Gamma Distribution with MLE Solution: Difference between revisions

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{{Reference Example}}
{{Reference Example}}


This example compares the MLE solution for the Gamma distribution.
This example validates the calculations for the MLE solution for the Gamma distribution in Weibull++ standard folios.




{{Reference_Example_Heading1}}
{{Reference_Example_Heading1}}


The data from Table 1.1 on page 4 in the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.
The data set from Table 1.1 on page 4 in the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.




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In Weibull++, the parameter <math>\mu\,\!</math> is used instead of <math>\theta \,\!</math>. The relation between them is <math>\theta = exp(\mu)\,\!</math>.
In Weibull++, the parameter <math>\mu\,\!</math> is used instead of <math>\theta \,\!</math>. The relation between them is <math>\theta = exp(\mu)\,\!</math>.
The following picture shows that in Weibull++, <math>\mu\,\!</math> = 2.8873. Therefore, <math>\theta = exp(\mu)\,\!</math> = 17.94, which is the same as the results in the book.


[[Image:Gamma_MLE.png|center]]
[[Image:Gamma_MLE.png|center]]
We can see <math>\mu\,\!</math> = 2.8873. Therefore, <math>\theta = exp(\mu)\,\!</math> = 17.94, which is the same as the results in the book.

Latest revision as of 16:19, 28 September 2015

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Gamma Distribution with MLE Solution

This example validates the calculations for the MLE solution for the Gamma distribution in Weibull++ standard folios.


Reference Case

The data set from Table 1.1 on page 4 in the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.


Data

Failures
17.88 28.92 33
41.52 42.12 45.6
48.4 51.84 51.96
54.12 55.56 67.8
68.64 68.64 68.88
84.12 93.12 98.64
105.12 105.84 127.92
128.04 173.4


Result

The solution is given on page 257, Example 11.1. The ML estimates are [math]\displaystyle{ \theta \,\! }[/math] = 17.94, k = 4.025.


Results in Weibull++

In Weibull++, the parameter [math]\displaystyle{ \mu\,\! }[/math] is used instead of [math]\displaystyle{ \theta \,\! }[/math]. The relation between them is [math]\displaystyle{ \theta = exp(\mu)\,\! }[/math].

The following picture shows that in Weibull++, [math]\displaystyle{ \mu\,\! }[/math] = 2.8873. Therefore, [math]\displaystyle{ \theta = exp(\mu)\,\! }[/math] = 17.94, which is the same as the results in the book.

Gamma MLE.png
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