Template:Fisher Matrix Confidence Bounds ED

Fisher Matrix Bounds

Bounds on the Parameters

For the failure rate [math]\displaystyle{ \hat{\lambda } }[/math] the upper ([math]\displaystyle{ {{\lambda }_{U}} }[/math]) and lower ([math]\displaystyle{ {{\lambda }_{L}} }[/math]) bounds are estimated by [30]:

[math]\displaystyle{ \begin{align} & {{\lambda }_{U}}= & \hat{\lambda }\cdot {{e}^{\left[ \tfrac{{{K}_{\alpha }}\sqrt{Var(\hat{\lambda })}}{\hat{\lambda }} \right]}} \\ & & \\ & {{\lambda }_{L}}= & \frac{\hat{\lambda }}{{{e}^{\left[ \tfrac{{{K}_{\alpha }}\sqrt{Var(\hat{\lambda })}}{\hat{\lambda }} \right]}}} \end{align} }[/math]

where [math]\displaystyle{ {{K}_{\alpha }} }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }}) }[/math]

If [math]\displaystyle{ \delta }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2} }[/math] for the two-sided bounds, and [math]\displaystyle{ \alpha =1-\delta }[/math] for the one-sided bounds. The variance of [math]\displaystyle{ \hat{\lambda }, }[/math] [math]\displaystyle{ Var(\hat{\lambda }), }[/math] is estimated from the Fisher matrix, as follows:

[math]\displaystyle{ Var(\hat{\lambda })={{\left( -\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} \right)}^{-1}} }[/math]

where [math]\displaystyle{ \Lambda }[/math] is the log-likelihood function of the exponential distribution, described in Appendix C.

Note that no true MLE solution exists for the case of the two-parameter exponential distribution. The mathematics simply break down while trying to simultaneously solve the partial derivative equations for both the [math]\displaystyle{ \gamma }[/math] and [math]\displaystyle{ \lambda }[/math] parameters, resulting in unrealistic conditions. The way around this conundrum involves setting [math]\displaystyle{ \gamma ={{T}_{1}}, }[/math] or the first time-to-failure, and calculating [math]\displaystyle{ \lambda }[/math] in the regular fashion for this methodology. Weibull++ treats [math]\displaystyle{ \gamma }[/math] as a constant when computing bounds, i.e. [math]\displaystyle{ Var(\hat{\gamma })=0. }[/math] (See the discussion in Appendix C for more information.)

Bounds on Reliability

The reliability of the two-parameter exponential distribution is:

[math]\displaystyle{ \hat{R}(T;\hat{\lambda })={{e}^{-\hat{\lambda }(T-\hat{\gamma })}} }[/math]

The corresponding confidence bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{R}_{L}}= & {{e}^{-{{\lambda }_{U}}(T-\hat{\gamma })}} \\ & {{R}_{U}}= & {{e}^{-{{\lambda }_{L}}(T-\hat{\gamma })}} \end{align} }[/math]

These equations hold true for the one-parameter exponential distribution, with [math]\displaystyle{ \gamma =0 }[/math].

Bounds on Time

The bounds around time for a given exponential percentile, or reliability value, are estimated by first solving the reliability equation with respect to time, or reliable life:

[math]\displaystyle{ \hat{T}=-\frac{1}{{\hat{\lambda }}}\cdot \ln (R)+\hat{\gamma } }[/math]

The corresponding confidence bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & -\frac{1}{{{\lambda }_{L}}}\cdot \ln (R)+\hat{\gamma } \\ & {{T}_{L}}= & -\frac{1}{{{\lambda }_{U}}}\cdot \ln (R)+\hat{\gamma } \end{align} }[/math]

The same equations apply for the one-parameter exponential with [math]\displaystyle{ \gamma =0. }[/math]