Template:Example: Lognormal Distribution MLE: Difference between revisions
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& \frac{\partial \Lambda }{\partial {{\sigma'}}}= & \underset{i=1}{\overset{14}{\mathop \sum }}\,\left( \frac{\ln ({{t}_{i}})-{\mu }'}{\sigma'^{3}}-\frac{1}{{{\sigma'}}} \right)=0 | & \frac{\partial \Lambda }{\partial {{\sigma'}}}= & \underset{i=1}{\overset{14}{\mathop \sum }}\,\left( \frac{\ln ({{t}_{i}})-{\mu }'}{\sigma'^{3}}-\frac{1}{{{\sigma'}}} \right)=0 | ||
\end{align}</math> | \end{align}</math> | ||
Substituting the values of <math>{{T}_{i}}</math> and solving the above system simultaneously, we get: | Substituting the values of <math>{{T}_{i}}</math> and solving the above system simultaneously, we get: | ||
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& {{{\hat{\mu }}}^{\prime }}= & 3.516 | & {{{\hat{\mu }}}^{\prime }}= & 3.516 | ||
\end{align}</math> | \end{align}</math> | ||
Using the equation for mean and standard deviation in section [[Lognormal Statistical Properties]] we get: | Using the equation for mean and standard deviation in section [[Lognormal Statistical Properties]] we get: | ||
::<math>\overline{T}=\hat{\mu }=48.25\text{ hours}</math> | ::<math>\overline{T}=\hat{\mu }=48.25\text{ hours}</math> | ||
and: | and: | ||
Revision as of 05:01, 8 August 2012
Lognormal Distribution MLE Example
Using the data of Example 2 and assuming a lognormal distribution, estimate the parameters using the MLE method.
Solution In this example we have only complete data. Thus, the partials reduce to:
- [math]\displaystyle{ \begin{align} & \frac{\partial \Lambda }{\partial {\mu }'}= & \frac{1}{\sigma'^{2}}\cdot \underset{i=1}{\overset{14}{\mathop \sum }}\,\ln ({{t}_{i}})-{\mu }'=0 \\ & \frac{\partial \Lambda }{\partial {{\sigma'}}}= & \underset{i=1}{\overset{14}{\mathop \sum }}\,\left( \frac{\ln ({{t}_{i}})-{\mu }'}{\sigma'^{3}}-\frac{1}{{{\sigma'}}} \right)=0 \end{align} }[/math]
Substituting the values of [math]\displaystyle{ {{T}_{i}} }[/math] and solving the above system simultaneously, we get:
- [math]\displaystyle{ \begin{align} & {{{\hat{\sigma' }}}}= & 0.849 \\ & {{{\hat{\mu }}}^{\prime }}= & 3.516 \end{align} }[/math]
Using the equation for mean and standard deviation in section Lognormal Statistical Properties we get:
- [math]\displaystyle{ \overline{T}=\hat{\mu }=48.25\text{ hours} }[/math]
and:
- [math]\displaystyle{ {{\hat{\sigma }}}=49.61\text{ hours}. }[/math]
The variance/covariance matrix is given by:
- [math]\displaystyle{ \left[ \begin{matrix} \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0515 & {} & \widehat{Cov}\left( {{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma'}}}} \right)=0.0000 \\ {} & {} & {} \\ \widehat{Cov}\left( {{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}} \right)=0.0000 & {} & \widehat{Var}\left( {{{\hat{\sigma' }}}} \right)=0.0258 \\ \end{matrix} \right] }[/math]