Template:Bounds on lambda rsa

Fisher Matrix Bounds
The parameter $$\lambda $$  must be positive, thus  $$\ln \lambda $$  is approximately treated as being normally distributed. These bounds are based on:


 * $$\frac{\ln (\widehat{\lambda })-\ln (\lambda )}{\sqrt{Var\left[ \ln (\widehat{\lambda }) \right]}}\ \tilde{\ }\ N(0,1)$$

The approximate confidence bounds on $$\lambda $$  are given as:


 * $$C{{B}_{\lambda }}=\widehat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{\lambda })}/\widehat{\lambda }}}$$

where $$\widehat{\lambda }=\tfrac{n}{T_{K}^}$$. The variance calculation is the same as Eqns. (var1), (var2) and (var3).

Crow Bounds
Time Terminated The confidence bounds on $$\lambda $$   for time terminated data are calculated using:


 * $$\begin{align}

& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\ & {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \end{align}$$

Failure Terminated The confidence bounds on $$\lambda $$  for failure terminated data are calculated using:


 * $$\begin{align}

& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \\ & {{\lambda }_{u}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot \underset{q=1}{\overset{K}{\mathop{\sum }}}\,T_{q}^{^{\beta }}} \end{align}$$