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
B
A real random variable
A
B
\Pr(A>x)\le\Pr(B>x)forallx\in(-infty,infty),
where
\Pr( ⋅ )
A\preceqB
A\lestB
If additionally
\Pr(A>x)<\Pr(B>x)
x
A
B
A\precB
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.
A\preceqB
u
{\rmE}[u(A)]\le{\rmE}[u(B)]
u
A\preceqB
u(A)\precequ(B)
u:Rn\toR
Ai
Bi
Ai\preceqBi
i
u(A1,...,An)\precequ(B1,...,Bn)
n | |
\sum | |
i=1 |
Ai\preceq
n | |
\sum | |
i=1 |
Bi
i
A(i)\preceqB(i)
Ai
Bi
Ai\preceqBi
i
A\preceqB
A
B
C
\sumc\Pr(C=c)=1
\Pr(A>u|C=c)\le\Pr(B>u|C=c)
u
c
\Pr(C=c)>0
A\preceqB
If
A\preceqB
{\rmE}[A]={\rmE}[B]
Al{\overset{d}{=}}B
Stochastic dominance relations are a family of stochastic orderings used in decision theory:[1]
A\prec(0)B
A\leB
A<B
A\prec(1)B
\Pr(A>x)\le\Pr(B>x)
x
x
\Pr(A>x)<\Pr(B>x)
A\prec(2)B
x | |
\int | |
-infty |
[\Pr(B>t)-\Pr(A>t)]dt\geq0
x
x
There also exist higher-order notions of stochastic dominance. With the definitions above, we have
A\prec(i)B\impliesA\prec(i+1)B
An
Rd
A
Rd
B
{\rmE}[f(A)]\le{\rmE}[f(B)]forallbounded,increasingfunctionsf\colonRd\longrightarrowR
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
B
\Pr(A>x)\le\Pr(B>x)forallx\inRd
and
A
B
\Pr(A\lex)\le\Pr(B\lex)forallx\inRd
All three order types also have integral representations, that is for a particular order
A
B
{\rmE}[f(A)]\le{\rmE}[f(B)]
f\colonRd\longrightarrowR
lG
lG
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 .
Deterministic dominance, denoted
A\succeqddB
A
B
\Pr[A=x] ⋅ \Pr[B=y]=0
\Pr[A\geqB]=1
0.6*30+0.4*20\succeqdd0.5*20+0.5*10
Bilinear dominance, denoted
A\succeqbdB
A
B
\Pr[A=x] ⋅ \Pr[B=y]\leq\Pr[A=y] ⋅ \Pr[B=x]
0.5*30+0.5*20\succeqbd0.33*30+0.33*20+0.34*10
Stochastic dominance (already mentioned above), denoted
A\succeqsdB
A
B
\Pr[A\geqx]\geq\Pr[B\geqx]
0.5*30+0.5*10\succeqsd0.5*20+0.5*10
Pairwise-comparison dominance, denoted
A\succeqpcB
A
B
\Pr[A\geqB]\geq\Pr[B\geqA]
0.67*30+0.33*10\succeqpc1.0*20
Downward-lexicographic dominance, denoted
A\succeqdlB
A
B
A
B
A
B
The hazard rate of a non-negative random variable
X
F
f
r(t)=
d | |
dt |
(-log(1-F(t)))=
f(t) | |
1-F(t) |
.
Given two non-negative variables
X
Y
F
G
r
q
X
Y
X\preceqhrY
r(t)\geq(t)
t\ge0
1-F(t) | |
1-G(t) |
t
Let
X
Y
f\left(t\right)
g\left(t\right)
g\left(t\right) | |
f\left(t\right) |
t
X
Y
X
Y
X\preceqlrY
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.
Convex order is a special kind of variability order. Under the convex ordering,
A
B
u
{\rmE}[u(A)]\leq{\rmE}[u(B)]
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(-\alphax)
\alpha
Considering a family of probability distributions
({P}\alpha)\alpha
(E,\preceq)
\alpha\inF
(F,\preceq)
(X\alpha)\alpha
X\alpha
{P}\alpha
X\alpha\preceqX\beta
\alpha\preceq\beta