Template:Example: 2 Parameter Exponential Distribution RRX

2 Parameter Exponential Distribution RRX

Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters and determine the correlation coefficient estimate, $$\hat{\rho }$$, using rank regression on X.

 Solution

The table constructed in Example 2 applies to this example also. Using the values from this table, we get:


 * $$\begin{align}

\hat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{t}_{i}}{{y}_{i}}-\tfrac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{t}_{i}}\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}}{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,y_{i}^{2}-\tfrac{14}} \\ \\  \hat{b}= & \frac{-927.4899-(630)(-13.2315)/14}{22.1148-{{(-13.2315)}^{2}}/14} \end{align}$$

or:


 * $$\hat{b}=-34.5563$$

and:


 * $$\hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{t}_{i}}}{14}-\hat{b}\frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}}}{14}$$

or:


 * $$\hat{a}=\frac{630}{14}-(-34.5563)\frac{(-13.2315)}{14}=12.3406$$

Therefore:


 * $$\hat{\lambda }=-\frac{1}{\hat{b}}=-\frac{1}{(-34.5563)}=0.0289\text{ failures/hour}$$

and:


 * $$\hat{\gamma }=\hat{a}=12.3406$$

The correlation coefficient is found to be:


 * $$\hat{\rho }=-0.9679$$

Note that the equation for regression on Y is not necessarily the same as that for the regression on X. The only time when the two regression methods yield identical results is when the data lie perfectly on a line. If this were the case, the correlation coefficient would be $$-1$$. The negative value of the correlation coefficient is due to the fact that the slope of the exponential probability plot is negative.

This example can be repeated using Weibull++, choosing two-parameter exponential and rank regression on X (RRX) methods for analysis, as shown below. The estimated parameters and the correlation coefficient using Weibull++ were found to be:


 * $$\begin{array}{*{35}{l}}

\hat{\lambda }= &0.0289 \text{failures/hour} \\ \hat{\gamma}= & 12.3395 \text{hours} \\ \hat{\rho} = &-0.9679 \\ \end{array}$$



The probability plot can be obtained simply by clicking the Plot icon.