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{{template:LDABOOK|13|The Gamma Distribution}}
{{template:LDABOOK|13|The Gamma Distribution}}
The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as a life distribution model for common failure mechanisms. The gamma distribution does arise naturally as the time-to-first-fail distribution for a system with standby exponentially distributed backups, and is also a good fit for the sum of independent exponential random variables. The gamma distribution is sometimes called the Erlang distribution, which is used frequently in queuing theory applications, as discussed in [[Appendix:_Life_Data_Analysis_References|[32]]].


==The Gamma Distribution==
===The Gamma Probability Density Function===
The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as a life distribution model for common failure mechanisms. The gamma distribution does arise naturally as the time-to-first-fail distribution for a system with standby exponentially distributed backups, and is also a good fit for the sum of independent exponential random variables. The gamma distribution is sometimes called the Erlang distribution, which is used frequently in queuing theory applications. [32]
The ''pdf'' of the gamma distribution is given by:


===Gamma Probability Density Function===
::<math>f(t)=\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)}\,\!</math>
The  <math>pdf</math>  of the gamma distribution is given by:
 
::<math>f(T)=\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)}</math>


where:  
where:  


::<math>z=\ln (t)-\mu </math>
::<math>\begin{align}
z=\ln (t)-\mu  
\end{align}\,\!</math>


and:  
and:  


::<math>\begin{align}
::<math>\begin{align}
   & {{e}^{\mu }}= & \text{scale parameter} \\  
   & {{e}^{\mu }}= \text{scale parameter} \\  
  & k= & \text{shape parameter}   
  & k= \text{shape parameter}   
\end{align}</math>
\end{align}\,\!</math>


where <math>0<t<\infty </math> , <math>-\infty <\mu <\infty </math> and <math>k>0</math> .
where <math>0<t<\infty \,\!</math>, <math>-\infty <\mu <\infty \,\!</math> and <math>k>0\,\!</math>.


===The Gamma Reliability Function===
===The Gamma Reliability Function===
The reliability for a mission of time <math>t\,\!</math> for the gamma distribution is:


The reliability for a mission of time  <math>T</math>  for the gamma distribution is:
::<math>\begin{align}
 
R=1-{{\Gamma }_{I}}(k;{{e}^{z}})
 
\end{align}\,\!</math>
::<math>R=1-{{\Gamma }_{1}}(k;{{e}^{z}})</math>
 


===The Gamma Mean, Median and Mode===
===The Gamma Mean, Median and Mode===
The gamma mean or MTTF is:
The gamma mean or MTTF is:


::<math>\overline{T}=k{{e}^{\mu }}\,\!</math>


::<math>\overline{T}=k{{e}^{\mu }}</math>
The mode exists if <math>k>1\,\!</math> and is given by:
 
 
The mode exists if <math>k>1</math> and is given by:
 
 
::<math>\tilde{T}=(k-1){{e}^{\mu }}</math>


::<math>\tilde{T}=(k-1){{e}^{\mu }}\,\!</math>


The median is:  
The median is:  


::<math>\widehat{T}={{e}^{\mu +\ln (\Gamma _{1}^{-1}(0.5;k))}}</math>
::<math>\widehat{T}={{e}^{\mu +\ln (\Gamma _{I}^{-1}(0.5;k))}}\,\!</math>


===The Gamma Standard Deviation===
===The Gamma Standard Deviation===
The standard deviation for the gamma distribution is:  
The standard deviation for the gamma distribution is:  


::<math>{{\sigma }_{T}}=\sqrt{k}{{e}^{\mu }}</math>
::<math>{{\sigma }_{T}}=\sqrt{k}{{e}^{\mu }}\,\!</math>
 


===The Gamma Reliable Life===
===The Gamma Reliable Life===
The gamma reliable life is:
The gamma reliable life is:


::<math>{{T}_{R}}={{e}^{\mu +\ln (\Gamma _{1}^{-1}(1-R;k))}}</math>
::<math>{{T}_{R}}={{e}^{\mu +\ln (\Gamma _{1}^{-1}(1-R;k))}}\,\!</math>


===The Gamma Failure Rate Function===
===The Gamma Failure Rate Function===
The instantaneous gamma failure rate is given by:  
The instantaneous gamma failure rate is given by:  


