Stochastic ordering

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

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 }}x\in (-\infty ,\infty ),}

where

Pr

(

⋅

)

{\displaystyle \Pr(\cdot )}

denotes the probability of an event. This is sometimes denoted

A

⪯

B

{\displaystyle A\preceq B}

or

A

≤

s

t

B

{\displaystyle A\leq _{\mathrm {st} }B}

.

If additionally

Pr

(

A

>

x

)

<

Pr

(

B

>

x

)

{\displaystyle \Pr(A>x)<\Pr(B>x)}

for some

x

{\displaystyle x}

, then

A

{\displaystyle A}

is stochastically strictly less than

B

{\displaystyle B}

, sometimes denoted

A

≺

B

{\displaystyle A\prec B}

. In decision theory, under this circumstance, B is said to be first-order stochastically dominant 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.

A

⪯

B

{\displaystyle A\preceq B}

if and only if for all non-decreasing functions

u

{\displaystyle u}

,

E

⁡

[

u

(

A

)

]

≤

E

⁡

[

u

(

B

)

]

{\displaystyle \operatorname {E} [u(A)]\leq \operatorname {E} [u(B)]}

.

If

u

{\displaystyle u}

is non-decreasing and

A

⪯

B

{\displaystyle A\preceq B}

then

u

(

A

)

⪯

u

(

B

)

{\displaystyle u(A)\preceq u(B)}

If

u

:

R

n

→

R

{\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} }

is increasing in each variable and

A

i

{\displaystyle A_{i}}

and

B

i

{\displaystyle B_{i}}

are independent sets of random variables with

A

i

⪯

B

i

{\displaystyle A_{i}\preceq B_{i}}

for each

i

{\displaystyle i}

, then

u

(

A

1

,

…

,

A

n

)

⪯

u

(

B

1

,

…

,

B

n

)

{\displaystyle u(A_{1},\dots ,A_{n})\preceq u(B_{1},\dots ,B_{n})}

and in particular

∑

i

=

1

n

A

i

⪯

∑

i

=

1

n

B

i

{\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)}}

.