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| ====Bounds on the Parameters====
| | #REDIRECT [[The Exponential Distribution]] |
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| For the failure rate <math>\hat{\lambda }</math> the upper (<math>{{\lambda }_{U}}</math>) and lower (<math>{{\lambda }_{L}}</math>) bounds are estimated by [30]:
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| ::<math>\begin{align}
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| & {{\lambda }_{U}}= & \hat{\lambda }\cdot {{e}^{\left[ \tfrac{{{K}_{\alpha }}\sqrt{Var(\hat{\lambda })}}{\hat{\lambda }} \right]}} \\
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| & & \\
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| & {{\lambda }_{L}}= & \frac{\hat{\lambda }}{{{e}^{\left[ \tfrac{{{K}_{\alpha }}\sqrt{Var(\hat{\lambda })}}{\hat{\lambda }} \right]}}}
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| \end{align}</math>
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| where <math>{{K}_{\alpha }}</math> is defined by:
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| ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
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| If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds.
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| The variance of <math>\hat{\lambda },</math> <math>Var(\hat{\lambda }),</math> is estimated from the Fisher matrix, as follows:
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| ::<math>Var(\hat{\lambda })={{\left( -\frac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} \right)}^{-1}}</math>
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| where <math>\Lambda </math> is the log-likelihood function of the exponential distribution, described in Appendix C.
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| 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>\gamma </math> and <math>\lambda </math> parameters, resulting in unrealistic conditions. The way around this conundrum involves setting <math>\gamma ={{T}_{1}},</math> or the first time-to-failure, and calculating <math>\lambda </math> in the regular fashion for this methodology. Weibull++ treats <math>\gamma </math> as a constant when computing bounds, i.e. <math>Var(\hat{\gamma })=0.</math> (See the discussion in Appendix C for more information.)
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