Template:Eyring-ex cb on mean life

Confidence Bounds on Mean Life
The mean life for the Eyring relationship l is given by Eqn. (eyring) by setting $$m=L(V)$$. The upper $$({{m}_{U}})$$  and lower  $$({{m}_{L}})$$  bounds on the mean life (ML estimate of the mean life) are estimated by:


 * $${{m}_{U}}=\widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}$$


 * $${{m}_{L}}=\widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}$$

where $${{K}_{\alpha }}$$  is defined by:


 * $$\alpha =\frac{1}{\sqrt{2\pi }}\mathop{}_^{\infty }{{e}^{-\tfrac{2}}}dt=1-\Phi ({{K}_{\alpha }})$$

If $$\delta $$  is the confidence level, then  $$\alpha =\tfrac{1-\delta }{2}$$  for the two-sided bounds, and  $$\alpha =1-\delta $$  for the one-sided bounds. The variance of $$\widehat{m}$$  is given by:


 * $$\begin{align}

& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial A} \right)}^{2}}Var(\widehat{A})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial m}{\partial A} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{A},\widehat{B}) \end{align}$$


 * or:


 * $$Var(\widehat{m})=\frac{1}{{e}^{-2\left( \widehat{A}-\tfrac{\widehat{B}}{V} \right)}}\left[ Var(\widehat{A})+\frac{1}Var(\widehat{B})-\frac{1}{V}Cov(\widehat{A},\widehat{B}) \right]$$

The variances and covariance of $$A$$  and  $$B$$  are estimated from the local Fisher matrix (evaluated at  $$\widehat{A}$$,  $$\widehat{B})$$  as follows:


 * $$\left[ \begin{matrix}

Var(\widehat{A}) & Cov(\widehat{A},\widehat{B}) \\ Cov(\widehat{B},\widehat{A}) & Var(\widehat{B}) \\ \end{matrix} \right]={{\left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\ \end{matrix} \right]}^{-1}}$$