Exponential Distribution for Grouped Data Example: Difference between revisions

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


Twenty units were reliability tested with the following results:
20 units were reliability tested with the following results:
 


{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|-
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|colspan="2" style="text-align:center"| Table - Life Test Data
|colspan="2" style="text-align:center"| '''Table - Life Test Data'''
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|-align="center"  
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|}  
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8-1. Assuming a two-parameter exponential distribution, estimate the parameters analytically using the MLE method.


8-2. Repeat part 8-1 using Weibull++ (enter the data as grouped data to duplicate the results of 8-1).
1. Assuming a 2-parameter exponential distribution, estimate the parameters by hand using the MLE analysis method.


8-3. Plot the exponential probability vs. time-to-failure using Weibull++.  
2. Repeat the above using Weibull++. (Enter the data as grouped data to duplicate the results.)


8-4. Plot <math>R(t)</math> vs. time using Weibull++.
3. Show the Probability plot for the analysis results.


8-5. Plot the <math>pdf</math> using Weibull++.
4. Show the Reliability vs. Time plot for the results.


8-6. Plot the failure rate vs. time using Weibull++.
5. Show the ''pdf'' plot for the results.


8-7. Estimate the parameters analytically using the RRY method (using grouped ranks).
6. Show the Failure Rate vs. Time plot for the results.
 
7. Estimate the parameters using the rank regression on Y (RRY) analysis method (and using grouped ranks).




'''Solution'''
'''Solution'''


8-1. For the two-parameter exponential distribution and for <math>\hat{\gamma }=100</math> hours (first failure), the partial of the log-likelihood function, <math>\Lambda </math>, becomes:
1. For the 2-parameter exponential distribution and for <math>\hat{\gamma }=100\,\!</math> hours (first failure), the partial of the log-likelihood function, <math>\lambda\,\!</math>, becomes:


::<math> \begin{align}  
::<math> \begin{align}  
Line 56: Line 58:


\end{align}
\end{align}
</math>
\,\!</math>
 


8-2. The data as entered in Weibull++ along with results are shown next.
2. Enter the data in a Weibull++ standard folio and calculate it as shown next.
<br>
<br>
<br>
<br>
[[Image:Exponential Distribution Example 8 Data.png|center|550px|]]
[[Image:Exponential Distribution Example 8 Data.png|center|750px|]]
 
 
3. On the Plot page of the folio, the exponential Probability plot will appear as shown next.
 
[[Image:Exponential Distribution Example 8 Plot.png|center|650px|]]
 
 
4. View the Reliability vs. Time plot.
 
 
[[Image:Exponential Distribution Example 8 Rel Plot.png|center|650px|]]


The exponential probability plot is shown next


[[Image:Exponential Distribution Example 8 Plot.png|center|550px|]]


The '''Plot Type''' drop-down box allows you to select different plot types, shown next.
5. View the ''pdf'' plot.


[[Image:Exponential Distribution Example 8 Plot Type.png|center]]


[[Image:Exponential Distribution Example 8 Pdf Plot.png|center|650px|]]


Select '''Reliability vs. Time'''.




[[Image:Exponential Distribution Example 8 Rel Plot.png|center|550px|]]
6. View the Failure Rate vs. Time plot.


The exponential <math>pdf</math> plot is shown next.


[[Image:Exponential Distribution Example 8 Failure Rate Plot.png|center|650px|]]


[[Image:Exponential Distribution Example 8 Pdf Plot.png|center|550px|]]


The exponential failure rate plot is shown next.
Note that, as described at the beginning of this chapter, the failure rate for the exponential distribution is constant. Also note that the Failure Rate vs. Time plot does show values for times before the location parameter, <math>\gamma \,\!</math>, at 100 hours.




[[Image:Exponential Distribution Example 8 Failure Rate Plot.png|center|550px|]]
7. In the case of grouped data, one must be cautious when estimating the parameters using a rank regression method. This is because the median rank values are determined from the total number of failures observed by time <math>{{T}_{i}}\,\!</math> where <math>i\,\!</math> indicates the group number. In this example, the total number of groups is <math>N=6\,\!</math> and the total number of units is <math>{{N}_{T}}=20\,\!</math>. Thus, the median rank values will be estimated for 20 units and for the total failed units (<math>{{N}_{{{F}_{i}}}}\,\!</math>) up to the <math>{{i}^{th}}\,\!</math> group, for the <math>{{i}^{th}}\,\!</math> rank value. The median ranks values can be found from rank tables or they can be estimated using ReliaSoft's Quick Statistical Reference tool.  




