Template:MLE Example (EXP1-2) - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

2012-08-15T06:52:52Z

Redirected page to The Exponential Distribution

← Older revision		Revision as of 06:52, 15 August 2012
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	~~====MLE Parameter Estimation====~~		#REDIRECT [[The_Exponential_Distribution]]

	~~<br>~~

	~~Let's assume six identical units are reliability tested at the same application and operation~~
	~~stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.~~

	~~The parameter of the exponential distribution can also be estimated using the maximum likelihood estimation (MLE) method.~~

	~~This log-likelihood function is:~~

	~~<br>~~
	~~::<math>\ln (L)=\Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left~~[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime }+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]</math>
	~~<br>~~
	~~where:~~
	~~<br>~~
	~~::<math>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}</math>~~

	~~<br>~~
	~~::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \prime }}}</math>~~

	~~<br>~~
	~~and~~
	~~<br>~~
	~~::• <math>{{F}_{e}}</math> is the number of groups of times-to-failure data points.~~
	~~::• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.~~
	~~::• <math>\lambda </math> is the failure rate parameter (unknown a priori, the only parameter to be found).~~
	~~::• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data.~~
	~~::• <math>S</math> is the number of groups of suspension data points.~~
	~~::• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.~~
	~~::• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group.~~
	~~::• <math>FI</math> is the number of interval data groups.~~
	~~::• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.~~
	~~::• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.~~
	~~::• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.~~

	~~<br>~~
	~~The solution will be found by solving for a parameter <math>\widehat{\lambda }</math> so that <math>\tfrac{\partial \Lambda }{\partial \lambda }=0</math> where:~~
	~~<br>~~
	::<math>\frac{\partial \Lambda }{\partial \lambda }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{1}{\lambda }-{{T}_{i}} \right)-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }T_{i}^{\prime }-\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}</math>

	~~<br>~~

	~~For the given data then the equation is~~
	~~<br>~~
	~~::<math>\frac{\partial \Lambda }{\partial \lambda }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,\left~~[ ~~\frac{1}{\lambda }-\left( {{T}_{i}} \right) \right~~]~~=\underset{i=1}{\overset{14}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}} \right) \right~~]~~=0</math>~~

	~~<br>~~
	~~Substituting the values for <math>T</math> we get:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~\frac{6}{\lambda }= & 4409,\text{ or:} \\~~
	~~\lambda = & 0.00136\text{ failure/hr}~~
	~~\end{align}</math>~~

Lisa Hacker at 10:43, 20 July 2012

2012-07-20T10:43:20Z

← Older revision		Revision as of 10:43, 20 July 2012
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	====MLE Parameter Estimation====		====MLE Parameter Estimation====

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	\lambda = & 0.00136\text{ failure/hr}		\lambda = & 0.00136\text{ failure/hr}
	\end{align}</math>		\end{align}</math>

	~~[[Category:Examples]]~~

Pantelis: /* Example 2 */

2011-10-29T12:53:34Z

Example 2

← Older revision		Revision as of 12:53, 29 October 2011
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	<br>		<br>

	~~====Example 2====~~		For the given data then the equation is

	~~<br>~~
	~~Using~~ the ~~same~~ data ~~as in~~ the ~~probability plotting example (Example 1), and assuming an exponential distribution, estimate the parameter using the MLE method.~~
	~~<br>~~
	~~''Solution''~~
	~~<br>~~
	~~In this example we have non-grouped data without suspensions. Thus Eqn. (exp-mle) becomes:~~
	<br>		<br>
	::<math>\frac{\partial \Lambda }{\partial \lambda }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}} \right) \right]=\underset{i=1}{\overset{14}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}} \right) \right]=0</math>		::<math>\frac{\partial \Lambda }{\partial \lambda }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}} \right) \right]=\underset{i=1}{\overset{14}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}} \right) \right]=0</math>

Pantelis: /* MLE Parameter Estimation */

2011-10-29T12:52:45Z

MLE Parameter Estimation

← Older revision		Revision as of 12:52, 29 October 2011
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	stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.		stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.

	The parameter of the exponential distribution can also be estimated using the maximum likelihood estimation (MLE) method. This log-likelihood function is:		The parameter of the exponential distribution can also be estimated using the maximum likelihood estimation (MLE) method.

			This log-likelihood function is:

	<br>		<br>

Pantelis: Created page with ' ====MLE Parameter Estimation====
Let's assume six identical units are reliability tested at the same application and operation stress levels. All of these units fail duri…'

2011-10-29T12:52:16Z

Created page with ' ====MLE Parameter Estimation==== Let's assume six identical units are reliability tested at the same application and operation stress levels. All of these units fail duri…'

New page

====MLE Parameter Estimation====

 

Let's assume six identical units are reliability tested at the same application and operation
stress levels. All of these units fail during the test after operating for the following times (in hours), <math>{{T}_{i}}</math> : 96, 257, 498, 763, 1051 and 1744.

The parameter of the exponential distribution can also be estimated using the maximum likelihood estimation (MLE) method. This log-likelihood function is:

 
::<math>\ln (L)=\Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime }+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]</math>
 
where:
 
::<math>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}</math>

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \prime }}}</math>

 
and
 
::• <math>{{F}_{e}}</math> is the number of groups of times-to-failure data points.
::• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.
::• <math>\lambda </math> is the failure rate parameter (unknown a priori, the only parameter to be found).
::• <math>{{T}_{i}}</math> is the time of the <math>{{i}^{th}}</math> group of time-to-failure data.
::• <math>S</math> is the number of groups of suspension data points.
::• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
::• <math>T_{i}^{\prime }</math> is the time of the <math>{{i}^{th}}</math> suspension data group.
::• <math>FI</math> is the number of interval data groups.
::• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.
::• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.
::• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.

 
The solution will be found by solving for a parameter <math>\widehat{\lambda }</math> so that <math>\tfrac{\partial \Lambda }{\partial \lambda }=0</math> where:
 
::<math>\frac{\partial \Lambda }{\partial \lambda }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{1}{\lambda }-{{T}_{i}} \right)-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }T_{i}^{\prime }-\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}</math>

 

====Example 2====

 
Using the same data as in the probability plotting example (Example 1), and assuming an exponential distribution, estimate the parameter using the MLE method.
 
''Solution''
 
In this example we have non-grouped data without suspensions. Thus Eqn. (exp-mle) becomes:
 
::<math>\frac{\partial \Lambda }{\partial \lambda }=\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}} \right) \right]=\underset{i=1}{\overset{14}{\mathop \sum }}\,\left[ \frac{1}{\lambda }-\left( {{T}_{i}} \right) \right]=0</math>

 
Substituting the values for <math>T</math> we get:

 
::<math>\begin{align}
\frac{6}{\lambda }= & 4409,\text{ or:} \\
\lambda = & 0.00136\text{ failure/hr}
\end{align}</math>

[[Category:Examples]]

Template:MLE Example (EXP1-2) - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution

Lisa Hacker at 10:43, 20 July 2012

Pantelis: /* Example 2 */

Pantelis: /* MLE Parameter Estimation */

Pantelis: Created page with ' ====MLE Parameter Estimation==== Let's assume six identical units are reliability tested at the same application and operation stress levels. All of these units fail duri…'

Pantelis: Created page with ' ====MLE Parameter Estimation====
Let's assume six identical units are reliability tested at the same application and operation stress levels. All of these units fail duri…'