::<math>\lambda =\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)(1-{{\Gamma }_{1}}(k;{{e}^{z}}))}</math>
::<math>\lambda =\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)(1-{{\Gamma }_{I}}(k;{{e}^{z}}))}\,\!</math>


===Characteristics of the Gamma Distribution===
==Characteristics of the Gamma Distribution==
Some of the specific characteristics of the gamma distribution are the following:
Some of the specific characteristics of the gamma distribution are the following:


For <math>k>1</math> :
For <math>k>1\,\!</math> :
 
:• As <math>t\to 0,\infty\,\!</math>, <math>f(t)\to 0.\,\!</math>  
• As <math>T\to 0,\infty </math> , <math>f(T)\to 0.</math>  
:• <math>f(t)\,\!</math> increases from 0 to the mode value and decreases thereafter.
 
:• If <math>k\le 2\,\!</math> then ''pdf'' has one inflection point at <math>t={{e}^{\mu }}\sqrt{k-1}(\,\!</math> <math>\sqrt{k-1}+1).\,\!</math>  
• <math>f(T)</math> increases from 0 to the mode value and decreases thereafter.
:• If <math>k>2\,\!</math> then ''pdf'' has two inflection points for <math>t={{e}^{\mu }}\sqrt{k-1}(\,\!</math> <math>\sqrt{k-1}\pm 1).\,\!</math>  
 
:• For a fixed <math>k\,\!</math>, as <math>\mu \,\!</math> increases, the ''pdf'' starts to look more like a straight angle.
• If   <math>k\le 2</math> then <math>pdf</math>  has one inflection point at <math>T={{e}^{\mu }}\sqrt{k-1}(</math>   <math>\sqrt{k-1}+1).</math>  
:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>  
 
• If   <math>k>2</math> then <math>pdf</math>  has two inflection points for <math>T={{e}^{\mu }}\sqrt{k-1}(</math>   <math>\sqrt{k-1}\pm 1).</math>  
 
• For a fixed <math>k</math> , as <math>\mu </math> increases, the <math>pdf</math> starts to look more like a straight angle.
 
As  <math>T\to \infty ,\lambda (T)\to \tfrac{1}{{{e}^{\mu }}}.</math>
 
[[Image:ldaGD10.1.gif|thumb|center|400px| ]]
 
For  <math>k=1</math> :
 
• Gamma becomes the exponential distribution.
 
• As  <math>T\to 0</math>  ,  <math>f(T)\to \tfrac{1}{{{e}^{\mu }}}.</math>
 
• As <math>T\to \infty ,f(T)\to 0.</math>
 
• The  <math>pdf</math>  decreases monotonically and is convex.
 
• <math>\lambda (T)\equiv \tfrac{1}{{{e}^{\mu }}}</math>  . <math>\lambda (T)</math>  is constant.
 
• The mode does not exist.
 
[[Image:ldaGD10.2.gif|thumb|center|400px| ]]
 
For  <math>0<k<1</math> :
 
• As  <math>T\to 0</math>  , <math>f(T)\to \infty .</math>
 
• As  <math>T\to \infty ,f(T)\to 0.</math>  


• As  <math>T\to \infty ,\lambda (T)\to \tfrac{1}{{{e}^{\mu }}}.</math>
[[Image:BSpdf1.png|center|400px| ]]


• The  <math>pdf</math> decreases monotonically and is convex.
For <math>k=1\,\!</math> :
:• Gamma becomes the exponential distribution.
:• As <math>t\to 0\,\!</math>, <math>f(T)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>
:• As <math>t\to \infty ,f(t)\to 0.\,\!</math>  
:• The ''pdf'' decreases monotonically and is convex.
:• <math>\lambda (t)\equiv \tfrac{1}{{{e}^{\mu }}}\,\!</math>. <math>\lambda (t)\,\!</math> is constant.
:• The mode does not exist.


• As  <math>\mu </math>  increases, the  <math>pdf</math>  gets stretched out to the right and its height decreases, while maintaining its shape.
[[Image:BSpdf2.png|center|400px| ]]


• As <math>\mu </math>  decreases, the <math>pdf</math> shifts towards the left and its height increases.
For <math>0<k<1\,\!</math> :
:• As <math>t\to 0\,\!</math>, <math>f(t)\to \infty .\,\!</math>
:• As <math>t\to \infty ,f(t)\to 0.\,\!</math>
:• As <math>t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\!</math>  
:• The ''pdf'' decreases monotonically and is convex.
:• As <math>\mu \,\!</math> increases, the ''pdf'' gets stretched out to the right and its height decreases, while maintaining its shape.
:• As <math>\mu \,\!</math> decreases, the ''pdf'' shifts towards the left and its height increases.
:• The mode does not exist.


