Generalized Gamma Distribution Example - Revision history

Lisa Hacker at 21:40, 18 September 2023

2023-09-18T21:40:10Z

← Older revision		Revision as of 21:40, 18 September 2023
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	<noinclude>{{Banner Weibull Examples}}		<noinclude>{{Banner Weibull Examples}}
	''This example appears in the [~~[The_Generalized_Gamma_Distribution#General_Example\|~~Life ~~Data Analysis Reference book]~~]''.</noinclude>		''This example appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference]''.</noinclude>

	The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test, as discussed in Lawless [[Appendix:_Life_Data_Analysis_References\|[21]]].		The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test, as discussed in Lawless [[Appendix:_Life_Data_Analysis_References\|[21]]].

Richard House at 16:10, 26 September 2012

2012-09-26T16:10:13Z

← Older revision		Revision as of 16:10, 26 September 2012
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	\text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84} \\		\text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84} \\
	\text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\		\text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\
	\end{array}</math>		\end{array}\,\!</math>

	When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:		When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:
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	& \widehat{\sigma }= & 0.509982 \\		& \widehat{\sigma }= & 0.509982 \\
	& \widehat{\lambda }= & 0.307639		& \widehat{\lambda }= & 0.307639
	\end{align}</math>		\end{align}\,\!</math>

	Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),\,\!</math> and the lognormal <math>(\lambda =0)\,\!</math> distributions. The value of <math>\lambda \,\!</math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .\,\!</math> For example, the 90% two-sided confidence bounds are:		Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),\,\!</math> and the lognormal <math>(\lambda =0)\,\!</math> distributions. The value of <math>\lambda \,\!</math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .\,\!</math> For example, the 90% two-sided confidence bounds are:

	::<math>\begin{align}		::<math>\begin{align}
	& {{\lambda }_{u}}= & -0.592087 \\		& {{\lambda }_{u}}= & -0.592087 \\
	& {{\lambda }_{u}}= & 1.20736		& {{\lambda }_{u}}= & 1.20736
	\end{align}</math>		\end{align}\,\!</math>

	We can then conclude that both distributions (i.e., Weibull and lognormal) are well supported by the data, with the lognormal being the better supported of the two.		We can then conclude that both distributions (i.e., Weibull and lognormal) are well supported by the data, with the lognormal being the better supported of the two.

Richard House at 16:37, 5 September 2012

2012-09-05T16:37:04Z

← Older revision		Revision as of 16:37, 5 September 2012
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	''This example appears in the [[The_Generalized_Gamma_Distribution#General_Example\|Life Data Analysis Reference book]]''.</noinclude>		''This example appears in the [[The_Generalized_Gamma_Distribution#General_Example\|Life Data Analysis Reference book]]''.</noinclude>

	The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [[Appendix: ~~Weibull References~~\|[21]]].		The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test, as discussed in Lawless [[Appendix:_Life_Data_Analysis_References\|[21]]].

	::<math>\begin{array}{*{35}{l}}		::<math>\begin{array}{*{35}{l}}

Richard House at 03:17, 28 August 2012

2012-08-28T03:17:59Z

← Older revision		Revision as of 03:17, 28 August 2012
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	\end{align}</math>		\end{align}</math>

	Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),</math> and the lognormal <math>(\lambda =0)</math> distributions. The value of <math>\lambda </math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .</math> For example, the 90% two-sided confidence bounds are:		Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),\,\!</math> and the lognormal <math>(\lambda =0)\,\!</math> distributions. The value of <math>\lambda \,\!</math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .\,\!</math> For example, the 90% two-sided confidence bounds are:

	::<math>\begin{align}		::<math>\begin{align}

Chris Kahn: moved Example: Generalized Gamma Distribution Example to Generalized Gamma Distribution Example

2012-08-20T00:51:06Z

moved Example: Generalized Gamma Distribution Example to Generalized Gamma Distribution Example

