Template:Example: MLE for Exponential Distribution: Difference between revisions

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'''MLE for Exponential Distribution'''
'''MLE for Exponential Distribution'''


Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.
Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.


'''Solution to Example 4'''
'''Solution'''


In this example we have complete data only. The partial derivative of the log-likelihood function, <math>\Lambda ,</math> is given by:
In this example we have complete data only. The partial derivative of the log-likelihood function, <math>\Lambda ,</math> is given by:

Revision as of 18:15, 9 February 2012

MLE for Exponential Distribution

Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.

Solution

In this example we have complete data only. The partial derivative of the log-likelihood function, [math]\displaystyle{ \Lambda , }[/math] is given by:


[math]\displaystyle{ \frac{\partial \Lambda }{\partial \lambda }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}}-\gamma \right) \right]=\underset{i=1}{\overset{14}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}}-\gamma \right) \right]=0 }[/math]


Complete descriptions of the partial derivatives can be found in Appendix. Recall that when using the MLE method for the exponential distribution, the value of [math]\displaystyle{ \gamma }[/math] is equal to that of the first failure time. The first failure occurred at 5 hours, thus [math]\displaystyle{ \gamma =5 }[/math] hours[math]\displaystyle{ . }[/math] Substituting the values for [math]\displaystyle{ T }[/math] and [math]\displaystyle{ \gamma }[/math] we get:

[math]\displaystyle{ \frac{14}{\hat{\lambda }}=560 }[/math]

or:

[math]\displaystyle{ \hat{\lambda }=0.025\text{ failures/hour}. }[/math]


Using Weibull++:

Weibullfolio1.png

The probability plot is:

Weibullfolioplot1.png