Note that, as described at the beginning of this chapter, the failure rate for the exponential distribution is constant. Also note that the failure rate plot does not exist for times before the location parameter, <math>\gamma </math>, at 100 hours.
For example, the median rank value of the fourth group will be the <math>{{17}^{th}}\,\!</math> rank out of a sample size of twenty units (or 81.945%).  


8-7. In the case of grouped data, one must be cautious when estimating the parameters using a rank regression method. That is because the median rank values are determined from the total number of failures observed by time <math>{{T}_{i}}</math> where <math>i</math> indicates the group number. In this example the total number of groups is <math>N=6</math> and the total number of units is <math>{{N}_{T}}=20</math>. Thus, the median rank values will be estimated for twenty units and for the total failed units (<math>{{N}_{{{F}_{i}}}}</math>) up to the <math>{{i}^{th}}</math> group, for the <math>{{i}^{th}}</math> rank value. The median ranks values can be found from rank tables or they can be estimated using ReliaSoft's Quick Statistical Reference.


For example, the median rank value of the fourth group will be the <math>{{17}^{th}}</math> rank out of a sample size of twenty units (or 81.945%).  
The following table is then constructed.


The following table is then constructed (as in Example 2).


<center><math>\begin{matrix}
<center><math>\begin{matrix}
Line 105: Line 113:
   \text{6} & \text{2} & \text{20} & \text{600} & \text{0}\text{.96594} & \text{-3}\text{.3795} & \text{360000} & \text{11}\text{.4211} & \text{-2027}\text{.7085}  \\
   \text{6} & \text{2} & \text{20} & \text{600} & \text{0}\text{.96594} & \text{-3}\text{.3795} & \text{360000} & \text{11}\text{.4211} & \text{-2027}\text{.7085}  \\
   \sum_{}^{} & {} & {} & \text{2100} & {} & \text{-9}\text{.6476} & \text{910000} & \text{20}\text{.9842} & \text{-4320}\text{.3362}  \\
   \sum_{}^{} & {} & {} & \text{2100} & {} & \text{-9}\text{.6476} & \text{910000} & \text{20}\text{.9842} & \text{-4320}\text{.3362}  \\
\end{matrix}</math></center>
\end{matrix}\,\!</math></center>




Given the values in the table above, calculate <math>\hat{a}</math> and <math>\hat{b}</math>:
Given the values in the table above, calculate <math>\hat{a}\,\!</math> and <math>\hat{b}\,\!</math>:




Line 115: Line 123:
  &  &  \\  
  &  &  \\  
  & \hat{b}= & \frac{-4320.3362-(2100)(-9.6476)/6}{910,000-{{(2100)}^{2}}/6}   
  & \hat{b}= & \frac{-4320.3362-(2100)(-9.6476)/6}{910,000-{{(2100)}^{2}}/6}   
\end{align}</math>
\end{align}\,\!</math>


or:
or:


::<math>\hat{b}=-0.005392</math>
::<math>\hat{b}=-0.005392\,\!</math>


and:
and:


::<math>\hat{a}=\overline{y}-\hat{b}\overline{t}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{t}_{i}}}{N}</math>
::<math>\hat{a}=\overline{y}-\hat{b}\overline{t}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{t}_{i}}}{N}\,\!</math>


or:
or:


::<math>\hat{a}=\frac{-9.6476}{6}-(-0.005392)\frac{2100}{6}=0.2793</math>
::<math>\hat{a}=\frac{-9.6476}{6}-(-0.005392)\frac{2100}{6}=0.2793\,\!</math>


Therefore:
Therefore:




::<math>\hat{\lambda }=-\hat{b}=-(-0.005392)=0.05392\text{ failures/hour}</math>
::<math>\hat{\lambda }=-\hat{b}=-(-0.005392)=0.05392\text{ failures/hour}\,\!</math>




and:
and:


::<math>\hat{\gamma }=\frac{\hat{a}}{\hat{\lambda }}=\frac{0.2793}{0.005392}</math>
::<math>\hat{\gamma }=\frac{\hat{a}}{\hat{\lambda }}=\frac{0.2793}{0.005392}\,\!</math>


or:
or:


::<math>\hat{\gamma }\simeq 51.8\text{ hours}</math>
::<math>\hat{\gamma }\simeq 51.8\text{ hours}\,\!</math>


