Template:Example: 2P Weibull Distribution RRY

2P-Weibull Distribution RRY Example

Consider the data in Example 1, where six units were tested to failure and the following failure times were recorded: 16, 34, 53, 75, 93 and 120 hours. Estimate the parameters and the correlation coefficient using rank regression on Y, assuming that the data follow the two-parameter Weibull distribution.

Solution

Construct a table as shown below.

Utilizing the values from the table, calculate $$ \hat{a} $$ and $$ \hat{b} $$ using the following equations:
 * $$ \hat{b} =\frac{\sum\limits_{i=1}^{6}(\ln t_{i})y_{i}-(\sum\limits_{i=1}^{6}\ln t_{i})(\sum\limits_{i=1}^{6}y_{i})/6}{ \sum\limits_{i=1}^{6}(\ln t_{i})^{2}-(\sum\limits_{i=1}^{6}\ln t_{i})^{2}/6}

$$


 * $$ \hat{b}=\frac{-8.0699-(23.9068)(-3.0070)/6}{97.9909-(23.9068)^{2}/6} $$

or


 * $$ \hat{b}=1.4301 $$

and:


 * $$ \hat{a}=\overline{y}-\hat{b}\overline{T}=\frac{\sum \limits_{i=1}^{N}y_{i}}{N}-\hat{b}\frac{\sum\limits_{i=1}^{N}\ln t_{i}}{N } $$

or:


 * $$ \hat{a}=\frac{(-3.0070)}{6}-(1.4301)\frac{23.9068}{6}=-6.19935 $$

Therefore:


 * $$ \hat{\beta }=\hat{b}=1.4301 $$

and:


 * $$ \hat{\eta }=e^{-\frac{\hat{a}}{\hat{b}}}=e^{-\frac{(-6.19935)}{ 1.4301}} $$

or:


 * $$ \hat{\eta }=76.318\text{ hr} $$

The correlation coefficient can be estimated as:


 * $$ \hat{\rho }=0.9956 $$

This example can be repeated in the Weibull++ software, as shown next. The following picture shows a Weibull++ standard folio data sheet calculated with the 2P-Weibull distribution and rank regression on Y.



The following plot shows the Weibull probability plot for the data set (with 90% two-sided confidence bounds).



If desired, the Weibull $$pdf$$ representing the data set can be written as:


 * $$ f(t)={\frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{t}{\eta }}\right) ^{\beta }} $$

or:


 * $$ f(t)={\frac{1.4302}{76.317}}\left( {\frac{t}{76.317}}\right) ^{0.4302}e^{-\left( {\frac{t}{76.317}}\right) ^{1.4302}} $$

You can also plot this result by selecting Pdf Plot on the Plot Type drop-down list on the control panel.



From this point on, different results, reports and plots can be obtained.