• The mode does not exist.
[[Image:BSpdf3.png|center|400px| ]]


[[Image:ldaGD10.3.gif|thumb|center|400px| ]]
==Confidence Bounds==
The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in the [[Confidence Bounds]] chapter.


===Confidence Bounds===
===Bounds on the Parameters===
The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in Chapter 5.
The lower and upper bounds on the mean, <math>\widehat{\mu }\,\!</math>, are estimated from:  
====Bounds on the Parameters====
The lower and upper bounds on the mean, <math>\widehat{\mu }</math> , are estimated from:  


::<math>\begin{align}
::<math>\begin{align}
   & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\  
   & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\  
  & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)}   
  & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)}   
\end{align}</math>
\end{align}\,\!</math>


 
Since the standard deviation, <math>\widehat{\sigma }\,\!</math>, must be positive, <math>\ln (\widehat{\sigma })\,\!</math> is treated as normally distributed and the bounds are estimated from:  
Since the standard deviation, <math>\widehat{\sigma }</math> , must be positive, <math>\ln (\widehat{\sigma })</math> is treated as normally distributed and the bounds are estimated from:  


::<math>\begin{align}
::<math>\begin{align}
   & {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{{\hat{k}}}}}\text{ (upper bound)} \\  
   & {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{{\hat{k}}}}}\text{ (upper bound)} \\  
  & {{k}_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{\widehat{k}}}}}\text{ (lower bound)}   
  & {{k}_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{\widehat{k}}}}}\text{ (lower bound)}   
\end{align}</math>
\end{align}\,\!</math>


where <math>{{K}_{\alpha }}</math> is defined by:
where <math>{{K}_{\alpha }}\,\!</math> is defined by:


::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})</math>
::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})\,\!</math>


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.
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.


The variances and covariances of <math>\widehat{\mu }</math> and <math>\widehat{k}</math> are estimated from the Fisher matrix, as follows:
The variances and covariances of <math>\widehat{\mu }\,\!</math> and <math>\widehat{k}\,\!</math> are estimated from the Fisher matrix, as follows:


::<math>\left( \begin{matrix}
::<math>\left( \begin{matrix}
Line 149: Line 124:
   {} & {}  \\
   {} & {}  \\
   -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{k}^{2}}}  \\
   -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{k}^{2}}}  \\
\end{matrix} \right)_{\mu =\widehat{\mu },k=\widehat{k}}^{-1}</math>
\end{matrix} \right)_{\mu =\widehat{\mu },k=\widehat{k}}^{-1}\,\!</math>




<math>\Lambda </math> is the log-likelihood function of the gamma distribution, described in Chapter 3 and Appendix C.
<math>\Lambda \,\!</math> is the log-likelihood function of the gamma distribution, described in [[Parameter Estimation]] and [[Appendix:_Log-Likelihood_Equations|Appendix D]]


===Bounds on Reliability===
===Bounds on Reliability===
The reliability of the gamma distribution is:  
The reliability of the gamma distribution is:  


::<math>\widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\widehat{k};{{e}^{\widehat{z}}})</math>
::<math>\widehat{R}(t;\hat{\mu },\hat{k})=1-{{\Gamma }_{I}}(\widehat{k};{{e}^{\widehat{z}}})\,\!</math>


where:
where:


::<math>\widehat{z}=\ln (t)-\widehat{\mu }</math>
::<math>\widehat{z}=\ln (t)-\widehat{\mu }\,\!</math>


The upper and lower bounds on reliability are:
The upper and lower bounds on reliability are:


::<math>{{R}_{U}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{  (upper bound)}</math>
::<math>{{R}_{U}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{  (upper bound)}\,\!</math>


::<math>{{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{  (lower bound)}</math>
::<math>{{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{  (lower bound)}\,\!</math>


where:
where:


::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})</math>
::<math>Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})\,\!</math>


===Bounds on Time===
===Bounds on Time===
The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:
The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:


 
::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z\,\!</math>
::<math>\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z</math>
 


where:
where:


