Template:Bounds on beta camsaa-cb

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

Fisher Matrix Bounds

The parameter [math]\displaystyle{ \beta }[/math] must be positive, thus [math]\displaystyle{ \ln \beta }[/math] is treated as being normally distributed as well.

[math]\displaystyle{ \frac{\ln \hat{\beta }-\ln \beta }{\sqrt{Var(\ln \hat{\beta }})}\ \tilde{\ }\ N(0,1) }[/math]

The approximate confidence bounds are given as:

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

[math]\displaystyle{ \alpha }[/math] in [math]\displaystyle{ {{z}_{\alpha }} }[/math] is different ( [math]\displaystyle{ \alpha /2 }[/math] , [math]\displaystyle{ \alpha }[/math] ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher Matrix.

[math]\displaystyle{ \left[ \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} \\ \end{matrix} \right]_{\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1}=\left[ \begin{matrix} Var(\widehat{\lambda }) & Cov(\widehat{\beta },\widehat{\lambda }) \\ Cov(\widehat{\beta },\widehat{\lambda }) & Var(\widehat{\beta }) \\ \end{matrix} \right] }[/math]

[math]\displaystyle{ \Lambda }[/math] is the natural log-likelihood function:

[math]\displaystyle{ \Lambda =N\ln \lambda +N\ln \beta -\lambda {{T}^{\beta }}+(\beta -1)\underset{i=1}{\overset{N}{\mathop \sum }}\,\ln {{T}_{i}} }[/math]

[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}}=-\frac{N}{{{\lambda }^{2}}} }[/math]

and:

[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}}=-\frac{N}{{{\beta }^{2}}}-\lambda {{T}^{\beta }}{{(\ln T)}^{2}} }[/math]

also:

[math]\displaystyle{ \frac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \beta }=-{{T}^{\beta }}\ln T }[/math]

Crow Bounds

Time Terminated Data

For the 2-sided [math]\displaystyle{ (1-\alpha ) }[/math] 100-percent confidence interval on [math]\displaystyle{ \beta }[/math] , calculate:

[math]\displaystyle{ \begin{align} & {{D}_{L}}= & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2(N-1)} \\ & {{D}_{U}}= & \frac{\chi _{1-\tfrac{\alpha }{2},2N}^{2}}{2(N-1)} \end{align} }[/math]

The fractiles can be found in the tables of the [math]\displaystyle{ {{\chi }^{2}} }[/math] distribution. Thus the confidence bounds on [math]\displaystyle{ \beta }[/math] are:

[math]\displaystyle{ \begin{align} & {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\ & {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta } \end{align} }[/math]

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

[math]\displaystyle{ \begin{align} & {{D}_{L}}= & \frac{N\cdot \chi _{\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)} \\ & {{D}_{U}}= & \frac{N\cdot \chi _{1-\tfrac{\alpha }{2},2(N-1)}^{2}}{2(N-1)(N-2)} \end{align} }[/math]

Thus the confidence bounds on [math]\displaystyle{ \beta }[/math] are:

[math]\displaystyle{ \begin{align} & {{\beta }_{L}}= & {{D}_{L}}\cdot \hat{\beta } \\ & {{\beta }_{U}}= & {{D}_{U}}\cdot \hat{\beta } \end{align} }[/math]