Monotone likelihood ratio

In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs). Formally, distributions

f

(

x

)

{\displaystyle \ f(x)\ }

and

g

(

x

)

{\displaystyle \ g(x)\ }

bear the property if

for every

x

2

>

x

1

,

f

(

x

2

)

g

(

x

2

)

≥

f

(

x

1

)

g

(

x

1

)

{\displaystyle \ {\text{for every }}x_{2}>x_{1},\quad {\frac {f(x_{2})}{\ g(x_{2})\ }}\geq {\frac {f(x_{1})}{\ g(x_{1})\ }}\ }

that is, if the ratio is nondecreasing in the argument

x

{\displaystyle x}

.

If the functions are first-differentiable, the property may sometimes be stated

∂

∂

x

(

f

(

x

)

g

(

x

)

)

≥

0

{\displaystyle {\frac {\ \partial }{\ \partial x}}\left({\frac {f(x)}{\ g(x)\ }}\right)\geq 0\ }

For two distributions that satisfy the definition with respect to some argument

x

,

{\displaystyle \ x\ ,}

we say they "have the MLRP in

x

.

{\displaystyle \ x~.}

" For a family of distributions that all satisfy the definition with respect to some statistic

T

(

X

)

,

{\displaystyle \ T(X)\ ,}

we say they "have the MLR in

T

(

X

)

.

{\displaystyle \ T(X)~.}

"

Intuition

The MLRP is used to represent a data-generating process that enjoys a straightforward relationship between the magnitude of some observed variable and the distribution it draws from. If

f

(

x

)

{\displaystyle \ f(x)\ }

satisfies the MLRP with respect to

g

(

x

)

{\displaystyle \ g(x)\ }

, the higher the observed value

x

{\displaystyle \ x\ }

, the more likely it was drawn from distribution

f

{\displaystyle \ f\ }

rather than

g

.

{\displaystyle \ g~.}

As usual for monotonic relationships, the likelihood ratio's monotonicity comes in handy in statistics, particularly when using maximum-likelihood estimation. Also, distribution families with MLR have a number of well-behaved stochastic properties, such as first-order stochastic dominance and increasing hazard ratios. Unfortunately, as is also usual, the strength of this assumption comes at the price of realism. Many processes in the world do not exhibit a monotonic correspondence between input and output.