**1. Introduction**

Data analysis has become of grea<sup>t</sup> interest in many fields of science such as health sciences, reliability analysis, industry, environmental studies, and others. The requirement of obtaining suitable models and statistical distributions has become essential, since defining new distributions will enable us to better describe and predict phenomenal and experimental data. See for example [1–9], among others.

Recently, several methods of obtaining new distributions from old ones have been developed. Many generalized classes of life time distributions have been discussed in the literature. It has been proven in many papers that the new generalizations are more flexible in modelling and better fit real-life data. These new distributions also have several desirable properties such as the asymptotic behavior of their probability density function and the hazard rate function's monotonicity, which has made them superior to the original distribution. All of this has encouraged authors to work more on developing new lifetime distributions using different generalization methods. Here, we refer to the papers of [10] for the Marshall–Olkin class, [11] for the Beta and Gamma classes, [12] for the odd exponentiated half-logistic-G (OEHL-G) family, [13] for the flexible Weibull class, [14] for the odd log-logistic Lindley class, [15] for the odd Chen class, [16] for the exponentiated odd Chen class, [17] for a new Kumaraswamy generalized class, [18,19] for the extended Gamma and log-Bilal models, respectively, [20] for type I half logistic odd Weibull-G and [21] for the Poisson transmuted-G family, among others.

**Citation:** Eliwa, M.S.; Alshammari, F.S.; Abualnaja, K.M.; El-Morshedy, M. A Flexible Extension to an Extreme Distribution. *Symmetry* **2021**, *13*, 745. https://doi.org/10.3390/sym13050745

Academic Editor: Zhivorad Tomovski

Received: 16 March 2021 Accepted: 19 April 2021 Published: 23 April 2021

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Here, we use the OEHL-G family of distributions to build a new flexible model with three parameters. The cumulative distribution function (CDF) of the OEHL-G family with two positive shape parameters *α* and *λ* can be reported as

$$\Pi(\mathbf{x};\lambda,\Theta,\mathfrak{a}) = \left\{ \frac{1 - e^{\frac{-\lambda}{1 - C(\mathbf{x};\Theta)}}}{1 + e^{\frac{-\lambda}{1 - C(\mathbf{x};\Theta)}}} \right\}^{d}; \mathfrak{x} \in \mathbb{N} \subset \mathbb{R} \tag{1}$$

where *<sup>G</sup>*(*x*; **Θ**) is the CDF of the baseline distribution under consideration (for more details, see Afify et al., 2017). In our study, the baseline CDF is the inverse exponential (IEx) distribution. The IEx model can be utilized to model datasets which have inverted bathtub failure rates (see, [22]), but it lacks model datasets that are highly skewed "asymmetric" (see, [23]). Therefore, it is essential to have a skewness property in the IEx distribution so that it would be able to fit asymmetry in the datasets that are heavily skewed. Hence, our goal is to obtain a generalized distribution of the IEx model such that it will extend the IEx distribution and also add more flexible features to this life-time model. Many authors have proposed some inverted models due to their flexibility in modeling various types of datasets in different fields (for instance, [24,25]). The basic motivations for using the odd exponentiated half-logistic inverse exponential (OEHLIEx) distribution in practice are the following:


### **2. The OEHLIEx Distribution**

A random variable *X* is said to have the IEx distribution with parameter *β* if its CDF *<sup>G</sup>*(*x*; *β*) = *e* −*β x* . Using the CDF of the IEx model in Equation (1), we obtain the CDF of the OEHLIEx distribution, which can be expressed as

$$F(\mathbf{x}; \lambda, \beta, a) = \left\{ 1 - e^{-\lambda \left( \varepsilon^{\frac{\beta}{\mathbf{y}}} - 1 \right)^{-1}} \right\}^{a} \left\{ 1 + e^{-\lambda \left( \varepsilon^{\frac{\beta}{\mathbf{y}}} - 1 \right)^{-1}} \right\}^{-a}; \mathbf{x} > 0,\tag{2}$$

where *α* and *λ* are the positive shape parameters, while *β* is the positive scale parameter. The corresponding probability density function (PDF) to Equation (2) is

$$f(\mathbf{x}; \boldsymbol{\lambda}, \boldsymbol{\beta}, \boldsymbol{a}) = 2a\lambda \boldsymbol{\beta} \mathbf{x}^{-2} e^{-\frac{\beta}{\pi}} e^{-\lambda \left(\boldsymbol{\varepsilon}^{\frac{\beta}{2}} - 1\right)^{-1}} \left\{ 1 - e^{-\frac{\beta}{\pi}} \right\}^{-2} \left\{ 1 - e^{-\lambda \left(\boldsymbol{\varepsilon}^{\frac{\beta}{2}} - 1\right)^{-1}} \right\}^{a-1} \left\{ 1 + e^{-\lambda \left(\boldsymbol{\varepsilon}^{\frac{\beta}{2}} - 1\right)^{-1}} \right\}^{-a-1} . \tag{3}$$

