*6.2. Precipitation Data Set*

The data set has thirty consecutive values of precipitation (in inches) in the month of March in Minneapolis, as provided by [29] and recently used by [30]. The data are: (0.77, 1.74, 0.81, 1.2, 1.95, 1.2, 0.47, 1.43, 3.37, 2.2, 3, 3.09, 1.51, 2.1, 0.52, 1.62, 1.31, 0.32, 0.59, 0.81, 2.81, 1.87, 1.18, 1.35, 4.75, 2.48, 0.96, 1.89, 0.9, 2.05). The descriptive statistical measures of these data are presented in Table 4.

**Table 4.** Descriptive statistical measures for the precipitation data set.


Based on the information in Table 4, the data are right-skewed and leptokurtic. The MLE, SE, and goodness-of-fit measures of the S-ML model and those of the other models for precipitation data set are given in Tables 5 and 6.

**Table 5.** MLEs, SEs, and goodness-of-fit measures for the precipitation data set with one parameter models.


**Table 6.** MLEs, SEs, and goodness-of-fit measures for the precipitation data set with models having more than one parameter.


We can observe from Table 5 that the S-ML model has the lowest statistics with the highest *p*-value, implying that it delivers a better fit than the other models studied. Comparing the models in Table 6, we can see that the lognormal model gives a better fit, while the S-ML model takes the second place, but with less modeling complexity in terms of the number of parameters. Figure 3 depicts the epdf and ecdf plots of the S-ML model for the precipitation data set.

**Figure 3.** Plots of the (**a**) epdf and (**b**) ecdf of the S-ML model for the precipitation data set.

From Figure 3, it is obvious that the S-ML model captures the histogram's overall pattern and illustrates the comparison of the cdf with the empirical cdf of the S-ML model. The suitable behaviour of the S-ML model is further confirmed by these graphs. Apart from the lognormal model, the S-ML model clearly fits better than the Lindley, ML, S-Expo and S-Lindley, and other models.

#### *6.3. Time between Failure Data Set*

This data set refers to the time between failures for repairable items. It was obtained from [31]. The data are: (1.43, 0.11, 0.71, 0.77, 2.63, 1.49, 3.46, 2.46, 0.59, 0.74, 1.23, 0.94, 4.36, 0.40, 1.74, 4.73, 2.23, 0.45, 0.70, 1.06, 1.46, 0.30, 1.82, 2.37, 0.63, 1.23, 1.24, 1.97, 1.86, 1.17). The descriptive statistical measures of these data are presented in Table 7.



From Table 7, we can observe that the failure time data set is right-skewed and leptokurtic.

The MLE, SE, and goodness-of-fit measures of the S-ML model and those of the other models for the failure time data set are given in Tables 8 and 9.

**Table 8.** MLEs, SEs, and goodness-of-fit measures for the failure time data set with one parameter models.



**Table 9.** MLEs, SEs, and goodness-of-fit measures for the failure time data set with models having more than one parameter.

Tables 8 and 9 show that, for the failure time data set, the S-ML model has the lowest statistics and the highest *p*-value, meaning that it provides a better match than the other models investigated.

Figure 4 depicts the epdf and ecdf plots of the S-ML model for the failure time data set.

**Figure 4.** Plots of the (**a**) epdf and (**b**) ecdf of the S-ML model for the failure time data set.

From Figure 4, it is obvious that the S-ML model captures the histogram's overall pattern and illustrates the comparison of the cdf with the empirical cdf of the S-ML model. The suitable behaviour of the S-ML model is further confirmed by these graphs.
