Template:Example: Lognormal General Example Suspension Data: Difference between revisions

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

Latest revision as of 07:57, 14 August 2012

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