::<math>z=\ln (-\ln (R))\,\!</math>


::<math>z=\ln (-\ln (R))</math>
::<math>Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })\,\!</math>
 
 
::<math>Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })</math>


or:
or:


 
::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })\,\!</math>
::<math>Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })</math>
 


The upper and lower bounds are then found by:
The upper and lower bounds are then found by:


::<math>\begin{align}
::<math>\begin{align}
   & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\  
   & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\  
  & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)}   
  & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)}   
\end{align}</math>
\end{align}\,\!</math>
 
====A Gamma Distribution Example====
Twenty four units were reliability tested and the following life test data were obtained:
 
 
<center><math>\begin{matrix}
  \text{61} & \text{50} & \text{67} & \text{49} & \text{53} & \text{62}  \\
  \text{53} & \text{61} & \text{43} & \text{65} & \text{53} & \text{56}  \\
  \text{62} & \text{56} & \text{58} & \text{55} & \text{58} & \text{48}  \\
  \text{66} & \text{44} & \text{48} & \text{58} & \text{43} & \text{40}  \\
\end{matrix}</math></center>
 
Fitting the gamma distribution to this data, using maximum likelihood as the analysis method, gives the following parameters:
 
::<math>\begin{align}
  & \hat{\mu }= & 7.72E-02 \\
& \hat{k}= & 50.4908 
\end{align}</math>
 
Using rank regression on  <math>X,</math>  the estimated parameters are:
 
::<math>\begin{align}
  & \hat{\mu }= & 0.2915 \\
& \hat{k}= & 41.1726 
\end{align}</math>
 


Using rank regression on  <math>Y,</math>  the estimated parameters are:
==General Example==
 
{{:Gamma Distribution Example}}
::<math>\begin{align}
  & \hat{\mu }= & 0.2915 \\
& \hat{k}= & 41.1726 
\end{align}</math>

Latest revision as of 17:10, 23 December 2015

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Chapter 13: The Gamma Distribution


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Chapter 13  
The Gamma Distribution  

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Available Software:
Weibull++

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More Resources:
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The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. It is not, however, widely used as a life distribution model for common failure mechanisms. The gamma distribution does arise naturally as the time-to-first-fail distribution for a system with standby exponentially distributed backups, and is also a good fit for the sum of independent exponential random variables. The gamma distribution is sometimes called the Erlang distribution, which is used frequently in queuing theory applications, as discussed in [32].

The Gamma Probability Density Function

The pdf of the gamma distribution is given by:

[math]\displaystyle{ f(t)=\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)}\,\! }[/math]

where:

[math]\displaystyle{ \begin{align} z=\ln (t)-\mu \end{align}\,\! }[/math]

and:

[math]\displaystyle{ \begin{align} & {{e}^{\mu }}= \text{scale parameter} \\ & k= \text{shape parameter} \end{align}\,\! }[/math]

where [math]\displaystyle{ 0\lt t\lt \infty \,\! }[/math], [math]\displaystyle{ -\infty \lt \mu \lt \infty \,\! }[/math] and [math]\displaystyle{ k\gt 0\,\! }[/math].

The Gamma Reliability Function

The reliability for a mission of time [math]\displaystyle{ t\,\! }[/math] for the gamma distribution is:

[math]\displaystyle{ \begin{align} R=1-{{\Gamma }_{I}}(k;{{e}^{z}}) \end{align}\,\! }[/math]

The Gamma Mean, Median and Mode

The gamma mean or MTTF is:

[math]\displaystyle{ \overline{T}=k{{e}^{\mu }}\,\! }[/math]

The mode exists if [math]\displaystyle{ k\gt 1\,\! }[/math] and is given by:

[math]\displaystyle{ \tilde{T}=(k-1){{e}^{\mu }}\,\! }[/math]

The median is:

[math]\displaystyle{ \widehat{T}={{e}^{\mu +\ln (\Gamma _{I}^{-1}(0.5;k))}}\,\! }[/math]

The Gamma Standard Deviation

The standard deviation for the gamma distribution is:

[math]\displaystyle{ {{\sigma }_{T}}=\sqrt{k}{{e}^{\mu }}\,\! }[/math]

The Gamma Reliable Life

The gamma reliable life is:

[math]\displaystyle{ {{T}_{R}}={{e}^{\mu +\ln (\Gamma _{1}^{-1}(1-R;k))}}\,\! }[/math]

