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| ===Bayesian Confidence Bounds===
| | #REDIRECT [[The_Lognormal_Distribution#Bayesian_Confidence_Bounds]] |
| ====Bounds on Parameters====
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| From Chapter [[Parameter Estimation]], we know that the marginal distribution of parameter <math>{\mu }'</math> is:
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| ::<math>\begin{align}
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| f({\mu }'|Data)= & \int_{0}^{\infty }f({\mu }',{{\sigma'}}|Data)d{{\sigma'}} \\
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| = & \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'}}}
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| \end{align}</math>
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| where:
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| ::<math>\varphi ({{\sigma '}})</math> is <math>\tfrac{1}{{{\sigma '}}}</math> , non-informative prior of <math>{{\sigma '}}</math> .
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| <math>\varphi ({\mu }')</math> is an uniform distribution from - <math>\infty </math> to + <math>\infty </math> , non-informative prior of <math>{\mu }'</math> .
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| With the above prior distributions, <math>f({\mu }'|Data)</math> can be rewritten as:
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| ::<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 one-sided upper bound of <math>{\mu }'</math> is:
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| ::<math>CL=P({\mu }'\le \mu _{U}^{\prime })=\int_{-\infty }^{\mu _{U}^{\prime }}f({\mu }'|Data)d{\mu }'</math>
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| The one-sided lower bound of <math>{\mu }'</math> is:
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| ::<math>1-CL=P({\mu }'\le \mu _{L}^{\prime })=\int_{-\infty }^{\mu _{L}^{\prime }}f({\mu }'|Data)d{\mu }'</math>
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| The two-sided bounds of <math>{\mu }'</math> is:
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| ::<math>CL=P(\mu _{L}^{\prime }\le {\mu }'\le \mu _{U}^{\prime })=\int_{\mu _{L}^{\prime }}^{\mu _{U}^{\prime }}f({\mu }'|Data)d{\mu }'</math>
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| The same method can be used to obtained the bounds of <math>{{\sigma '}}</math> .
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| ====Bounds on Time (Type 1)====
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| The reliable life of the lognormal distribution is:
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| ::<math>\ln T={\mu }'+{{\sigma '}}{{\Phi }^{-1}}(1-R)</math>
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| The one-sided upper on time bound is given by:
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| ::<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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| The above equation can be rewritten in terms of <math>{\mu }'</math> as:
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| ::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{t}_{U}}-{{\sigma '}}{{\Phi }^{-1}}(1-R)</math>
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| From the posterior distribution of <math>{\mu }'</math> get:
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| ::<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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| The above equation is solved w.r.t. <math>{{t}_{U}}.</math> The same method can be applied for one-sided lower bounds and two-sided bounds on Time.
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| ====Bounds on Reliability (Type 2)====
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| The one-sided upper bound on reliability is given by:
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| ::<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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| From the posterior distribution of <math>{\mu }'</math> is:
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| ::<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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| The above equation is solved w.r.t. <math>{{R}_{U}}.</math> The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.
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| '''Example 8:'''
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| {{Example: Lognormal Distribution Bayesian Bound (Parameters)}}
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