Template:Bounds on lambda camsaa-cb

Fisher Matrix Bounds
The parameter $$\lambda $$  must be positive, thus  $$\ln \lambda $$  is treated as being normally distributed as well. These bounds are based on:
 * $$\frac{\ln \hat{\lambda }-\ln \lambda }{\sqrt{Var(\ln \hat{\lambda }})}\ \tilde{\ }\ N(0,1)$$

The approximate confidence bounds on $$\lambda $$  are given as:
 * $$C{{B}_{\lambda }}=\hat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}}$$


 * where:


 * $$\hat{\lambda }=\frac{n}$$

The variance calculation is the same as Eqn. (variance1).

Crow Bounds
Time Terminated Data For the 2-sided $$(1-\alpha )$$ 100-percent confidence interval, the confidence bounds on  $$\lambda $$  are:


 * $$\begin{align}

& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^}} \\ & {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2{{T}^}} \end{align}$$

The fractiles can be found in the tables of the $${{\chi }^{2}}$$  distribution. Failure Terminated Data For the 2-sided $$(1-\alpha )$$ 100-percent confidence interval, the confidence bounds on  $$\lambda $$  are:


 * $$\begin{align}

& {{\lambda }_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2{{T}^}} \\ & {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2{{T}^}} \end{align}$$