**6. Data Analysis**

*6.1. Data Analysis and Discussion Based on Complete Samples*

In this section, we illustrate the empirical importance of the OEHLIEx distribution using three applications on real data. These data are used to compare the fits of the OEHLIEx distribution with some competitive models such as the inverse exponential (IEx), exponential (Ex), exponentiated half-logistic (EHL), generalized half-logistic (GHL), and normal (N) models. For the comparison of the models, we should use the values of LL, and Kolmogorov–Smirnov (K-S) test with its *p*-value.

**The first data set (I)** represents the relief times of twenty patients receiving an analgesic (see, [27]).

**The second data set (II)** represents the strengths of glass fibers (see, [28]).

**The third data set (III)** represents the failure times (in minutes) for a sample of 15 electronic components in an accelerated life test (see, [29]).

Tables 18–20 list the MLEs with their corresponding standard errors (in parentheses), and goodness-of-fit (GoF) measures for the datasets.


**Table 18.** The MLE(s) and GoF statistics for data set I.

The (1 − δ)100% CIs of the parameters *α*, *β*, and *λ* are, respectively, [0, 42.752], [0, 0.413], and [0, 0.943].

### **Table 19.** The MLE(s) and GoF statistics for data set II.


The (1 − δ)100% CIs of the parameters *α*, *β*, and *λ* are, respectively, [1.754, 163.435], [0.006, 0.020], and [0.021, 0.073].


**Table 20.** The MLE(s) and GoF statistics for data set III.

The (1 − δ)100% CIs of the parameters *α*, *β*, and *λ* are, respectively, [0.437, 2.410], [0, 7.803], and [0, 0.694].

> Regarding Tables 18–20, it is clear that the OEHLIEx model is the best model among all tested models. Regarding data set I, it is noted that the TrIEx and EHL models work quite well besides the OEHLIEx model where *p*-value > 0.05, but we always search for the most fitting model. Thus, we recommend using the OEHLIEx model to analyze data set I. Similarly, for dataset III, it is found that the TrIEx, IEx, Ex, and GHL models work quite well besides the OEHLIEx model, but we also recommend utilizing the OEHLIEx model to analyze these data.

> Figures 3–5 show the empirical estimated CDF "ECDF", probability–probability (PP), and fitted PDF "FPDF" plots for data sets I, II, and III, respectively, which support the results of Tables 18–20. Moreover, it is noted that the datasets plausibly came from the OEHLIEx model.

**Figure 3.** The ECDF "left panel", PP "middle panel", FPDF "right panel" plots for data set I.

**Figure 4.** The ECDF "left panel", PP "middle panel", FPDF "right panel" plots for data set II.

**Figure 5.** The ECDF "left panel", PP "middle panel", FPDF "right panel" plots for data set III.

Figures 6–8 show the profiles of the LL function "PLLF" based on data sets I, II, and III.

**Figure 6.** The PLLF for data set I.

**Figure 7.** The PLLF for data set II.

**Figure 8.** The PLLF for data set III.

Regarding Figures 6–8, it clear that the estimators have a unique solution where the profiles of the LL function are unimodal shaped. Figure 9 shows the total time in test (TTT) plots for data sets I, II, and III. It is clear that the datasets suffer from an increasing hazard rate. Thus, the proposed model can be used to model the HRF for these data sets.

**Figure 9.** The TTT plots for data I "left panel", data II "middle panel", and data III "right panel".

Table 21 lists some computational statistics for the three datasets by utilizing the OEHLIEx model.

Based on the model parameters, data sets I and II suffer from under-dispersed phenomena (index of dispersion < 1), whereas data set III suffers from over-dispersion (index of dispersion > 1). Moreover, data sets I, II, and III represent positive-skewed data with a platykurtic shape.


**Table 21.** Some computational statistics for data sets I, II, and III.

### *6.2. Dataset IV: Analysis and Discussion Based on Type-II Right Censored Samples*

The censored data have been obtained from (http://www.biochemia-medica.com, accessed on 6 March 2021). They represent the recovery time of 50 patients suffering from cancer, monthly. The MLEs of the unknown parameters, −*LL*\*, K-S, and *p*-value for the proposed model are given in Table 22.


**Table 22.** The MLE(s) and GoF statistics for data set IV.

Depending on −*LL*\*, K-S, and *p*-values, it is noted that the OEHLIEx model is appropriate to analyze data set IV. The (1 − δ)100% CIs of the parameters *α*, *β*, and *λ* are, respectively, [126.999, 137.201], [0, 0.019], and [0, 0.107]. Figure 10 shows the TTT and PP plots based on the type-II right censored sample.

**Figure 10.** The TTT "left panel" and PP "right panel" plots for data set IV.

According to Figure 10, it is observed that the proposed model fits the data well. Moreover, the shape of the HRF increases. Table 23 shows some computational statistics for data IV utilizing the OEHLIEx model.



Regarding Table 23, it is noted that the dataset IV suffers from under-dispersed phenomena which are positive skewed and platykurtic shaped.
