Lifetime analysis: Survival analysis and reliability Monotone likelihood ratio

1 lifetime analysis: survival analysis , reliability

1.1 proofs
1.2 first-order stochastic dominance
1.3 monotone hazard rate

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}