Relation to other statistical properties Monotone likelihood ratio
1 relation other statistical properties
1.1 exponential families
1.2 powerful tests: karlin–rubin theorem
1.3 median unbiased estimation
1.4 lifetime analysis: survival analysis , reliability
1.4.1 proofs
1.4.2 first-order stochastic dominance
1.4.3 monotone hazard rate
1.5 example
relation other statistical properties
monotone likelihoods used in several areas of statistical theory, including point estimation , hypothesis testing, in probability models.
exponential families
one-parameter exponential families have monotone likelihood-functions. in particular, one-dimensional exponential family of probability density functions or probability mass functions with
f
θ
(
x
)
=
c
(
θ
)
h
(
x
)
exp
(
π
(
θ
)
t
(
x
)
)
{\displaystyle f_{\theta }(x)=c(\theta )h(x)\exp(\pi (\theta )t(x))}
has monotone non-decreasing likelihood ratio in sufficient statistic t(x), provided
π
(
θ
)
{\displaystyle \pi (\theta )}
non-decreasing.
most powerful tests: karlin–rubin theorem
monotone likelihood functions used construct uniformly powerful tests, according karlin–rubin theorem. consider scalar measurement having probability density function parameterized scalar parameter θ, , define likelihood ratio
ℓ
(
x
)
=
f
θ
1
(
x
)
/
f
θ
0
(
x
)
{\displaystyle \ell (x)=f_{\theta _{1}}(x)/f_{\theta _{0}}(x)}
. if
ℓ
(
x
)
{\displaystyle \ell (x)}
monotone non-decreasing, in
x
{\displaystyle x}
, pair
θ
1
≥
θ
0
{\displaystyle \theta _{1}\geq \theta _{0}}
(meaning greater
x
{\displaystyle x}
is, more
h
1
{\displaystyle h_{1}}
is), threshold test:
φ
(
x
)
=
{
1
if
x
>
x
0
0
if
x
<
x
0
{\displaystyle \varphi (x)={\begin{cases}1&{\text{if }}x>x_{0}\\0&{\text{if }}x<x_{0}\end{cases}}}
where
x
0
{\displaystyle x_{0}}
chosen
e
θ
0
φ
(
x
)
=
α
{\displaystyle \operatorname {e} _{\theta _{0}}\varphi (x)=\alpha }
is ump test of size α testing
h
0
:
θ
≤
θ
0
vs.
h
1
:
θ
>
θ
0
.
{\displaystyle h_{0}:\theta \leq \theta _{0}{\text{ vs. }}h_{1}:\theta >\theta _{0}.}
note same test ump testing
h
0
:
θ
=
θ
0
vs.
h
1
:
θ
>
θ
0
.
{\displaystyle h_{0}:\theta =\theta _{0}{\text{ vs. }}h_{1}:\theta >\theta _{0}.}
median unbiased estimation
monotone likelihood-functions used construct median-unbiased estimators, using methods specified johann pfanzagl , others. 1 such procedure analogue of rao–blackwell procedure mean-unbiased estimators: procedure holds smaller class of probability distributions rao–blackwell procedure mean-unbiased estimation larger class of loss functions.
lifetime analysis: survival analysis , reliability
if family of distributions
f
θ
(
x
)
{\displaystyle f_{\theta }(x)}
has monotone likelihood ratio property in
t
(
x
)
{\displaystyle t(x)}
,
but not conversely: neither monotone hazard rates nor stochastic dominance imply mlrp.
proofs
let distribution family
f
θ
{\displaystyle f_{\theta }}
satisfy mlr in x,
θ
1
>
θ
0
{\displaystyle \theta _{1}>\theta _{0}}
,
x
1
>
x
0
{\displaystyle x_{1}>x_{0}}
:
f
θ
1
(
x
1
)
f
θ
0
(
x
1
)
≥
f
θ
1
(
x
0
)
f
θ
0
(
x
0
)
,
{\displaystyle {\frac {f_{\theta _{1}}(x_{1})}{f_{\theta _{0}}(x_{1})}}\geq {\frac {f_{\theta _{1}}(x_{0})}{f_{\theta _{0}}(x_{0})}},}
or equivalently:
f
θ
1
(
x
1
)
f
θ
0
(
x
0
)
≥
f
θ
1
(
x
0
)
f
θ
0
(
x
1
)
.
{\displaystyle f_{\theta _{1}}(x_{1})f_{\theta _{0}}(x_{0})\geq f_{\theta _{1}}(x_{0})f_{\theta _{0}}(x_{1}).\,}
integrating expression twice, obtain:
first-order stochastic dominance
combine 2 inequalities above first-order dominance:
f
θ
1
(
x
)
≤
f
θ
0
(
x
)
∀
x
{\displaystyle f_{\theta _{1}}(x)\leq f_{\theta _{0}}(x)\ \forall x}
monotone hazard rate
use second inequality above monotone hazard rate:
f
θ
1
(
x
)
1
−
f
θ
1
(
x
)
≤
f
θ
0
(
x
)
1
−
f
θ
0
(
x
)
∀
x
{\displaystyle {\frac {f_{\theta _{1}}(x)}{1-f_{\theta _{1}}(x)}}\leq {\frac {f_{\theta _{0}}(x)}{1-f_{\theta _{0}}(x)}}\ \forall x}
example
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