← Older revision	Revision as of 00:51, 20 August 2012
(No difference)

Melinda Caroline at 06:34, 15 August 2012

2012-08-15T06:34:00Z

← Older revision		Revision as of 06:34, 15 August 2012
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	[[Image:GGamma Example 1 Plot.png\|center\|500px\| ]]		[[Image:GGamma Example 1 Plot.png\|center\|500px\| ]]

	It is also important to note that, as in the case of the mixed Weibull distribution, the choice of regression analysis (i.e., RRX or ~~RRy~~) is of no consequence in the generalized gamma model because it uses non-linear regression.		It is also important to note that, as in the case of the mixed Weibull distribution, the choice of regression analysis (i.e., RRX or RRY) is of no consequence in the generalized gamma model because it uses non-linear regression.

Richard House at 06:00, 6 August 2012

2012-08-06T06:00:07Z

← Older revision		Revision as of 06:00, 6 August 2012
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	In Weibull++ the generalized gamma probability is plotted on a gamma probability paper, as shown next.		In Weibull++ the generalized gamma probability is plotted on a gamma probability paper, as shown next.

	[[Image:GGamma Example 1 Plot.png~~\|thumb~~\|center\|500px\| ]]		[[Image:GGamma Example 1 Plot.png\|center\|500px\| ]]

	It is also important to note that, as in the case of the mixed Weibull distribution, the choice of regression analysis (i.e., RRX or RRy) is of no consequence in the generalized gamma model because it uses non-linear regression.		It is also important to note that, as in the case of the mixed Weibull distribution, the choice of regression analysis (i.e., RRX or RRy) is of no consequence in the generalized gamma model because it uses non-linear regression.

Richard House at 03:49, 6 August 2012

2012-08-06T03:49:01Z

← Older revision		Revision as of 03:49, 6 August 2012
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	The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [[Appendix: Weibull References\|[21]]].		The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [[Appendix: Weibull References\|[21]]].


	::<math>\begin{array}{*{35}{l}}		::<math>\begin{array}{*{35}{l}}
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	\text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\		\text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\
	\end{array}</math>		\end{array}</math>


	When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:		When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:


	::<math>\begin{align}		::<math>\begin{align}
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	& \widehat{\lambda }= & 0.307639		& \widehat{\lambda }= & 0.307639
	\end{align}</math>		\end{align}</math>


	Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),</math> and the lognormal <math>(\lambda =0)</math> distributions. The value of <math>\lambda </math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .</math> For example, the 90% two-sided confidence bounds are:		Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),</math> and the lognormal <math>(\lambda =0)</math> distributions. The value of <math>\lambda </math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .</math> For example, the 90% two-sided confidence bounds are:


	::<math>\begin{align}		::<math>\begin{align}
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	& {{\lambda }_{u}}= & 1.20736		& {{\lambda }_{u}}= & 1.20736
	\end{align}</math>		\end{align}</math>


	We can then conclude that both distributions (i.e., Weibull and lognormal) are well supported by the data, with the lognormal being the better supported of the two.		We can then conclude that both distributions (i.e., Weibull and lognormal) are well supported by the data, with the lognormal being the better supported of the two.

Kate Racaza: proofread

2012-07-23T09:18:15Z

proofread

← Older revision		Revision as of 09:18, 23 July 2012
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	{~~\| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;"~~		<noinclude>{{Banner Weibull Examples}}
	\|-		''This example appears in the [[The_Generalized_Gamma_Distribution#General_Example\|Life Data Analysis Reference book]]''.</noinclude>
	~~\| valign="middle" align="left" bgcolor=EEEDF7\|~~[[~~Image:Weibull-Examples-banner.png~~\|~~400px\|center~~]]
	\|}
	<br>
	The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [[Appendix: Weibull References\|[21]]].		The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [[Appendix: Weibull References\|[21]]].

