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| {{characteristics of the generalized gamma distribution}} | | {{characteristics of the generalized gamma distribution}} |
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| ===Confidence Bounds===
| | {{confidence bounds gd}} |
| The only method available in Weibull++ for confidence bounds for the generalized gamma distribution is the Fisher matrix, which is described next.
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| ====Bounds on the Parameters====
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| The lower and upper bounds on the parameter <math>\mu </math> are estimated from:
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| ::<math>\begin{align}
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| & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\
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| & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)}
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| \end{align}</math>
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| For the parameter <math>\widehat{\sigma }</math> , <math>\ln (\widehat{\sigma })</math> is treated as normally distributed, and the bounds are estimated from:
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| ::<math>\begin{align}
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| & {{\sigma }_{U}}= & \widehat{\sigma }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}\text{ (upper bound)} \\
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| & {{\sigma }_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\sigma })}}{\widehat{\sigma }}}}}\text{ (lower bound)}
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| \end{align}</math>
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| For the parameter <math>\lambda ,</math> the bounds are estimated from:
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| ::<math>\begin{align}
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| & {{\lambda }_{U}}= & \widehat{\lambda }+{{K}_{\alpha }}\sqrt{Var(\widehat{\lambda })}\text{ (upper bound)} \\
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| & {{\lambda }_{L}}= & \widehat{\lambda }-{{K}_{\alpha }}\sqrt{Var(\widehat{\lambda })}\text{ (lower bound)}
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| \end{align}</math>
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| where <math>{{K}_{\alpha }}</math> is defined by:
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| ::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
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|
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| If <math>\delta </math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}</math> for the two-sided bounds, and <math>\alpha =1-\delta </math> for the one-sided bounds.
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| The variances and covariances of <math>\widehat{\mu }</math> and <math>\widehat{\sigma }</math> are estimated as follows:
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| ::<math>\begin{align}
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| & & \left( \begin{matrix}
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| \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\sigma } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{\lambda } \right) \\
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| \widehat{Cov}\left( \widehat{\sigma },\widehat{\mu } \right) & \widehat{Var}\left( \widehat{\sigma } \right) & \widehat{Cov}\left( \widehat{\sigma },\widehat{\lambda } \right) \\
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| \widehat{Cov}\left( \widehat{\lambda },\widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\lambda },\widehat{\sigma } \right) & \widehat{Var}\left( \widehat{\lambda } \right) \\
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| \end{matrix} \right) \\
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| & = & \left( \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \lambda } \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \sigma } \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial \lambda } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \lambda \partial \sigma } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\lambda }^{2}}} \\
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| \end{matrix} \right)_{\mu =\widehat{\mu },\sigma =\widehat{\sigma },\lambda =\hat{\lambda }}^{-1}
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| \end{align}</math>
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| Where <math>\Lambda </math> is the log-likelihood function of the generalized gamma distribution.
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| ====Bounds on Reliability====
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| The upper and lower bounds on reliability are given by:
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| ::<math>\begin{align}
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| & {{R}_{U}}= & \frac{{\hat{R}}}{\hat{R}+(1-\hat{R}){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\hat{R}(1-\hat{R})}}}} \\
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| & {{R}_{L}}= & \frac{{\hat{R}}}{\hat{R}+(1-\hat{R}){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\hat{R}(1-\hat{R})}}}}
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| \end{align}</math>
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| :where:
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| ::<math>\begin{align}
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| & Var(\widehat{R})= & {{\left( \frac{\partial R}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial R}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial R}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+ \\
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| & & +2\left( \frac{\partial R}{\partial \mu } \right)\left( \frac{\partial R}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial R}{\partial \mu } \right)\left( \frac{\partial R}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })+ \\
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| & & +2\left( \frac{\partial R}{\partial \lambda } \right)\left( \frac{\partial R}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
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| \end{align}</math>
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| ====Bounds on Time====
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| The bounds around time for a given percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, given by Eqn. (GGamma Time). Since <math>T</math> is a positive variable, the transformed variable <math>\hat{u}=\ln (\widehat{T})</math> is treated as normally distributed and the bounds are estimated from:
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| ::<math>\begin{align}
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| & {{u}_{u}}= & \ln {{T}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})} \\
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| & {{u}_{L}}= & \ln {{T}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}
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| \end{align}</math>
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| Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> we get:
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{{{T}_{U}}}}\text{ (upper bound)} \\
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| & {{T}_{L}}= & {{e}^{{{T}_{L}}}}\text{ (lower bound)}
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| \end{align}</math>
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| The variance of <math>u</math> is estimated from:
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & {{\left( \frac{\partial u}{\partial \mu } \right)}^{2}}Var(\widehat{\mu })+{{\left( \frac{\partial u}{\partial \sigma } \right)}^{2}}Var(\widehat{\sigma })+{{\left( \frac{\partial u}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+ \\
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| & & +2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\mu },\widehat{\sigma })+2\left( \frac{\partial u}{\partial \mu } \right)\left( \frac{\partial u}{\partial \lambda } \right)Cov(\widehat{\mu },\widehat{\lambda })+ \\
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| & & +2\left( \frac{\partial u}{\partial \lambda } \right)\left( \frac{\partial u}{\partial \sigma } \right)Cov(\widehat{\lambda },\widehat{\sigma })
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| \end{align}</math>
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| ====A Generalized Gamma Distribution Example====
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| The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [21].
