (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

(Xα)α{displaystyle (X_{alpha })_{alpha }}

on the same probability space, such that the distribution of

Xα{displaystyle X_{alpha }}

is

Pα{displaystyle {P}_{alpha }}

and

Xα⪯Xβ{displaystyle X_{alpha }preceq X_{beta }}

almost surely whenever

α⪯β{displaystyle alpha preceq beta }

. 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.