Template:Bounds on lambda camsaa-gd

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{\ }\

$$\hat{\beta }(1\pm S)$$
 * $$N(0,1)$$

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:


 * $$\hat{\lambda }=\frac{n}{T_{k}^}$$

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

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\cdot T_{k}^{\beta }} \\ & {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot T_{k}^{\beta }} \end{align}$$

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\cdot T_{k}^{\beta }} \\ & {{\lambda }_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \end{align}$$