Gamma Distribution Example: Difference between revisions

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<noinclude>{{Banner Weibull Examples}}
<noinclude>{{Banner Weibull Examples}}
''This example appears in the [[The_Gamma_Distribution#General_Example|Life Data Analysis Reference book]]''.</noinclude>
''This example appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference]''.</noinclude>


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


<center><math>\begin{matrix}
{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
  \text{61} & \text{50} & \text{67} & \text{49} & \text{53} & \text{62}  \\
|- align="center"
  \text{53} & \text{61} & \text{43} & \text{65} & \text{53} & \text{56}  \\
|61 ||50||67||49||53||62
  \text{62} & \text{56} & \text{58} & \text{55} & \text{58} & \text{48}  \\
|- align="center"
  \text{66} & \text{44} & \text{48} & \text{58} & \text{43} & \text{40}  \\
|53||61||43||65||53||56
\end{matrix}</math></center>
|- align="center"
|62||56||58||55||58||48
|- align="center"
|66||44||48||58||43||40
|}  


Fitting the gamma distribution to this data, using maximum likelihood as the analysis method, gives the following parameters:  
Fitting the gamma distribution to this data, using maximum likelihood as the analysis method, gives the following parameters:  
Line 16: Line 20:
   & \hat{\mu }=  7.72E-02 \\  
   & \hat{\mu }=  7.72E-02 \\  
  & \hat{k}=  50.4908   
  & \hat{k}=  50.4908   
\end{align}</math>
\end{align}\,\!</math>


Using rank regression on <math>X,</math> the estimated parameters are:  
Using rank regression on <math>X,\,\!</math> the estimated parameters are:  


::<math>\begin{align}
::<math>\begin{align}
   & \hat{\mu }=  0.2915 \\  
   & \hat{\mu }=  0.2915 \\  
  & \hat{k}=  41.1726   
  & \hat{k}=  41.1726   
\end{align}</math>
\end{align}\,\!</math>


Using rank regression on <math>Y,</math> the estimated parameters are:  
Using rank regression on <math>Y,\,\!</math> the estimated parameters are:  


::<math>\begin{align}
::<math>\begin{align}
   & \hat{\mu }=  0.2915 \\  
   & \hat{\mu }=  0.2915 \\  
  & \hat{k}=  41.1726   
  & \hat{k}=  41.1726   
\end{align}</math>
\end{align}\,\!</math>

Latest revision as of 21:38, 18 September 2023

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This example appears in the Life data analysis reference.

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

61 50 67 49 53 62
53 61 43 65 53 56
62 56 58 55 58 48
66 44 48 58 43 40

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]