Then:
Then:


::<math>f(T)=(0.005392){{e}^{-0.005392(T-51.8)}}</math>
::<math>f(T)=(0.005392){{e}^{-0.005392(T-51.8)}}\,\!</math>


Using Weibull++ , the estimated parameters are:
 
Using Weibull++, the estimated parameters are:


::<math>\begin{align}
::<math>\begin{align}
   \hat{\lambda }= & 0.0054\text{ failures/hour} \\  
   \hat{\lambda }= & 0.0054\text{ failures/hour} \\  
   \hat{\gamma }= & 51.82\text{ hours}   
   \hat{\gamma }= & 51.82\text{ hours}   
\end{align}</math>
\end{align}\,\!</math>


The small difference in the values from Weibull++ is due to rounding. In Weibull++ the calculations and the rank values are carried out up to the <math>{{15}^{th}}</math> decimal point.
The small difference in the values from Weibull++ is due to rounding. In the application, the calculations and the rank values are carried out up to the <math>15^{th}\,\!</math> decimal point.
<br>

Latest revision as of 21:46, 18 September 2023

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This example appears in the Life data analysis reference.


20 units were reliability tested with the following results:


Table - Life Test Data
Number of Units in Group Time-to-Failure
7 100
5 200
3 300
2 400
1 500
2 600


1. Assuming a 2-parameter exponential distribution, estimate the parameters by hand using the MLE analysis method.

2. Repeat the above using Weibull++. (Enter the data as grouped data to duplicate the results.)

3. Show the Probability plot for the analysis results.

4. Show the Reliability vs. Time plot for the results.

5. Show the pdf plot for the results.

6. Show the Failure Rate vs. Time plot for the results.

7. Estimate the parameters using the rank regression on Y (RRY) analysis method (and using grouped ranks).


Solution

1. For the 2-parameter exponential distribution and for [math]\displaystyle{ \hat{\gamma }=100\,\! }[/math] hours (first failure), the partial of the log-likelihood function, [math]\displaystyle{ \lambda\,\! }[/math], becomes:

[math]\displaystyle{ \begin{align} \frac{\partial \Lambda }{\partial \lambda }= &\underset{i=1}{\overset{6}{\mathop \sum }}\,{N_i} \left[ \frac{1}{\lambda }-\left( {{T}_{i}}-100 \right) \right]=0\\ \Rightarrow & 7[\frac{1}{\lambda }-(100-100)]+5[\frac{1}{\lambda}-(200-100)] + \ldots +2[\frac{1}{\lambda}-(600-100)]\\ = & 0\\ \Rightarrow & \hat{\lambda}=\frac{20}{3100}=0.0065 \text{fr/hr} \end{align} \,\! }[/math]


2. Enter the data in a Weibull++ standard folio and calculate it as shown next.

Exponential Distribution Example 8 Data.png


3. On the Plot page of the folio, the exponential Probability plot will appear as shown next.

Exponential Distribution Example 8 Plot.png


4. View the Reliability vs. Time plot.


Exponential Distribution Example 8 Rel Plot.png


5. View the pdf plot.


Exponential Distribution Example 8 Pdf Plot.png


6. View the Failure Rate vs. Time plot.


Exponential Distribution Example 8 Failure Rate Plot.png


Note that, as described at the beginning of this chapter, the failure rate for the exponential distribution is constant. Also note that the Failure Rate vs. Time plot does show values for times before the location parameter, [math]\displaystyle{ \gamma \,\! }[/math], at 100 hours.


7. In the case of grouped data, one must be cautious when estimating the parameters using a rank regression method. This is because the median rank values are determined from the total number of failures observed by time [math]\displaystyle{ {{T}_{i}}\,\! }[/math] where [math]\displaystyle{ i\,\! }[/math] indicates the group number. In this example, the total number of groups is [math]\displaystyle{ N=6\,\! }[/math] and the total number of units is [math]\displaystyle{ {{N}_{T}}=20\,\! }[/math]. Thus, the median rank values will be estimated for 20 units and for the total failed units ([math]\displaystyle{ {{N}_{{{F}_{i}}}}\,\! }[/math]) up to the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] group, for the [math]\displaystyle{ {{i}^{th}}\,\! }[/math] rank value. The median ranks values can be found from rank tables or they can be estimated using ReliaSoft's Quick Statistical Reference tool.


