Families of distributions satisfying MLR Monotone likelihood ratio
1 families of distributions satisfying mlr
1.1 list of families
1.2 hypothesis testing
1.3 example: effort , output
families of distributions satisfying mlr
statistical models assume data generated distribution family of distributions , seek determine distribution. task simplified if family has monotone likelihood ratio property (mlrp).
a family of density functions
{
f
θ
(
x
)
}
θ
∈
Θ
{\displaystyle \{f_{\theta }(x)\}_{\theta \in \theta }}
indexed parameter
θ
{\displaystyle \theta }
taking values in ordered set
Θ
{\displaystyle \theta }
said have monotone likelihood ratio (mlr) in statistic
t
(
x
)
{\displaystyle t(x)}
if
θ
1
<
θ
2
{\displaystyle \theta _{1}<\theta _{2}}
,
f
θ
2
(
x
=
x
1
,
x
2
,
x
3
,
…
)
f
θ
1
(
x
=
x
1
,
x
2
,
x
3
,
…
)
{\displaystyle {\frac {f_{\theta _{2}}(x=x_{1},x_{2},x_{3},\dots )}{f_{\theta _{1}}(x=x_{1},x_{2},x_{3},\dots )}}}
non-decreasing function of
t
(
x
)
{\displaystyle t(x)}
.
then family of distributions has mlr in
t
(
x
)
{\displaystyle t(x)}
.
list of families
hypothesis testing
if family of random variables has mlrp in
t
(
x
)
{\displaystyle t(x)}
, uniformly powerful test can determined hypotheses
h
0
:
θ
≤
θ
0
{\displaystyle h_{0}:\theta \leq \theta _{0}}
versus
h
1
:
θ
>
θ
0
{\displaystyle h_{1}:\theta >\theta _{0}}
.
example: effort , output
example: let
e
{\displaystyle e}
input stochastic technology – worker s effort, instance – ,
y
{\displaystyle y}
output, likelihood of described probability density function
f
(
y
;
e
)
.
{\displaystyle f(y;e).}
monotone likelihood ratio property (mlrp) of family
f
{\displaystyle f}
expressed follows:
e
1
,
e
2
{\displaystyle e_{1},e_{2}}
, fact
e
2
>
e
1
{\displaystyle e_{2}>e_{1}}
implies ratio
f
(
y
;
e
2
)
/
f
(
y
;
e
1
)
{\displaystyle f(y;e_{2})/f(y;e_{1})}
increasing in
y
{\displaystyle y}
.
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