Generalized Gamma Distribution Example: Difference between revisions
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{ | <noinclude>{{Banner Weibull Examples}} | ||
''This example appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference]''.</noinclude> | |||
The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test, as discussed in Lawless [[Appendix:_Life_Data_Analysis_References|[21]]]. | |||
::<math>\begin{array}{*{35}{l}} | ::<math>\begin{array}{*{35}{l}} | ||
| Line 11: | Line 8: | ||
\text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84} \\ | \text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84} \\ | ||
\text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\ | \text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\ | ||
\end{array}</math> | \end{array}\,\!</math> | ||
When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained: | When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained: | ||
| Line 20: | Line 16: | ||
& \widehat{\sigma }= & 0.509982 \\ | & \widehat{\sigma }= & 0.509982 \\ | ||
& \widehat{\lambda }= & 0.307639 | & \widehat{\lambda }= & 0.307639 | ||
\end{align}</math> | \end{align}\,\!</math> | ||
Note that for this data, the generalized gamma offers a compromise between the Weibull | Note that for this data, the generalized gamma offers a compromise between the Weibull <math>(\lambda =1),\,\!</math> and the lognormal <math>(\lambda =0)\,\!</math> distributions. The value of <math>\lambda \,\!</math> indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on <math>\lambda .\,\!</math> For example, the 90% two-sided confidence bounds are: | ||
::<math>\begin{align} | ::<math>\begin{align} | ||
& {{\lambda }_{u}}= & -0.592087 \\ | & {{\lambda }_{u}}= & -0.592087 \\ | ||
& {{\lambda }_{u}}= & 1.20736 | & {{\lambda }_{u}}= & 1.20736 | ||
\end{align}</math> | \end{align}\,\!</math> | ||
We can then conclude that both distributions (i.e., Weibull and lognormal) are well supported by the data, with the lognormal being the better supported of the two. | |||
In Weibull++ the generalized gamma probability is plotted on gamma probability paper, as shown next. | In Weibull++ the generalized gamma probability is plotted on a gamma probability paper, as shown next. | ||
[[Image:GGamma Example 1 Plot.png | [[Image:GGamma Example 1 Plot.png|center|500px| ]] | ||
It is important to | It is also important to note that, as in the case of the mixed Weibull distribution, the choice of regression analysis (i.e., RRX or RRY) is of no consequence in the generalized gamma model because it uses non-linear regression. | ||
Latest revision as of 21:40, 18 September 2023
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This example appears in the Life data analysis reference.
The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test, as discussed in Lawless [21].
- [math]\displaystyle{ \begin{array}{*{35}{l}} \text{17}\text{.88} & \text{28}\text{.92} & \text{33} & \text{41}\text{.52} & \text{42}\text{.12} & \text{45}\text{.6} & \text{48}\text{.4} & \text{51}\text{.84} & \text{51}\text{.96} & \text{54}\text{.12} \\ \text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84} \\ \text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {} \\ \end{array}\,\! }[/math]
When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:
- [math]\displaystyle{ \begin{align} & \widehat{\mu }= & 4.23064 \\ & \widehat{\sigma }= & 0.509982 \\ & \widehat{\lambda }= & 0.307639 \end{align}\,\! }[/math]
Note that for this data, the generalized gamma offers a compromise between the Weibull [math]\displaystyle{ (\lambda =1),\,\! }[/math] and the lognormal [math]\displaystyle{ (\lambda =0)\,\! }[/math] distributions. The value of [math]\displaystyle{ \lambda \,\! }[/math] indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on [math]\displaystyle{ \lambda .\,\! }[/math] For example, the 90% two-sided confidence bounds are:
- [math]\displaystyle{ \begin{align} & {{\lambda }_{u}}= & -0.592087 \\ & {{\lambda }_{u}}= & 1.20736 \end{align}\,\! }[/math]
We can then conclude that both distributions (i.e., Weibull and lognormal) are well supported by the data, with the lognormal being the better supported of the two. In Weibull++ the generalized gamma probability is plotted on a gamma probability paper, as shown next.
It is also important to note that, as in the case of the mixed Weibull distribution, the choice of regression analysis (i.e., RRX or RRY) is of no consequence in the generalized gamma model because it uses non-linear regression.