Template:Lognormal distribution bayesian confidence bounds

Bounds on Parameters
From Chapter 5, we know that the marginal distribution of parameter $${\mu }'$$  is:


 * $$\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}$$


 * where:
 * $$\varphi ({{\sigma }_})$$ is  $$\tfrac{1}$$, non-informative prior of  $${{\sigma }_}$$.

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


 * $$f({\mu }'|Data)=\frac{\int_{0}^{\infty }L(Data|{\mu }',{{\sigma }_})\tfrac{1}d{{\sigma }_}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|{\mu }',{{\sigma }_})\tfrac{1}d{\mu }'d{{\sigma }_}}$$

The one-sided upper bound of  $${\mu }'$$  is:


 * $$CL=P({\mu }'\le \mu _{U}^{\prime })=\int_{-\infty }^{\mu _{U}^{\prime }}f({\mu }'|Data)d{\mu }'$$

The one-sided lower bound of $${\mu }'$$  is:


 * $$1-CL=P({\mu }'\le \mu _{L}^{\prime })=\int_{-\infty }^{\mu _{L}^{\prime }}f({\mu }'|Data)d{\mu }'$$

The two-sided bounds of $${\mu }'$$  is:


 * $$CL=P(\mu _{L}^{\prime }\le {\mu }'\le \mu _{U}^{\prime })=\int_{\mu _{L}^{\prime }}^{\mu _{U}^{\prime }}f({\mu }'|Data)d{\mu }'$$

The same method can be used to obtained the bounds of $${{\sigma }_}$$.

Bounds on Time (Type 1)
The reliable life of the lognormal distribution is:


 * $$\ln T={\mu }'+{{\sigma }_}{{\Phi }^{-1}}(1-R)$$

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


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\ln T\le \ln {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'+{{\sigma }_}{{\Phi }^{-1}}(1-R)\le \ln {{T}_{U}})$$

Eqn. (1SBT) can be rewritten in terms of $${\mu }'$$  as:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln {{T}_{U}}-{{\sigma }_}{{\Phi }^{-1}}(1-R)$$

From the posterior distribution of $${\mu }'$$  get:


 * $$CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln {{T}_{U}}-{{\sigma }_}{{\Phi }^{-1}}(1-R)}L({{\sigma }_},{\mu }')\tfrac{1}d{\mu }'d{{\sigma }_}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma }_},{\mu }')\tfrac{1}d{\mu }'d{{\sigma }_}}$$

Eqn. (1SCBT) is solved w.r.t. $${{T}_{U}}.$$  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:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,({\mu }'\le \ln T-{{\sigma }_}{{\Phi }^{-1}}(1-{{R}_{U}}))$$

From the posterior distribution of $${\mu }'$$  is:


 * $$CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{\ln T-{{\sigma }_}{{\Phi }^{-1}}(1-{{R}_{U}})}L({{\sigma }_},{\mu }')\tfrac{1}d{\mu }'d{{\sigma }_}}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L({{\sigma }_},{\mu }')\tfrac{1}d{\mu }'d{{\sigma }_}}$$

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

Example 8: