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

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'''Lognormal Distribution General Example Suspension Data'''
'''Lognormal Distribution General Example Suspension Data'''


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 (at the end of this chapter) shows their times-to-failure.
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'''
'''Solution'''

Revision as of 23:51, 13 February 2012

Lognormal Distribution General Example Suspension Data

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 below shows their times-to-failure.

Solution

The distribution used in the publication was the base-10 lognormal. Published results (using MLE):

[math]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix} }[/math]


Published 95% confidence limits on the parameters:


[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \begin{matrix} {{\widehat{\mu }}^{\prime }}=2.2223 \\ {{\widehat{\sigma' }}}=0.3064 \\ \end{matrix} }[/math]

• Weibull++ computed 95% confidence limits on the parameters:


[math]\displaystyle{ \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]\displaystyle{ \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]


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