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| ===Bayesian Confidence Bounds===
| | #REDIRECT [[The_Lognormal_Distribution#Bayesian_Confidence_Bounds]] |
| ====Bounds on Parameters====
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| From Chapter 5, 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 }_{{{T}'}}}|Data)d{{\sigma }_{{{T}'}}} \\
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| = & \frac{\int_{0}^{\infty }L(Data|{\mu }',{{\sigma }_{{{T}'}}})\varphi ({\mu }')\varphi ({{\sigma }_{{{T}'}}})d{{\sigma }_{{{T}'}}}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|{\mu }',{{\sigma }_{{{T}'}}})\varphi ({\mu }')\varphi ({{\sigma }_{{{T}'}}})d{\mu }'d{{\sigma }_{{{T}'}}}}
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| \end{align}</math>
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| :where:
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| ::<math>\varphi ({{\sigma }_{{{T}'}}})</math> is <math>\tfrac{1}{{{\sigma }_{{{T}'}}}}</math> , non-informative prior of <math>{{\sigma }_{{{T}'}}}</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 }_{{{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>
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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 }_{{{T}'}}}</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 }_{{{T}'}}}{{\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 }_{{{T}'}}}{{\Phi }^{-1}}(1-R)\le \ln {{T}_{U}})</math>
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| Eqn. (1SBT) 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 }_{{{T}'}}}{{\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 }_{{{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>
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| Eqn. (1SCBT) 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 }_{{{T}'}}}{{\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 }_{{{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>
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| Eqn. (1SCBR) 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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