Template:Bounds on instantaneous mtbf camsaa-gd

Fisher Matrix Bounds
The instantaneous MTBF, $${{m}_{i}}(t)$$, must be positive, thus  $$\ln {{m}_{i}}(t)$$  is approximately treated as being normally distributed as well.


 * $$\frac{\ln {{{\hat{m}}}_{i}}(t)-\ln {{m}_{i}}(t)}{\sqrt{Var(\ln {{{\hat{m}}}_{i}}(t)})}\ \tilde{\ }\ N(0,1)$$

The approximate confidence bounds on the instantaneous MTBF are then estimated from:


 * $$CB={{\hat{m}}_{i}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}}$$


 * where:


 * $${{\hat{m}}_{i}}(t)=\frac{1}{\lambda \beta {{t}^{\beta -1}}}$$


 * $$\begin{align}

& Var({{{\hat{m}}}_{i}}(t))= & {{\left( \frac{\partial {{m}_{i}}(t)}{\partial \beta } \right)}^{2}}Var(\hat{\beta })+{{\left( \frac{\partial {{m}_{i}}(t)}{\partial \lambda } \right)}^{2}}Var(\hat{\lambda }) \\ & & +2\left( \frac{\partial {{m}_{i}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{i}}(t)}{\partial \lambda } \right)cov(\hat{\beta },\,\,\,\hat{\lambda }) \end{align}$$

The variance calculation is the same as Eqn. (variances) and:


 * $$\begin{align}

& \frac{\partial {{m}_{i}}(t)}{\partial \beta }= & -\frac{1}{\hat{\lambda }{{{\hat{\beta }}}^{2}}}{{t}^{1-\hat{\beta }}}-\frac{1}{\hat{\lambda }\hat{\beta }}{{t}^{1-\hat{\beta }}}\ln t \\ & \frac{\partial {{m}_{i}}(t)}{\partial \lambda }= & -\frac{1}{{{{\hat{\lambda }}}^{2}}\hat{\beta }}{{t}^{1-\hat{\beta }}} \end{align}$$

Crow Bounds

 * Step 1: Calculate $$P(i)=\tfrac,\,\,i=1,2,\ldots ,K$$.
 * Step 2: Calculate:


 * $$A=\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac{\left[ P{{(i)}^}-P{{(i-1)}^} \right]}$$


 * Step 3: Calculate $$D=\sqrt{\tfrac{1}{A}+1}$$  and  $$W=\tfrac{({{z}_{1-\alpha /2}})\cdot D}{\sqrt{N}}$$ . Thus an approximate 2-sided  $$(1-\alpha )$$ 100-percent confidence interval on  $${{\hat{m}}_{i}}(t)$$  is:


 * $$MTB{{F}_{i}}={{\widehat{m}}_{i}}(1\pm W)$$