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| '''Lognormal Distribution General Example Suspension Data'''
| | #REDIRECT [[The Lognormal Distribution]] |
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| From [[Appendix: Weibull References|Nelson [30, p. 324]]]. Ninety-six locomotive controls were tested, 37 failed and 59 were suspended after running for 135,000 miles. Table 9.6 below shows their times-to-failure.
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| '''Solution'''
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| The distribution used in the publication was the base-10 lognormal. | |
| Published results (using MLE):
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=2.2223 \\
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| {{\widehat{\sigma' }}}=0.3064 \\
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| \end{matrix}</math>
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| Published 95% confidence limits on the parameters:
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\
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| {{\widehat{\sigma'}}}=\left\{ 0.2365,0.3970 \right\} \\
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| \end{matrix}</math>
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| Published variance/covariance matrix:
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| ::<math>\left[ \begin{matrix}
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| \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0020 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 \\
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| {} & {} & {} \\
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| \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma '}}}} \right)=0.0016 \\
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| \end{matrix} \right]</math>
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| To replicate the published results (since Weibull++ uses a lognormal to the base <math>e</math> ), take the base-10 logarithm of the data and estimate the parameters using the Normal distribution and MLE.
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| • Weibull++ computed parameters for maximum likelihood are:
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=2.2223 \\
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| {{\widehat{\sigma' }}}=0.3064 \\
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| \end{matrix}</math>
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| • Weibull++ computed 95% confidence limits on the parameters:
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| ::<math>\begin{matrix}
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| {{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\
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| {{\widehat{\sigma'}}}=\left\{ 0.2395,0.3920 \right\} \\
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| \end{matrix}</math>
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| • Weibull++ computed/variance covariance matrix:
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| ::<math>\left[ \begin{matrix}
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| \widehat{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0019 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.0009 \\
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| {} & {} & {} \\
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| \widehat{Cov}({\mu }',{{{\hat{\sigma' }}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma' }}}} \right)=0.0015 \\
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| \end{matrix} \right]</math>
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| {|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
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| |-
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| |colspan="4" style="text-align:center"|Table - Nelson's Locomotive Data
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| |-
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| !
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| !Number in State
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| !F or S
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| !Time
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| |-
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| |1||1||F||22.5
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| |-
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| |2||1||F||37.5
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| |-
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| |3||1||F||46
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| |-
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| |4||1||F||48.5
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| |-
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| |5||1||F||51.5
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| |-
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| |6||1||F||53
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| |-
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| |7||1||F||54.5
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| |-
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| |8||1||F||57.5
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| |-
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| |9||1||F||66.5
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| |-
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| |10||1||F||68
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| |-
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| |11||1||F||69.5
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| |-
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| |12||1||F||76.5
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| |-
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| |13||1||F||77
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| |-
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| |14||1||F||78.5
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| |-
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| |15||1||F||80
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| |-
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| |16||1||F||81.5
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| |-
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| |17||1||F||82
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| |-
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| |18||1||F||83
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| |-
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| |19||1||F||84
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| |-
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| |20||1||F||91.5
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| |-
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| |21||1||F||93.5
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| |-
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| |22||1||F||102.5
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| |-
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| |23||1||F||107
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| |-
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| |24||1||F||108.5
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| |-
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| |25||1||F||112.5
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| |-
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| |26||1||F||113.5
| |
| |-
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| |27||1||F||116
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| |-
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| |28||1||F||117
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| |-
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| |29||1||F||118.5
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| |-
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| |30||1||F||119
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| |-
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| |31||1||F||120
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| |-
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| |32||1||F||122.5
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| |-
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| |33||1||F||123
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| |-
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| |34||1||F||127.5
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| |-
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| |35||1||F||131
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| |-
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| |36||1||F||132.5
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| |-
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| |37||1||F||134
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| |-
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| |38||59||S||135
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| |}
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