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]

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}$$