Template:Example: Lognormal General Example Suspension Data - Revision history

Chris Kahn: Redirected page to The Lognormal Distribution

2012-08-14T07:57:03Z

Redirected page to The Lognormal Distribution

← Older revision		Revision as of 07:57, 14 August 2012
Line 1:		Line 1:
	~~'''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>~~

	~~{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"~~
	\|-
	~~\|colspan="4" style="text-align:center"\|Table - 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~~
	\|}

Richard House at 05:10, 8 August 2012

2012-08-08T05:10:19Z

← Older revision		Revision as of 05:10, 8 August 2012
Line 30:		Line 30:
	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.		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:		*Weibull++ computed parameters for maximum likelihood are:

	::<math>\begin{matrix}		::<math>\begin{matrix}
Line 37:		Line 37:
	\end{matrix}</math>		\end{matrix}</math>

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

	::<math>\begin{matrix}		::<math>\begin{matrix}
Line 44:		Line 44:
	\end{matrix}</math>		\end{matrix}</math>

	• Weibull++ computed/variance covariance matrix:		*Weibull++ computed/variance covariance matrix:

	::<math>\left[ \begin{matrix}		::<math>\left[ \begin{matrix}

Richard House at 05:09, 8 August 2012

2012-08-08T05:09:40Z

← Older revision		Revision as of 05:09, 8 August 2012
Line 12:		Line 12:
	{{\widehat{\sigma' }}}=0.3064 \\		{{\widehat{\sigma' }}}=0.3064 \\
	\end{matrix}</math>		\end{matrix}</math>


	Published 95% confidence limits on the parameters:		Published 95% confidence limits on the parameters:


	::<math>\begin{matrix}		::<math>\begin{matrix}
Line 21:		Line 19:
	{{\widehat{\sigma'}}}=\left\{ 0.2365,0.3970 \right\} \\		{{\widehat{\sigma'}}}=\left\{ 0.2365,0.3970 \right\} \\
	\end{matrix}</math>		\end{matrix}</math>


	Published variance/covariance matrix:		Published variance/covariance matrix:


	::<math>\left[ \begin{matrix}		::<math>\left[ \begin{matrix}
Line 35:		Line 31:

	• Weibull++ computed parameters for maximum likelihood are:		• Weibull++ computed parameters for maximum likelihood are:


	::<math>\begin{matrix}		::<math>\begin{matrix}
Line 43:		Line 38:

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


	::<math>\begin{matrix}		::<math>\begin{matrix}
Line 49:		Line 43:
	{{\widehat{\sigma'}}}=\left\{ 0.2395,0.3920 \right\} \\		{{\widehat{\sigma'}}}=\left\{ 0.2395,0.3920 \right\} \\
	\end{matrix}</math>		\end{matrix}</math>



	• Weibull++ computed/variance covariance matrix:		• Weibull++ computed/variance covariance matrix:


	::<math>\left[ \begin{matrix}		::<math>\left[ \begin{matrix}
Line 61:		Line 52:
	\end{matrix} \right]</math>		\end{matrix} \right]</math>

			{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"

	{\|align="center" border="1" cellspacing="1"
	\|-		\|-
	\|colspan="4" style="text-align:center"\|Table - Nelson's Locomotive Data		\|colspan="4" style="text-align:center"\|Table - Nelson's Locomotive Data

Harry Guo at 17:05, 2 March 2012

2012-03-02T17:05:29Z

← Older revision		Revision as of 17:05, 2 March 2012
Line 65:		Line 65:
	{\|align="center" border="1" cellspacing="1"		{\|align="center" border="1" cellspacing="1"
	\|-		\|-
	\|colspan="4" style="text-align:center"\|Table ~~9.6~~ - Nelson's Locomotive Data		\|colspan="4" style="text-align:center"\|Table - Nelson's Locomotive Data
	\|-		\|-
	!		!