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	When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:		When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:


	::<math>\begin{align}		::<math>\begin{align}
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	& \widehat{\lambda }= & 0.307639		& \widehat{\lambda }= & 0.307639
	\end{align}</math>		\end{align}</math>


	Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),</math> and the lognormal <math>(\lambda =0)</math> distributions. The value of <math>\lambda </math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .</math> For example, the 90% two-sided confidence bounds are:		Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),</math> and the lognormal <math>(\lambda =0)</math> distributions. The value of <math>\lambda </math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .</math> For example, the 90% two-sided confidence bounds are:


	::<math>\begin{align}		::<math>\begin{align}
Line 29:		Line 30:
	\end{align}</math>		\end{align}</math>

	It can be then ~~concluded~~ that both distributions (i.e. Weibull and lognormal) are well supported by the data, with the lognormal being the ,better supported of the two.
	In Weibull++ the generalized gamma probability is plotted on gamma probability paper, as shown next.		We can then conclude that both distributions (i.e., Weibull and lognormal) are well supported by the data, with the lognormal being the better supported of the two.
			In Weibull++ the generalized gamma probability is plotted on a gamma probability paper, as shown next.

	[[Image:GGamma Example 1 Plot.png\|thumb\|center\|500px\| ]]		[[Image:GGamma Example 1 Plot.png\|thumb\|center\|500px\| ]]

	It is important to ~~also~~ note that as in the case of the mixed Weibull distribution, in the ~~case~~ of regression analysis~~, using a generalized gamma model, the choice of regression axis,~~ i.e. ~~<math>~~RRX~~</math>~~ or ~~<math>RRY,</math>~~ is of no consequence ~~since~~ non-linear regression ~~is utilized~~.		It is also important to note that, as in the case of the mixed Weibull distribution, the choice of regression analysis (i.e., RRX or RRy) is of no consequence in the generalized gamma model because it uses non-linear regression.

Nicolette Young: Created page with '{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;" |- | valign="middle" align="left" bgcolor=EEEDF7|[[Image:Weibull-E…'

2012-03-21T23:39:31Z

Created page with '{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;" |- | valign="middle" align="left" bgcolor=EEEDF7|[[Image:Weibull-E…'

New page

{| class="FCK__ShowTableBorders" border="0" cellspacing="0" cellpadding="0" align="center"; style="width:100%;"
|-
| valign="middle" align="left" bgcolor=EEEDF7|[[Image:Weibull-Examples-banner.png|400px|center]]
|}
<br>
The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [[Appendix: Weibull References|[21]]].

::<math>\begin{array}{*{35}{l}}
\text{17}\text{.88} & \text{28}\text{.92} & \text{33} & \text{41}\text{.52} & \text{42}\text{.12} & \text{45}\text{.6} & \text{48}\text{.4} & \text{51}\text{.84} & \text{51}\text{.96} & \text{54}\text{.12} \\
\text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84} \\
\text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\
\end{array}</math>

When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:

::<math>\begin{align}
& \widehat{\mu }= & 4.23064 \\
& \widehat{\sigma }= & 0.509982 \\
& \widehat{\lambda }= & 0.307639
\end{align}</math>

Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),</math> and the lognormal <math>(\lambda =0)</math> distributions. The value of <math>\lambda </math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .</math> For example, the 90% two-sided confidence bounds are:

::<math>\begin{align}
& {{\lambda }_{u}}= & -0.592087 \\
& {{\lambda }_{u}}= & 1.20736
\end{align}</math>

It can be then concluded that both distributions (i.e. Weibull and lognormal) are well supported by the data, with the lognormal being the ,better supported of the two.
In Weibull++ the generalized gamma probability is plotted on gamma probability paper, as shown next.

[[Image:GGamma Example 1 Plot.png|thumb|center|500px| ]]

It is important to also note that as in the case of the mixed Weibull distribution, in the case of regression analysis, using a generalized gamma model, the choice of regression axis, i.e. <math>RRX</math> or <math>RRY,</math> is of no consequence since non-linear regression is utilized.