**6. Applications of the NHLx Model**

*6.1. Heavy-Tailed Data Applications*

Two real data sets taken from [31], namely the theft and claim data, are considered to illustrate the proposed methodology. These data sets are known to have heavy tail features. Table 2 presents the estimation of the tails of several standard distributions, namely the lognormal, Weibull, gamma, and exponential distributions, and the proposed NHLx distribution, taken at several values. The survival function, denoted by *S*(*x*) for all distributions in full generality, determines the tail probabilities at the point *x*.


**Table 2.** Estimation of the tail probabilities of various distributions for the considered data sets.

It is obvious from Table 2 that the NHLx model has a better fit in both data sets, and its corresponding tail probabilities are also fairly high. This means that the proposed distribution is also a heavy-tailed distribution, which was compared to other heavy-tailed distributions and contains more mass at the tail ends than the other distributions considered for comparison.

The rest of the study is devoted to the in-depth analysis of two famous data sets in the literature, highlighting the efficiency of the estimated NHLx model under real-life scenarios.

#### *6.2. Practical Applications*

The first data set contains 65 successive eruptions of the waiting times (in seconds) of the Kiama Blowhole data. It was studied in [32,33]. The second data set is about intensive care unit (ICU) patients for varying time periods of 37 patients. It was analyzed in [34] and, more recently, in [35].

The descriptive measures such as mean, median, skewness, and kurtosis have been computed for both the eruption data and ICU data sets. The results are presented in Table 3.

**Table 3.** Descriptive measures for the two data sets.


From the measures of skewness and kurtosis, it is clear that the data are highly skewed and heavy-tailed. Furthermore, the mean value is larger than the median.

For comparison purposes, we consider some of the most accurate extended Lx models: the WL, EXL, and Lx models.

The MLEs and the corresponding SEs of these models are listed in Table 4.


**Table 4.** MLEs with SEs in parentheses of the considered models for the two data sets.

The measures of goodness of fit are used to verify whether a data set is distributionally compatible with a given model. To judge the accuracy of a model, we use the Cramér– von Mises (W\*), Anderson–Darling (A\*), and Kolmogorov–Smirnov (K-S) statistics (D), along with the K-S *p*-Value related to D. Adequacy measures are widely used to determine which model is best. Here, we traditionally consider the Akaike information criterion (AIC), consistent AIC (CAIC), Bayesian information criterion (BIC), and Hannan–Quinn information criterion (HQIC), which are based on the MLEs of the models. The model with the minimum W\*, A\*, D, AIC, CAIC, BIC, and HQIC value and maximum *p*-Value is chosen as the best one that fits the data. We may refer to [36] for the precise definitions of these measures. Their values for the considered models and the two data sets are collected in Table 5.

From Table 5, it is witnessed that the two data sets have a better fit for the proposed NHLx model than the other three models.



The histogram plots and estimated pdfs of the considered models are reported in Figure 3.

From Figure 3, we see that both histograms exhibit the skewed nature of the two data sets, and the estimated pdf curves depict that the NHLx model is observed to have a better pattern of closeness to the histogram plot when compared to the other three models.

**Figure 3.** Curves of the estimated pdfs of the considered models for the two data sets. (**a**) Eruption data. (**b**) ICU data.

#### **7. Conclusions**

In this paper, we propose a new four-parameter Lomax distribution called the Nadarajah–Haghighi Lomax distribution. It aims to provide a new lower-bounded distribution that combines the functionalities of the Nadarajah–Haghighi and Lomax distributions, and extends the modeling scope of the so-called exponential Lomax distribution. We have derived various properties, including the expression of the probability density, hazard and quantile functions, and diverse kinds of moments. The maximum likelihood method is used for estimating the model parameters. Simulation studies show its effectiveness by considering different sets of parameters. Furthermore, the support of two real data sets is taken to illustrate the applications of the Nadarajah–Haghighi Lomax distribution and it is compared with other Lomax-based distributions. From the obtained results, it is very easy to understand that the Nadarajah–Haghighi Lomax distribution has a better fit than the other Lomax models. The perspectives of new work based on the Nadarajah–Haghighi Lomax distribution are numerous, including:


**Author Contributions:** Conceptualization, V.B.V.N., R.V.V. and C.C.; methodology, V.B.V.N., R.V.V. and C.C.; software, V.B.V.N., R.V.V. and C.C.; validation, V.B.V.N., R.V.V. and C.C.; formal analysis, V.B.V.N., R.V.V. and C.C.; investigation, V.B.V.N., R.V.V. and C.C.; data curation, V.B.V.N., R.V.V. and C.C.; writing—original draft preparation, V.B.V.N., R.V.V. and C.C.; writing—review and editing, V.B.V.N., R.V.V. and C.C.; visualization, V.B.V.N., R.V.V. and C.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We thank the three reviewers and the associate editor for their in-depth comments on the first version of the article.

**Conflicts of Interest:** The authors declare no conflict of interest.
