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]

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]

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, [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 Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\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]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \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]

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]

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]

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, [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 Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\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]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \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]

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]

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]

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, [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 Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\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]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \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]

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]

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]

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, [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 Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\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]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \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]

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]

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]

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, [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 Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\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]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \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]

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]

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]

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, [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 Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\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]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \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]

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]

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]

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, [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 Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\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]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \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]

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]

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, [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 Chapter 3 and Appendix C.

Bounds on Reliability

The reliability of the gamma distribution is:

[math]\displaystyle{ \widehat{R}(T;\hat{\mu },\hat{k})=1-{{\Gamma }_{1}}(\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]

A Gamma Distribution Example

Twenty four units were reliability tested and the following life test data were obtained:

[math]\displaystyle{ \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]

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]