The Gamma Failure Rate Function

The instantaneous gamma failure rate is given by:

[math]\displaystyle{ \lambda =\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)(1-{{\Gamma }_{I}}(k;{{e}^{z}}))}\,\! }[/math]

Characteristics of the Gamma Distribution

Some of the specific characteristics of the gamma distribution are the following:

For [math]\displaystyle{ k\gt 1\,\! }[/math] :

• As [math]\displaystyle{ t\to 0,\infty\,\! }[/math], [math]\displaystyle{ f(t)\to 0.\,\! }[/math]
• [math]\displaystyle{ f(t)\,\! }[/math] increases from 0 to the mode value and decreases thereafter.
• If [math]\displaystyle{ k\le 2\,\! }[/math] then pdf has one inflection point at [math]\displaystyle{ t={{e}^{\mu }}\sqrt{k-1}(\,\! }[/math] [math]\displaystyle{ \sqrt{k-1}+1).\,\! }[/math]
• If [math]\displaystyle{ k\gt 2\,\! }[/math] then pdf has two inflection points for [math]\displaystyle{ t={{e}^{\mu }}\sqrt{k-1}(\,\! }[/math] [math]\displaystyle{ \sqrt{k-1}\pm 1).\,\! }[/math]
• For a fixed [math]\displaystyle{ k\,\! }[/math], as [math]\displaystyle{ \mu \,\! }[/math] increases, the pdf starts to look more like a straight angle.
• As [math]\displaystyle{ t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\! }[/math]
BSpdf1.png

For [math]\displaystyle{ k=1\,\! }[/math] :

• Gamma becomes the exponential distribution.
• As [math]\displaystyle{ t\to 0\,\! }[/math], [math]\displaystyle{ f(T)\to \tfrac{1}{{{e}^{\mu }}}.\,\! }[/math]
• As [math]\displaystyle{ t\to \infty ,f(t)\to 0.\,\! }[/math]
• The pdf decreases monotonically and is convex.
• [math]\displaystyle{ \lambda (t)\equiv \tfrac{1}{{{e}^{\mu }}}\,\! }[/math]. [math]\displaystyle{ \lambda (t)\,\! }[/math] is constant.
• The mode does not exist.
BSpdf2.png

For [math]\displaystyle{ 0\lt k\lt 1\,\! }[/math] :

• As [math]\displaystyle{ t\to 0\,\! }[/math], [math]\displaystyle{ f(t)\to \infty .\,\! }[/math]
• As [math]\displaystyle{ t\to \infty ,f(t)\to 0.\,\! }[/math]
• As [math]\displaystyle{ t\to \infty ,\lambda (t)\to \tfrac{1}{{{e}^{\mu }}}.\,\! }[/math]
• The pdf decreases monotonically and is convex.
• As [math]\displaystyle{ \mu \,\! }[/math] increases, the pdf gets stretched out to the right and its height decreases, while maintaining its shape.
• As [math]\displaystyle{ \mu \,\! }[/math] decreases, the pdf shifts towards the left and its height increases.
• The mode does not exist.
BSpdf3.png

Confidence Bounds

The only method available in Weibull++ for confidence bounds for the gamma distribution is the Fisher matrix, which is described next. The complete derivations were presented in detail (for a general function) in the Confidence Bounds chapter.

Bounds on the Parameters

The lower and upper bounds on the mean, [math]\displaystyle{ \widehat{\mu }\,\! }[/math], are estimated from:

[math]\displaystyle{ \begin{align} & {{\mu }_{U}}= & \widehat{\mu }+{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (upper bound)} \\ & {{\mu }_{L}}= & \widehat{\mu }-{{K}_{\alpha }}\sqrt{Var(\widehat{\mu })}\text{ (lower bound)} \end{align}\,\! }[/math]

Since the standard deviation, [math]\displaystyle{ \widehat{\sigma }\,\! }[/math], must be positive, [math]\displaystyle{ \ln (\widehat{\sigma })\,\! }[/math] is treated as normally distributed and the bounds are estimated from:

[math]\displaystyle{ \begin{align} & {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{{\hat{k}}}}}\text{ (upper bound)} \\ & {{k}_{L}}= & \frac{\widehat{\sigma }}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}{\widehat{k}}}}}\text{ (lower bound)} \end{align}\,\! }[/math]

where [math]\displaystyle{ {{K}_{\alpha }}\,\! }[/math] is defined by:

[math]\displaystyle{ \alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})\,\! }[/math]

If [math]\displaystyle{ \delta \,\! }[/math] is the confidence level, then [math]\displaystyle{ \alpha =\tfrac{1-\delta }{2}\,\! }[/math] for the two-sided bounds and [math]\displaystyle{ \alpha =1-\delta \,\! }[/math] for the one-sided bounds.

