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#REDIRECT [[Lognormal Distribution Examples]]
==General Examples==
 
'''Example 9:'''
{{Example: Lognormal General Example Interval Data}}
 
'''Example 10:'''
{{Example: Lognormal General Example Complete Data}}
 
 
 
===Example 11===
From Kececioglu [19, p. 406]. Nine identical units are tested continuously to failure and their times-to-failure were recorded at 30.4, 36.7, 53.3, 58.5, 74.0, 99.3, 114.3, 140.1, and 257.9 hours.
====Solution to Example 11====
The results published were obtained by using the unbiased model.
Published Results (using MLE):
 
::<math>\begin{matrix}
  {{\widehat{\mu }}^{\prime }}=4.3553  \\
  {{\widehat{\sigma }}_{{{T}'}}}=0.67677  \\
\end{matrix}</math>
 
 
This same data set can be entered into Weibull++ by creating a data sheet capable of handling non-grouped time-to-failure data. Since the results shown above are unbiased, the Use Unbiased Std on Normal Data option in the User Setup must be selected in order to duplicate these results.
Weibull++ computed parameters for maximum likelihood are:
 
 
::<math>\begin{matrix}
  {{\widehat{\mu }}^{\prime }}=4.3553  \\
  {{\widehat{\sigma }}_{{{T}'}}}=0.6768  \\
\end{matrix}</math>
 
===Example 12===
From Kececioglu [20, p. 347]. Fifteen identical units were tested to failure and following is a table of their times-to-failure:
 
 
<center><math>\text{Table 9}\text{.5 - Data of Example 11}</math></center>
 
<center><math>\begin{matrix}
  \text{Data Point Index} & \text{Time-to-Failure, hr}  \\
  \text{1} & \text{62}\text{.5}  \\
  \text{2} & \text{91}\text{.9}  \\
  \text{3} & \text{100}\text{.3}  \\
  \text{4} & \text{117}\text{.4}  \\
  \text{5} & \text{141}\text{.1}  \\
  \text{6} & \text{146}\text{.8}  \\
  \text{7} & \text{172}\text{.7}  \\
  \text{8} & \text{192}\text{.5}  \\
  \text{9} & \text{201}\text{.6}  \\
  \text{10} & \text{235}\text{.8}  \\
  \text{11} & \text{249}\text{.2}  \\
  \text{12} & \text{297}\text{.5}  \\
  \text{13} & \text{318}\text{.3}  \\
  \text{14} & \text{410}\text{.6}  \\
  \text{15} & \text{550}\text{.5}  \\
\end{matrix}</math></center>
 
 
====Solution to Example 12====
Published results (using probability plotting):
 
::<math>\begin{matrix}
  {{\widehat{\mu }}^{\prime }}=5.22575  \\
  {{\widehat{\sigma }}_{{{T}'}}}=0.62048.  \\
\end{matrix}</math>
 
 
Weibull++ computed parameters for rank regression on X are:
 
 
::<math>\begin{matrix}
  {{\widehat{\mu }}^{\prime }}=5.2303  \\
  {{\widehat{\sigma }}_{{{T}'}}}=0.6283.  \\
\end{matrix}</math>
 
 
The small differences are due to the precision errors when fitting a line manually, whereas in Weibull++ the line was fitted mathematically.
 
===Example 13===
From 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 (at the end of this chapter) shows their times-to-failure.
====Solution to Example 13====
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 }}_{{{T}'}}}=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 }}_{{{T}'}}}=\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 }}}_{{{T}'}}})=0.001  \\
  {} & {} & {}  \\
  \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma }}}_{{{T}'}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma }}}_{{{T}'}}} \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 }}_{{{T}'}}}=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 }}_{{{T}'}}}=\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 }}}_{{{T}'}}})=0.0009  \\
  {} & {} & {}  \\
  \widehat{Cov}({\mu }',{{{\hat{\sigma }}}_{{{T}'}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma }}}_{{{T}'}}} \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 07:32, 14 August 2012

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