Template:Bounds on Parameters.LRCB.FMB.ED: Difference between revisions

Latest revision as of 06:54, 10 August 2012

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-====Bounds on Parameters====
+#REDIRECT [[The Exponential Distribution]]
-For one-parameter distributions such as the exponential, the likelihood confidence bounds are calculated by finding values for <math>\theta </math> that satisfy:
-::<math>-2\cdot \text{ln}\left( \frac{L(\theta )}{L(\hat{\theta })} \right)=\chi _{\alpha ;1}^{2}</math>
-This equation can be rewritten as:
-::<math>L(\theta )=L(\hat{\theta })\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}</math>
-For complete data, the likelihood function for the exponential distribution is given by:
-::<math>L(\lambda )=\underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};\lambda )=\underset{i=1}{\overset{N}{\mathop \prod }}\,\lambda \cdot {{e}^{-\lambda \cdot {{x}_{i}}}}</math>
-where the <math>{{x}_{i}}</math> values represent the original time-to-failure data.  For a given value of <math>\alpha </math>, values for <math>\lambda </math> can be found which represent the maximum and minimum values that satisfy Eqn. (lratio3). These represent the confidence bounds for the parameters at a confidence level <math>\delta ,</math> where <math>\alpha =\delta </math> for two-sided bounds and <math>\alpha =2\delta -1</math> for one-sided.