For example, the median rank value of the fourth group will be the [math]\displaystyle{ {{17}^{th}}\,\! }[/math] rank out of a sample size of twenty units (or 81.945%).


The following table is then constructed.


[math]\displaystyle{ \begin{matrix} N & {{N}_{F}} & {{N}_{{{F}_{i}}}} & {{T}_{i}} & F({{T}_{i}}) & {{y}_{i}} & T_{i}^{2} & y_{i}^{2} & {{T}_{i}}{{y}_{i}} \\ \text{1} & \text{7} & \text{7} & \text{100} & \text{0}\text{.32795} & \text{-0}\text{.3974} & \text{10000} & \text{0}\text{.1579} & \text{-39}\text{.7426} \\ \text{2} & \text{5} & \text{12} & \text{200} & \text{0}\text{.57374} & \text{-0}\text{.8527} & \text{40000} & \text{0}\text{.7271} & \text{-170}\text{.5402} \\ \text{3} & \text{3} & \text{15} & \text{300} & \text{0}\text{.72120} & \text{-1}\text{.2772} & \text{90000} & \text{1}\text{.6313} & \text{-383}\text{.1728} \\ \text{4} & \text{2} & \text{17} & \text{400} & \text{0}\text{.81945} & \text{-1}\text{.7117} & \text{160000} & \text{2}\text{.9301} & \text{-684}\text{.6990} \\ \text{5} & \text{1} & \text{18} & \text{500} & \text{0}\text{.86853} & \text{-2}\text{.0289} & \text{250000} & \text{4}\text{.1166} & \text{-1014}\text{.4731} \\ \text{6} & \text{2} & \text{20} & \text{600} & \text{0}\text{.96594} & \text{-3}\text{.3795} & \text{360000} & \text{11}\text{.4211} & \text{-2027}\text{.7085} \\ \sum_{}^{} & {} & {} & \text{2100} & {} & \text{-9}\text{.6476} & \text{910000} & \text{20}\text{.9842} & \text{-4320}\text{.3362} \\ \end{matrix}\,\! }[/math]


Given the values in the table above, calculate [math]\displaystyle{ \hat{a}\,\! }[/math] and [math]\displaystyle{ \hat{b}\,\! }[/math]:


[math]\displaystyle{ \begin{align} & \hat{b}= & \frac{\underset{i=1}{\overset{6}{\mathop{\sum }}}\,{{t}_{i}}{{y}_{i}}-(\underset{i=1}{\overset{6}{\mathop{\sum }}}\,{{t}_{i}})(\underset{i=1}{\overset{6}{\mathop{\sum }}}\,{{y}_{i}})/6}{\underset{i=1}{\overset{6}{\mathop{\sum }}}\,t_{i}^{2}-{{(\underset{i=1}{\overset{6}{\mathop{\sum }}}\,{{t}_{i}})}^{2}}/6} \\ & & \\ & \hat{b}= & \frac{-4320.3362-(2100)(-9.6476)/6}{910,000-{{(2100)}^{2}}/6} \end{align}\,\! }[/math]

or:

[math]\displaystyle{ \hat{b}=-0.005392\,\! }[/math]

and:

[math]\displaystyle{ \hat{a}=\overline{y}-\hat{b}\overline{t}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{t}_{i}}}{N}\,\! }[/math]

or:

[math]\displaystyle{ \hat{a}=\frac{-9.6476}{6}-(-0.005392)\frac{2100}{6}=0.2793\,\! }[/math]

Therefore:


[math]\displaystyle{ \hat{\lambda }=-\hat{b}=-(-0.005392)=0.05392\text{ failures/hour}\,\! }[/math]


and:

[math]\displaystyle{ \hat{\gamma }=\frac{\hat{a}}{\hat{\lambda }}=\frac{0.2793}{0.005392}\,\! }[/math]

or:

[math]\displaystyle{ \hat{\gamma }\simeq 51.8\text{ hours}\,\! }[/math]

Then:

[math]\displaystyle{ f(T)=(0.005392){{e}^{-0.005392(T-51.8)}}\,\! }[/math]


Using Weibull++, the estimated parameters are:

[math]\displaystyle{ \begin{align} \hat{\lambda }= & 0.0054\text{ failures/hour} \\ \hat{\gamma }= & 51.82\text{ hours} \end{align}\,\! }[/math]

The small difference in the values from Weibull++ is due to rounding. In the application, the calculations and the rank values are carried out up to the [math]\displaystyle{ 15^{th}\,\! }[/math] decimal point.