The variances and covariances of [math]\displaystyle{ \widehat{\mu }\,\! }[/math] and [math]\displaystyle{ \widehat{k}\,\! }[/math] are estimated from the Fisher matrix, as follows:

[math]\displaystyle{ \left( \begin{matrix} \widehat{Var}\left( \widehat{\mu } \right) & \widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) \\ \widehat{Cov}\left( \widehat{\mu },\widehat{k} \right) & \widehat{Var}\left( \widehat{k} \right) \\ \end{matrix} \right)=\left( \begin{matrix} -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\mu }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} \\ {} & {} \\ -\tfrac{{{\partial }^{2}}\Lambda }{\partial \mu \partial k} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{k}^{2}}} \\ \end{matrix} \right)_{\mu =\widehat{\mu },k=\widehat{k}}^{-1}\,\! }[/math]


[math]\displaystyle{ \Lambda \,\! }[/math] is the log-likelihood function of the gamma distribution, described in Parameter Estimation and Appendix D

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(t;\hat{\mu },\hat{k})=1-{{\Gamma }_{I}}(\widehat{k};{{e}^{\widehat{z}}})\,\! }[/math]

where:

[math]\displaystyle{ \widehat{z}=\ln (t)-\widehat{\mu }\,\! }[/math]

The upper and lower bounds on reliability are:

[math]\displaystyle{ {{R}_{U}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{-{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (upper bound)}\,\! }[/math]
[math]\displaystyle{ {{R}_{L}}=\frac{\widehat{R}}{\widehat{R}+(1-\widehat{R})\exp (\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})\text{ }}}{\widehat{R}(1-\widehat{R})})}\text{ (lower bound)}\,\! }[/math]

where:

[math]\displaystyle{ Var(\widehat{R})={{(\frac{\partial R}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial R}{\partial \mu })(\frac{\partial R}{\partial k})Cov(\widehat{\mu },\widehat{k})+{{(\frac{\partial z}{\partial k})}^{2}}Var(\widehat{k})\,\! }[/math]

Bounds on Time

The bounds around time for a given gamma percentile (unreliability) are estimated by first solving the reliability equation with respect to time, as follows:

[math]\displaystyle{ \widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z\,\! }[/math]

where:

[math]\displaystyle{ z=\ln (-\ln (R))\,\! }[/math]
[math]\displaystyle{ Var(\widehat{T})={{(\frac{\partial T}{\partial \mu })}^{2}}Var(\widehat{\mu })+2(\frac{\partial T}{\partial \mu })(\frac{\partial T}{\partial \sigma })Cov(\widehat{\mu },\widehat{\sigma })+{{(\frac{\partial T}{\partial \sigma })}^{2}}Var(\widehat{\sigma })\,\! }[/math]

or:

[math]\displaystyle{ Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })\,\! }[/math]

The upper and lower bounds are then found by:

[math]\displaystyle{ \begin{align} & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\ & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)} \end{align}\,\! }[/math]

General Example

24 units were reliability tested, and the following life test data were obtained:

61 50 67 49 53 62
53 61 43 65 53 56
62 56 58 55 58 48
66 44 48 58 43 40

Fitting the gamma distribution to this data, using maximum likelihood as the analysis method, gives the following parameters:

[math]\displaystyle{ \begin{align} & \hat{\mu }= 7.72E-02 \\ & \hat{k}= 50.4908 \end{align}\,\! }[/math]

Using rank regression on [math]\displaystyle{ X,\,\! }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= 0.2915 \\ & \hat{k}= 41.1726 \end{align}\,\! }[/math]

Using rank regression on [math]\displaystyle{ Y,\,\! }[/math] the estimated parameters are:

[math]\displaystyle{ \begin{align} & \hat{\mu }= 0.2915 \\ & \hat{k}= 41.1726 \end{align}\,\! }[/math]
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