Template:Acb on the mean life

Confidence Bounds on the Mean Life
The Arrhenius-exponential distribution is given by Eqn. (arrhenius) by setting $$m=L(V)$$  as shown in Eqn. (Arrean). The upper $$({{m}_{U}})$$  and lower  $$({{m}_{L}})$$  bounds on the mean life are then estimated by:


 * $$\begin{align}

& {{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}}}} \end{align}$$

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 (i.e., 95%=0.95), 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 C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) \\ & & +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C}) \end{align}$$

or:


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

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


 * $$\left[ \begin{matrix}

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