Template:Bounds on lambda camsaa-gd

Bounds on [math]\displaystyle{ \lambda }[/math]

Fisher Matrix Bounds

The parameter [math]\displaystyle{ \lambda }[/math] must be positive, thus [math]\displaystyle{ \ln \lambda }[/math] is treated as being normally distributed as well. These bounds are based on:

[math]\displaystyle{ \frac{\ln \hat{\lambda }-\ln \lambda }{\sqrt{Var(\ln \hat{\lambda }})}\ \tilde{\ }\ \lt math\gt \hat{\beta }(1\pm S) }[/math]

[math]\displaystyle{ N(0,1) }[/math]

The approximate confidence bounds on [math]\displaystyle{ \lambda }[/math] are given as:

[math]\displaystyle{ C{{B}_{\lambda }}=\hat{\lambda }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} }[/math]

where:

[math]\displaystyle{ \hat{\lambda }=\frac{n}{T_{k}^{{\hat{\beta }}}} }[/math]

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

Crow Bounds

Time Terminated Data
For the 2-sided [math]\displaystyle{ (1-\alpha ) }[/math] 100-percent confidence interval, the confidence bounds on [math]\displaystyle{ \lambda }[/math] are:

[math]\displaystyle{ \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} }[/math]

Failure Terminated Data
For the 2-sided [math]\displaystyle{ (1-\alpha ) }[/math] 100-percent confidence interval, the confidence bounds on [math]\displaystyle{ \lambda }[/math] are:

[math]\displaystyle{ \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} }[/math]