The PDF of the OEHLIEx model can be represented as an infinite mixture of an exponentiated IEx (Exp-IEx) distribution:

$$f(\mathbf{x}; \lambda, \beta, \mathbf{a}) = \sum\_{k,l=0}^{\infty} \varphi\_{k,l} \, \mathbf{r}\_{k+l+1}(\mathbf{x}),\tag{4}$$

where

$$q\_{k,l} = 2a\lambda \sum\_{j,i=0}^{\infty} \frac{(-1)^{j+k+l} (\lambda(i+j+1))^k}{k!(k+l+1)} \binom{-a-1}{i} \binom{a-1}{j} \binom{-k-2}{l}$$

and

$$\pi\_{k+l+1}(x) = \frac{(k+l+1)\beta}{x^2} \ e^{-\frac{(k+l+1)\beta}{x}}$$

represents the Exp-IEx density with the power parameter (*k* + *l* + <sup>1</sup>). The corresponding reliability function to Equation (2) can be obtained as

$$R(\mathbf{x}; \lambda, \beta, a) = 1 - \left\{ 1 - e^{-\lambda \left( e^{\frac{\beta}{\mathbf{x}}} - 1 \right)^{-1}} \right\}^{\mu} \left\{ 1 + e^{-\lambda \left( e^{\frac{\beta}{\mathbf{x}}} - 1 \right)^{-1}} \right\}^{-\mu}; \mathbf{x} > 0. \tag{5}$$

For the proposed model, the hazard rate function (HRF) and its cumulative can be reported, respectively, as

$$h(\mathbf{x};\lambda,\boldsymbol{\beta},a) = \frac{2a\lambda\boldsymbol{\beta}\mathbf{x}^{-2}e^{-\frac{\beta}{\pi}}\left\{1-e^{-\frac{\beta}{\pi}}\right\}^{-2}\left\{1-e^{-\lambda\left(e^{\frac{\beta}{\pi}}-1\right)^{-1}}\right\}^{a-1}\left\{1+\left.e^{-\lambda\left(e^{\frac{\beta}{\pi}}-1\right)^{-1}}\right\}^{-1}\right\}^{-1}}{e^{\lambda\left(e^{\frac{\beta}{\pi}}-1\right)^{-1}}\left(\left\{1+e^{-\lambda\left(e^{\frac{\beta}{\pi}}-1\right)^{-1}}\right\}^{a}-\left\{1-\left.e^{-\lambda\left(e^{\frac{\beta}{\pi}}-1\right)^{-1}}\right\}\right)^{a}}\tag{6}$$

and

$$H(\mathbf{x}; \boldsymbol{\lambda}, \boldsymbol{\beta}, \boldsymbol{a}) = -\log\left(1 - \left\{1 - e^{-\boldsymbol{\lambda}\left(\boldsymbol{e}^{\frac{\boldsymbol{\beta}}{\mathbf{x}}} - 1\right)^{-1}}\right\}^{a} \left\{1 + e^{-\boldsymbol{\lambda}\left(\boldsymbol{e}^{\frac{\boldsymbol{\beta}}{\mathbf{x}}} - 1\right)^{-1}}\right\}^{-a}\right),\tag{7}$$

where the cumulative hazard function is the integral of the hazard function. Figure 1 shows the PDF plots and its HRF for different parameter values. It is immediate that the PDF is unimodal shaped and can be used to discuss right- and left-skewed datasets, whereas the HRF can be increasing, constant, increasing–constant, or unimodal shaped.Therefore, the OEHLIEx distribution can be utilized to analyze various types of data in several practical fields.

**Figure 1.** The PDF "left panel" and HRF "right panel" plots.

### **3. Statistical and Reliability Properties**

*3.1. Quantile Function (QF), Skewness, and Kurtosis*

The QF of the OEHLIEx distribution is given as follows: if *U* has a uniform random variable on *U*(0, <sup>1</sup>), then

$$X\_{lI} = -\beta \left\{ \ln \left( \frac{-\ln \left( 1 - \mu^{\frac{1}{a}} \right) + \ln \left( 1 + \mu^{\frac{1}{a}} \right)}{\lambda - \ln \left( 1 - \mu^{\frac{1}{a}} \right) + \ln \left( 1 + \mu^{\frac{1}{a}} \right)} \right) \right\}^{-1}.\tag{8}$$

Equation (8) can be used to generate a random sample, and the median can be derived at *u* = 0.5. Moreover, it can be used to obtain the skewness and kurtosis, where skewness = X3/4 + X1/4−2X2/4 X3/4− X1/4 and Kurtosis = X3/8− X1/8+ X7/8− X5/8 X6/8− X2/8 . Tables 1–3 report some numerical values of the quantiles, skewness, and kurtosis of the OEHLIEx model using Maple software.


**Table 1.** The quantiles, skewness, and kurtosis for *λ*→∞.

**Table 2.** The quantiles, skewness, and kurtosis for *β*<sup>→</sup>∞.


**Table 3.** The quantiles, skewness, and kurtosis for *α*→∞.


Regarding Tables 1–3, the proposed model is suitable for modelling asymmetric as well as symmetric datasets that are platykurtic shaped.
