Template:Lognormal distribution bayesian confidence bounds: Difference between revisions

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:where:
:where:
::<math>\varphi ({{\sigma }_{{{T}'}}})</math>  is  <math>\tfrac{1}{{{\sigma }_{{{T}'}}}}</math> , non-informative prior of  <math>{{\sigma }_{{{T}'}}}</math> .
::<math>\varphi ({{\sigma }})</math>  is  <math>\tfrac{1}{{{\sigma }}}</math> , non-informative prior of  <math>{{\sigma }}</math> .
<math>\varphi ({\mu }')</math>  is an uniform distribution from - <math>\infty </math>  to + <math>\infty </math> , non-informative prior of  <math>{\mu }'</math> .
<math>\varphi ({\mu }')</math>  is an uniform distribution from - <math>\infty </math>  to + <math>\infty </math> , non-informative prior of  <math>{\mu }'</math> .
With the above prior distributions,  <math>f({\mu }'|Data)</math>  can be rewritten as:
With the above prior distributions,  <math>f({\mu }'|Data)</math>  can be rewritten as:




::<math>f({\mu }'|Data)=\frac{\int_{0}^{\infty }L(Data|{\mu }',{{\sigma }_{{{T}'}}})\tfrac{1}{{{\sigma }_{{{T}'}}}}d{{\sigma }_{{{T}'}}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|{\mu }',{{\sigma }_{{{T}'}}})\tfrac{1}{{{\sigma }_{{{T}'}}}}d{\mu }'d{{\sigma }_{{{T}'}}}}</math>
::<math>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>




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The same method can be used to obtained the bounds of  <math>{{\sigma }_{{{T}'}}}</math> .
The same method can be used to obtained the bounds of  <math>{{\sigma }}</math> .


====Bounds on Time (Type 1)====
====Bounds on Time (Type 1)====
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::<math>\ln T={\mu }'+{{\sigma }_{{{T}'}}}{{\Phi }^{-1}}(1-R)</math>
::<math>\ln T={\mu }'+{{\sigma }}{{\Phi }^{-1}}(1-R)</math>




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




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




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




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




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




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'''Example 8:'''
'''Example 8:'''
{{Example: Lognormal Distribution Bayesian Bound (Parameters)}}
{{Example: Lognormal Distr

Revision as of 23:31, 13 February 2012

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 ‘}}{{\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