Template:Lognormal distribution general examples - Revision history

Chris Kahn: Redirected page to Lognormal Distribution Examples

2012-08-14T07:32:10Z

Redirected page to Lognormal Distribution Examples

← Older revision		Revision as of 07:32, 14 August 2012
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	~~==General~~ Examples==		#REDIRECT [[Lognormal Distribution Examples]]
	~~{{:Lognormal_Distribution_Examples}}~~

Chris Kahn: /* General Examples */

2012-08-14T07:23:38Z

General Examples

← Older revision		Revision as of 07:23, 14 August 2012
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	==General Examples==		==General Examples==
			{{:Lognormal_Distribution_Examples}}
	~~'''Example 9:'''~~
	~~{{Example: Lognormal General Example Interval Data}}~~


	~~'''Example 10:'''~~
	~~{{Example: Lognormal General Example Complete Data}}~~


	~~'''Example 11:'''~~
	~~{{Example: Lognormal General Example Complete Data Unbiased MLE}}~~


	~~'''Example 12:'''~~
	~~{{Example: Lognormal General Example Complete Data RRX}}~~


	~~'''Example 13:'''~~
	{{~~Example~~: ~~Lognormal General Example Suspension Data~~}}

Harry Guo at 19:03, 13 February 2012

2012-02-13T19:03:55Z

← Older revision		Revision as of 19:03, 13 February 2012
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	==General Examples==		==General Examples==

	'''Example 9:'''		'''Example 9:'''
	{{Example: Lognormal General Example Interval Data}}		{{Example: Lognormal General Example Interval Data}}


	'''Example 10:'''		'''Example 10:'''
	{{Example: Lognormal General Example Complete Data}}		{{Example: Lognormal General Example Complete Data}}


	'''Example 11:'''		'''Example 11:'''
	{{Example: Lognormal General Example Complete Data Unbiased MLE}}		{{Example: Lognormal General Example Complete Data Unbiased MLE}}


	'''Example 12:'''		'''Example 12:'''
	{{Example: Lognormal General Example Complete Data RRX}}		{{Example: Lognormal General Example Complete Data RRX}}


	'''Example 13:'''		'''Example 13:'''
	{{Example: Lognormal General Example Suspension Data}}		{{Example: Lognormal General Example Suspension Data}}

Harry Guo: /* General Examples */

2012-02-13T19:00:20Z

General Examples

Show changes

Harry Guo: /* General Examples */

2012-02-13T18:57:47Z

General Examples

← Older revision		Revision as of 18:57, 13 February 2012
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	{{Example: Lognormal General Example Interval Data}}		{{Example: Lognormal General Example Interval Data}}

			'''Example 10:'''
			{{Example: Lognormal General Example Complete Data}}

	~~===Example 10===~~
	~~Determine the lognormal parameter estimates for the data given in Table 9.4.~~
	~~{\|align="center" border=1 cellspacing=1~~
	\|-
	~~\|colspan="3" style="text-align:center"\| Table 9.4 - Non-Grouped Data for Example 12~~
	\|-
	~~!Data point index~~
	~~!State F or S~~
	~~!State End Time~~
	\|-
	~~\|1 \|\|F\|\|2~~
	\|-
	~~\|2 \|\|F\|\|5~~
	\|-
	~~\|3 \|\|F\|\|11~~
	\|-
	~~\|4 \|\|F\|\|23~~
	\|-
	~~\|5 \|\|F\|\|29~~
	\|-
	~~\|6 \|\|F\|\|37~~
	\|-
	~~\|7\|\|F\|\|43~~
	\|-
	~~\|8\|\|F\|\|59~~
	\|}

	~~====Solution to Example 10====~~
	~~Using Weibull++, the computed parameters for maximum likelihood are:~~

	~~::<math>\begin{align}~~
	~~& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\~~
	~~& {{{\hat{\sigma }}}_{{{T}'}}}= & 1.10~~
	~~\end{align}</math>~~


	~~For rank regression on <math>X\ \ :</math>~~

	~~::<math>\begin{align}~~
	~~& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\~~
	~~& {{{\hat{\sigma }}}_{{{T}'}}}= & 1.24~~
	~~\end{align}</math>~~


	~~For rank regression on <math>Y\ \ :</math>~~

	~~::<math>\begin{align}~~
	~~& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\~~
	~~& {{{\hat{\sigma }}}_{{{T}'}}}= & 1.36~~
	~~\end{align}</math>~~

