The Gamma Distribution

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. [32]

Gamma Probability Density Function
The $$pdf$$  of the gamma distribution is given by:

$$f(T)=\frac{t\Gamma (k)}$$

where:

$$z=\ln (t)-\mu $$

and:

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

where $$00$$. The Gamma Reliability Function The reliability for a mission of time $$T$$  for the gamma distribution is:

$$R=1-{{\Gamma }_{1}}(k;{{e}^{z}})$$

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:

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

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:

$$\lambda =\frac{t\Gamma (k)(1-{{\Gamma }_{1}}(k;{{e}^{z}}))}$$

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 $$T\to \infty ,\lambda (T)\to \tfrac{1}.$$

For $$k=1$$ :

•	Gamma becomes the exponential distribution.

•	As $$T\to 0$$ ,  $$f(T)\to \tfrac{1}.$$

•	As $$T\to \infty ,f(T)\to 0.$$

•	The $$pdf$$  decreases monotonically and is convex.

•	 $$\lambda (T)\equiv \tfrac{1}$$. $$\lambda (T)$$ 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 $$T\to \infty ,\lambda (T)\to \tfrac{1}.$$

•	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 Chapter 5.

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:

$$\begin{align} & {{k}_{U}}= & \widehat{k}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{k})}}}}\text{ (upper bound)} \\ & {{k}_{L}}= & \frac{\widehat{\sigma }}\text{ (lower bound)} \end{align}$$

where $${{K}_{\alpha }}$$  is defined by:

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

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 Chapter 3 and Appendix C.

Bounds on Reliability
The reliability of the gamma distribution is:

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

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:

$$\begin{align} & {{T}_{U}}= & \hat{T}+{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Upper bound)} \\ & {{T}_{L}}= & \hat{T}-{{K}_{\alpha }}\sqrt{Var(\hat{T})}\text{ (Lower bound)} \end{align}$$

A Gamma Distribution Example
Twenty four 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:

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