mlrp

In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs). Formally, distributions ƒ(x) and g(x) bear the property if





for every


x

1


>

x

0


,




f
(

x

1


)


g
(

x

1


)



≥



f
(

x

0


)


g
(

x

0


)





{\displaystyle {\text{for every }}x_{1}>x_{0},\quad {\frac {f(x_{1})}{g(x_{1})}}\geq {\frac {f(x_{0})}{g(x_{0})}}}
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, we say they "have the MLRP in x." For a family of distributions that all satisfy the definition with respect to some statistic T(X), we say they "have the MLR in T(X)."

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