Gamma Distribution with MLE Solution: Difference between revisions

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{{Reference_Example_Heading1}}
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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.

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Gamma Distribution with MLE Solution

This example compares the MLE solution for the Gamma distribution.


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