Template:Lognormal distribution bayesian confidence bounds

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


 * $$\begin{align}

f({\mu }'|Data)= & \int_{0}^{\infty }f({\mu }',|Data)d \\ = & \frac{\int_{0}^{\infty }L(Data|{\mu }',)\varphi ({\mu }')\varphi d}{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|{\mu }',)\varphi ({\mu }')\varphi d{\mu }'d} \end{align}$$

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

$$\varphi ({\mu }')$$ 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:


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

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 $$$$.

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


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

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


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

The above equation can be rewritten in terms of $${\mu }'$$  as:


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

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


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

The above equation 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:


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

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


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

The above equation 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: