The Gamma Distribution

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Chapter 13: The Gamma Distribution



Index

Chapter 13  
The Gamma Distribution  

Contents

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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:

f(t)=\frac{{{e}^{kz-{{e}^{z}}}}}{t\Gamma (k)}\,\!

where:

\begin{align}
z=\ln (t)-\mu 
\end{align}\,\!

and:

\begin{align}
  & {{e}^{\mu }}=  \text{scale parameter} \\ 
 & k=  \text{shape parameter}  
\end{align}\,\!

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

The Gamma Reliability Function

The reliability for a mission of time t\,\! for the gamma distribution is:

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

The Gamma Mean, Median and Mode

The gamma mean or MTTF is:

\overline{T}=k{{e}^{\mu }}\,\!

The mode exists if k>1\,\! and is given by:

\tilde{T}=(k-1){{e}^{\mu }}\,\!

The median is:

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

The Gamma Standard Deviation

The standard deviation for the gamma distribution is:

{{\sigma }_{T}}=\sqrt{k}{{e}^{\mu }}\,\!

The Gamma Reliable Life

The gamma reliable life is:

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

The Gamma Failure Rate Function

The instantaneous gamma failure rate is given by:

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

Characteristics of the Gamma Distribution

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

For k>1\,\! :

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

For k=1\,\! :

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

For 0<k<1\,\! :

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

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, \widehat{\mu }\,\!, are estimated from:

\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}\,\!

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

\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}\,\!

where {{K}_{\alpha }}\,\! is defined by:

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

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

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

\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}\,\!


\Lambda \,\! 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:

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

where:

\widehat{z}=\ln (t)-\widehat{\mu }\,\!

The upper and lower bounds on reliability are:

{{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)}\,\!
{{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)}\,\!

where:

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})\,\!

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:

\widehat{T}(\widehat{\mu },\widehat{\sigma })=\widehat{\mu }+\widehat{\sigma }z\,\!

where:

z=\ln (-\ln (R))\,\!
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 })\,\!

or:

Var(\widehat{T})=Var(\widehat{\mu })+2\widehat{z}Cov(\widehat{\mu },\widehat{\sigma })+{{\widehat{z}}^{2}}Var(\widehat{\sigma })\,\!

The upper and lower bounds are then found by:

\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}\,\!

General Example

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

\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}\,\!

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

\begin{align}
  & \hat{\mu }=  7.72E-02 \\ 
 & \hat{k}=  50.4908  
\end{align}\,\!

Using rank regression on X,\,\! the estimated parameters are:

\begin{align}
  & \hat{\mu }=  0.2915 \\ 
 & \hat{k}=  41.1726  
\end{align}\,\!

Using rank regression on Y,\,\! the estimated parameters are:

\begin{align}
  & \hat{\mu }=  0.2915 \\ 
 & \hat{k}=  41.1726  
\end{align}\,\!
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