Template:Lognormal distribution bayesian confidence bounds

Bayesian Confidence Bounds

Bounds on Parameters

From Chapter Parameter Estimation, we know that the marginal distribution of parameter [math]\displaystyle{ {\mu }' }[/math] is:

[math]\displaystyle{ \begin{align} f({\mu }'|Data)= & \int_{0}^{\infty }f({\mu }',{{\sigma'}}|Data)d{{\sigma'}} \\ = & \frac{\int_{0}^{\infty }L(Data|{\mu }',{{\sigma'}})\varphi ({\mu }')\varphi ({{\sigma'}})d{{\sigma'}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|{\mu }',{{\sigma'}})\varphi ({\mu }')\varphi ({{\sigma'}})d{\mu }'d{{\sigma'}}} \end{align} }[/math]

where:

[math]\displaystyle{ \varphi ({{\sigma '}}) }[/math] is [math]\displaystyle{ \tfrac{1}{{{\sigma '}}} }[/math] , non-informative prior of [math]\displaystyle{ {{\sigma '}} }[/math] .

[math]\displaystyle{ \varphi ({\mu }') }[/math] is an uniform distribution from - [math]\displaystyle{ \infty }[/math] to + [math]\displaystyle{ \infty }[/math] , non-informative prior of [math]\displaystyle{ {\mu }' }[/math] . With the above prior distributions, [math]\displaystyle{ f({\mu }'|Data) }[/math] can be rewritten as:

[math]\displaystyle{ f({\mu }'|Data)=\frac{\int_{0}^{\infty }L(Data|{\mu }',{{\sigma '}})\tfrac{1}{{{\sigma '}}}d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|{\mu }',{{\sigma '}})\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}} }[/math]

The one-sided upper bound of [math]\displaystyle{ {\mu }' }[/math] is:

[math]\displaystyle{ CL=P({\mu }'\le \mu _{U}^{\prime })=\int_{-\infty }^{\mu _{U}^{\prime }}f({\mu }'|Data)d{\mu }' }[/math]

The one-sided lower bound of [math]\displaystyle{ {\mu }' }[/math] is:

[math]\displaystyle{ 1-CL=P({\mu }'\le \mu _{L}^{\prime })=\int_{-\infty }^{\mu _{L}^{\prime }}f({\mu }'|Data)d{\mu }' }[/math]

The two-sided bounds of [math]\displaystyle{ {\mu }' }[/math] is:

[math]\displaystyle{ CL=P(\mu _{L}^{\prime }\le {\mu }'\le \mu _{U}^{\prime })=\int_{\mu _{L}^{\prime }}^{\mu _{U}^{\prime }}f({\mu }'|Data)d{\mu }' }[/math]

The same method can be used to obtained the bounds of [math]\displaystyle{ {{\sigma '}} }[/math] .

Bounds on Time (Type 1)

The reliable life of the lognormal distribution is:

[math]\displaystyle{ \ln T={\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R) }[/math]

The one-sided upper on time bound is given by:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln T\le \ln {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R)\le \ln {{T}_{U}}) }[/math]

Eqn. (1SBT) can be rewritten in terms of [math]\displaystyle{ {\mu }' }[/math] as:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{T}_{U}}-{{\sigma '}}{{\Phi }^{-1}}(1-R) }[/math]

From the posterior distribution of [math]\displaystyle{ {\mu }' }[/math] get:

[math]\displaystyle{ CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{T}_{U}}-{{\sigma ‘}}{{\Phi }^{-1}}(1-R)}L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma '}},{\mu }')\tfrac{1}{{{\sigma '}}}d{\mu }'d{{\sigma '}}} }[/math]

Eqn. (1SCBT) is solved w.r.t. [math]\displaystyle{ {{T}_{U}}. }[/math] The same method can be applied for one-sided lower bounds and two-sided bounds on Time.

Bounds on Reliability (Type 2)

The one-sided upper bound on reliability is given by:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln T-{{\sigma '}}{{\Phi }^{-1}}(1-{{R}_{U}})) }[/math]

From the posterior distribution of [math]\displaystyle{ {\mu }' }[/math] is:

[math]\displaystyle{ CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma 1}}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma ‘}},{\mu }')\tfrac{1}{{{\sigma ‘}}}d{\mu }'d{{\sigma ‘}}} }[/math]

Eqn. (1SCBR) is solved w.r.t. [math]\displaystyle{ {{R}_{U}}. }[/math] The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.

Example 8: {{Example: Lognormal Distr