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| ::<math>\begin{array}{*{35}{l}}
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| \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} \\
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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} \\
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| \text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\
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| \end{array}</math>
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| When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:
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| ::<math>\begin{align}
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| & \widehat{\mu }= & 4.23064 \\
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| & \widehat{\sigma }= & 0.509982 \\
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| & \widehat{\lambda }= & 0.307639
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| \end{align}</math>
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| 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:
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| ::<math>\begin{align}
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| & {{\lambda }_{u}}= & -0.592087 \\
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| & {{\lambda }_{u}}= & 1.20736
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| \end{align}</math>
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| 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.
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| In Weibull++ the generalized gamma probability is plotted on gamma probability paper, as shown next.
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| [[Image:ldagamma10.2.gif|thumb|center|500px| ]]
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| 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.
| |
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Chapter 12: The Generalized Gamma Distribution
|
The Generalized Gamma Distribution
While not as frequently used for modeling life data as the previous distributions, the generalized gamma distribution does have the ability to mimic the attributes of other distributions such as the Weibull or lognormal, based on the values of the distribution's parameters. While the generalized gamma distribution is not often used to model life data by itself , its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis
|
Chapter 12: The Generalized Gamma Distribution
|
The Generalized Gamma Distribution
While not as frequently used for modeling life data as the previous distributions, the generalized gamma distribution does have the ability to mimic the attributes of other distributions such as the Weibull or lognormal, based on the values of the distribution's parameters. While the generalized gamma distribution is not often used to model life data by itself , its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
Template loop detected: Template:Generalized gamma probability density function
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Template loop detected: Template:Characteristics of the generalized gamma distribution
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|
Chapter 12: The Generalized Gamma Distribution
|
The Generalized Gamma Distribution
While not as frequently used for modeling life data as the previous distributions, the generalized gamma distribution does have the ability to mimic the attributes of other distributions such as the Weibull or lognormal, based on the values of the distribution's parameters. While the generalized gamma distribution is not often used to model life data by itself , its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
Template loop detected: Template:Generalized gamma probability density function
Template loop detected: Template:Generalized gamma reliability function
Template loop detected: Template:Generalized gamma failure rate function
Template loop detected: Template:Generalized gamma reliable life
Template loop detected: Template:Characteristics of the generalized gamma distribution
Template loop detected: Template:Confidence bounds gd
New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis
|
Chapter 12: The Generalized Gamma Distribution
|
The Generalized Gamma Distribution
While not as frequently used for modeling life data as the previous distributions, the generalized gamma distribution does have the ability to mimic the attributes of other distributions such as the Weibull or lognormal, based on the values of the distribution's parameters. While the generalized gamma distribution is not often used to model life data by itself , its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
Template loop detected: Template:Generalized gamma probability density function
Template loop detected: Template:Generalized gamma reliability function
Template loop detected: Template:Generalized gamma failure rate function
Template loop detected: Template:Generalized gamma reliable life
Template loop detected: Template:Characteristics of the generalized gamma distribution
Template loop detected: Template:Confidence bounds gd
New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis
|
Chapter 12: The Generalized Gamma Distribution
|
The Generalized Gamma Distribution
While not as frequently used for modeling life data as the previous distributions, the generalized gamma distribution does have the ability to mimic the attributes of other distributions such as the Weibull or lognormal, based on the values of the distribution's parameters. While the generalized gamma distribution is not often used to model life data by itself , its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
Template loop detected: Template:Generalized gamma probability density function
Template loop detected: Template:Generalized gamma reliability function
Template loop detected: Template:Generalized gamma failure rate function
Template loop detected: Template:Generalized gamma reliable life
Template loop detected: Template:Characteristics of the generalized gamma distribution
Template loop detected: Template:Confidence bounds gd
New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis
|
Chapter 12: The Generalized Gamma Distribution
|
The Generalized Gamma Distribution
While not as frequently used for modeling life data as the previous distributions, the generalized gamma distribution does have the ability to mimic the attributes of other distributions such as the Weibull or lognormal, based on the values of the distribution's parameters. While the generalized gamma distribution is not often used to model life data by itself , its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
Template loop detected: Template:Generalized gamma probability density function
Template loop detected: Template:Generalized gamma reliability function
Template loop detected: Template:Generalized gamma failure rate function
Template loop detected: Template:Generalized gamma reliable life
Template loop detected: Template:Characteristics of the generalized gamma distribution
Template loop detected: Template:Confidence bounds gd
New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images, more targeted search and the latest content available as a PDF. As of September 2023, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest reference at help.reliasoft.com/reference/life_data_analysis
|
Chapter 12: The Generalized Gamma Distribution
|
The Generalized Gamma Distribution
While not as frequently used for modeling life data as the previous distributions, the generalized gamma distribution does have the ability to mimic the attributes of other distributions such as the Weibull or lognormal, based on the values of the distribution's parameters. While the generalized gamma distribution is not often used to model life data by itself , its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
Template loop detected: Template:Generalized gamma probability density function
Template loop detected: Template:Generalized gamma reliability function
Template loop detected: Template:Generalized gamma failure rate function
Template loop detected: Template:Generalized gamma reliable life
Template loop detected: Template:Characteristics of the generalized gamma distribution
Template loop detected: Template:Confidence bounds gd