	===Example 11===		===Example 11===

Harry Guo: /* General Examples */

2012-02-13T18:56:13Z

General Examples

@@ Line 1: / Line 1: @@
 ==General Examples==
-====Solution to Example 9====
+'''Example 9:'''
-This is a sequence of interval times-to-failure where the intervals vary substantially in length. Using Weibull++, the computed parameters for maximum likelihood are calculated to be:
+{{Example: Lognormal General Example Interval Data}}
 ===Example 10===

Harry Guo: /* Example 9 */

2012-02-10T23:29:14Z

Example 9

← Older revision		Revision as of 23:29, 10 February 2012
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	==General Examples==		==General Examples==
	===Example 9===		===Example 9===
	Determine the lognormal parameter estimates for the data given in Table ~~9.3~~.		Determine the lognormal parameter estimates for the data given in Table below.
	{\|align="center" border=1 cellspacing=1		{\|align="center" border=1 cellspacing=1
	\|-		\|-
	\|colspan="3" style="text-align:center"\| Table ~~9.3~~ - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)		\|colspan="3" style="text-align:center"\| Table - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
	\|-		\|-
	!Data point index		!Data point index

Harry Guo: Created page with ' ==General Examples== ===Example 9=== Determine the lognormal parameter estimates for the data given in Table 9.3. {|align="center" border=1 cellspacing=1 |- |colspan="3" style=…'

2012-02-10T23:25:36Z

Created page with ' ==General Examples== ===Example 9=== Determine the lognormal parameter estimates for the data given in Table 9.3. {|align="center" border=1 cellspacing=1 |- |colspan="3" style=…'

New page

==General Examples==
===Example 9===
Determine the lognormal parameter estimates for the data given in Table 9.3.
{|align="center" border=1 cellspacing=1
|-
|colspan="3" style="text-align:center"| Table 9.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
|-
!Data point index
!Last Inspected
!State End Time
|-
|1 ||30||32
|-
|2 ||32||35
|-
|3 ||35||37
|-
|4 ||37||40
|-
|5 ||42||42
|-
|6 ||45||45
|-
|7||50||50
|-
|8||55||55
|}

====Solution to Example 9====
This is a sequence of interval times-to-failure where the intervals vary substantially in length. Using Weibull++, the computed parameters for maximum likelihood are calculated to be:

::<math>\begin{align}
& {{{\hat{\mu }}}^{\prime }}= & 3.64 \\
& {{{\hat{\sigma }}}_{{{T}'}}}= & 0.18
\end{align}</math>

For rank regression on <math>X\ \ :</math>

::<math>\begin{align}
& {{{\hat{\mu }}}^{\prime }}= & 3.64 \\
& {{{\hat{\sigma }}}_{{{T}'}}}= & 0.17
\end{align}</math>

For rank regression on <math>Y\ \ :</math>

::<math>\begin{align}
& {{{\hat{\mu }}}^{\prime }}= & 3.64 \\
& {{{\hat{\sigma }}}_{{{T}'}}}= & 0.21
\end{align}</math>

===Example 10===
Determine the lognormal parameter estimates for the data given in Table 9.4.
{|align="center" border=1 cellspacing=1
|-
|colspan="3" style="text-align:center"| Table 9.4 - Non-Grouped Data for Example 12
|-
!Data point index
!State F or S
!State End Time
|-
|1 ||F||2
|-
|2 ||F||5
|-
|3 ||F||11
|-
|4 ||F||23
|-
|5 ||F||29
|-
|6 ||F||37
|-
|7||F||43
|-
|8||F||59
|}

====Solution to Example 10====
Using Weibull++, the computed parameters for maximum likelihood are:

::<math>\begin{align}
& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\
& {{{\hat{\sigma }}}_{{{T}'}}}= & 1.10
\end{align}</math>

For rank regression on <math>X\ \ :</math>

::<math>\begin{align}
& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\
& {{{\hat{\sigma }}}_{{{T}'}}}= & 1.24
\end{align}</math>

For rank regression on <math>Y\ \ :</math>

::<math>\begin{align}
& {{{\hat{\mu }}}^{\prime }}= & 2.83 \\
& {{{\hat{\sigma }}}_{{{T}'}}}= & 1.36
\end{align}</math>

===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
|}