Harry Guo at 23:51, 13 February 2012

2012-02-13T23:51:16Z

← Older revision		Revision as of 23:51, 13 February 2012
Line 1:		Line 1:
	'''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'''

Harry Guo at 23:50, 13 February 2012

2012-02-13T23:50:48Z

← Older revision		Revision as of 23:50, 13 February 2012
Line 1:		Line 1:
	'''Lognormal Distribution General Example Suspension Data'''		'''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 (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 (at the end of this chapter) shows their times-to-failure.

	'''Solution'''		'''Solution'''
Line 10:		Line 10:
	::<math>\begin{matrix}		::<math>\begin{matrix}
	{{\widehat{\mu }}^{\prime }}=2.2223 \\		{{\widehat{\mu }}^{\prime }}=2.2223 \\
	{{\widehat{\sigma ~~}}_{{{T}~~'}}}=0.3064 \\		{{\widehat{\sigma' }}}=0.3064 \\
	\end{matrix}</math>		\end{matrix}</math>

Line 19:		Line 19:
	::<math>\begin{matrix}		::<math>\begin{matrix}
	{{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\		{{\widehat{\mu }}^{\prime }}=\left\{ 2.1336,2.3109 \right\} \\
	{{\widehat{\sigma ~~}}_{{{T}~~'}}}=\left\{ 0.2365,0.3970 \right\} \\		{{\widehat{\sigma'}}}=\left\{ 0.2365,0.3970 \right\} \\
	\end{matrix}</math>		\end{matrix}</math>

Line 27:		Line 27:

	::<math>\left[ \begin{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{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0020 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 \\
	{} & {} & {} \\		{} & {} & {} \\
	\widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma }~~}}_{{{T}'~~}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma }~~}}_{{{T}'~~}}} \right)=0.0016 \\		\widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.001 & {} & \widehat{Var}\left( {{{\hat{\sigma '}}}} \right)=0.0016 \\
	\end{matrix} \right]</math>		\end{matrix} \right]</math>

Line 39:		Line 39:
	::<math>\begin{matrix}		::<math>\begin{matrix}
	{{\widehat{\mu }}^{\prime }}=2.2223 \\		{{\widehat{\mu }}^{\prime }}=2.2223 \\
	{{\widehat{\sigma ~~}}_{{{T}~~'}}}=0.3064 \\		{{\widehat{\sigma' }}}=0.3064 \\
	\end{matrix}</math>		\end{matrix}</math>

Line 47:		Line 47:
	::<math>\begin{matrix}		::<math>\begin{matrix}
	{{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\		{{\widehat{\mu }}^{\prime }}=\left\{ 2.1364,2.3081 \right\} \\
	{{\widehat{\sigma ~~}}_{{{T}~~'}}}=\left\{ 0.2395,0.3920 \right\} \\		{{\widehat{\sigma'}}}=\left\{ 0.2395,0.3920 \right\} \\
	\end{matrix}</math>		\end{matrix}</math>

Line 56:		Line 56:

	::<math>\left[ \begin{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{Var}\left( {{{\hat{\mu }}}^{\prime }} \right)=0.0019 & {} & \widehat{Cov}({{{\hat{\mu }}}^{\prime }},{{{\hat{\sigma' }}}})=0.0009 \\
	{} & {} & {} \\		{} & {} & {} \\
	\widehat{Cov}({\mu }',{{{\hat{\sigma }~~}}_{{{T}'~~}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma }~~}}_{{{T}'~~}}} \right)=0.0015 \\		\widehat{Cov}({\mu }',{{{\hat{\sigma' }}}})=0.0009 & {} & \widehat{Var}\left( {{{\hat{\sigma' }}}} \right)=0.0015 \\
	\end{matrix} \right]</math>		\end{matrix} \right]</math>

Harry Guo: Created page with ''''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 …'

2012-02-13T19:03:23Z

Created page with ''''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 …'

New page

'''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 (at the end of this chapter) 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 }}_{{{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
|}