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