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| ====Confidence Bounds on Time====
| | #REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Exponential]] |
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| The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:
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| ::<math>\widehat{T}=-\widehat{m}\cdot \ln (R)</math>
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| The corresponding confidence bounds are then estimated from:
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| ::<math>\begin{align}
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| & {{T}_{U}}= -{{m}_{U}}\cdot \ln (R) \\
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| & {{T}_{L}}= -{{m}_{L}}\cdot \ln (R)
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| \end{align}</math>
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| where <math>{{m}_{U}}</math> and <math>{{m}_{L}}</math> are estimated estimated by:
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| ::<math>\begin{align}
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| & {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
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| & {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
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| \end{align}</math>
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| <br>
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