**1. Introduction**

Appropriate data modeling is believed to provide greater insight into the data, divulging its properties and allowing for tracking its characteristics. Consequently, there is a potential for developing efficient methods for clearer grasp of real-world occurrences. We developed a coherent model to help meet the aspirations of applied practitioners in a wide range of scientific domains, inspired by the application of theoretical probability models in applied research. Tahir and Nadarajah [1] provided a deep review of novel approaches that can be adopted to develop new generalized classes ("G-classes" for short) of distributions. In parallel to G-classes, Tahir and Cordiero [2] presented a review on compounding univariate distributions, their expansions, and classes to detect anomaly scenarios under series and parallel structures. In the current article, we adopted the approach extensively discussed in Section 7 of [2], by integrating two continuous cumulative distribution functions (cdfs) together. Cordeiro et al. [3] initiated this idea and proposed the Exponential-Weibull distribution. In the same vein, we proposed minimum Lindley Lomax (minLLx) distribution by compounding the Lindley and Lomax distributions.

The Lindley (L) and Lomax (Lx) distributions are indispensable models for characterizing data, notably in engineering, for the replacement and maintenance of various goods, systems, and reliability processes. For the stated reason, researchers have found ample evidence of studies that conformed to these distributions, namely, Ghitany et al. [4], Ramos and Louzada [5], Singh et al. [6], Oguntunde et al. [7], Wei et al. [8], and Elgarhy et al. [9], just to mention a few. It is an intriguing fact that both the Lindley and the Lomax distributions emerged from an extension of the exponential model, which is commonly used

**Citation:** Khan, S.; Hamedani, G.G.; Reyad, H.M.; Jamal, F.; Shafiq, S.; Othman, S. The Minimum Lindley Lomax Distribution: Properties and Applications. *Math. Comput. Appl.* **2022**, *27*, 16. https://doi.org/ 10.3390/mca27010016

Academic Editor: Sandra Ferreira

Received: 6 January 2022 Accepted: 15 February 2022 Published: 18 February 2022

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to quantify the lifetime of a process or device. Assume that a system comprises of two sub-systems that are operating in tandem at the same time, and that the system will collapse if the first sub-system falters. Let us assume further that the failure times of subsystems follow the Lindley and Lomax distributions with *Y* and *Z* independent variables having cdfs, respectively, as follows

$$G(y) = 1 - \left(\frac{1+\theta+\theta y}{1+\theta}\right)e^{-\theta y}, \ y \ge 0, \ \theta > 0$$

$$H(z) = 1 - (1+\lambda z)^{-\beta}, \ z \ge 0, \ \lambda, \ \beta > 0.$$

Then, the new arbitrary variable (av) *X* = min(*Y*, *Z*) will be called the min Lindley Lomax (minLLx) to determine the system's failure mechanism. The cdf of the minLLx av is follows as

$$F(\mathbf{x}) = 1 - \frac{e^{-\theta \mathbf{x}}}{\left(1 + \lambda \mathbf{x}\right)^{\beta}} \left(\frac{1 + \theta + \theta \mathbf{x}}{1 + \theta}\right), \qquad \mathbf{x} \ge \mathbf{0}, \quad \theta, \lambda, \beta > 0. \tag{1}$$

The probability density function (pdf), survival function (sf), and hazard rate function (hrf) in harmony with Equation (1) are given, respectively, by

$$f(\mathbf{x}) = \frac{e^{-\theta \mathbf{x}}}{(1+\theta)(1+\lambda \mathbf{x})^{\beta+1}} \Big[ \lambda \theta (1+\theta + \theta \mathbf{x}) + \theta^2 (1+\mathbf{x})(1+\lambda \mathbf{x}) \Big], \text{ x > 0, } \theta, \lambda, \ \beta > 0 \,, \tag{2}$$

$$S(\mathbf{x}) = \frac{e^{-\theta \mathbf{x}}}{(1+\lambda \mathbf{x})^{\beta}} \left( \frac{1+\theta+\theta \mathbf{x}}{1+\theta} \right)$$

and

$$h(\mathbf{x}) = \frac{\lambda \beta (1 + \theta + \theta \mathbf{x}) + \theta^2 (1 + \mathbf{x})(1 + \lambda \mathbf{x})}{(1 + \lambda \mathbf{x})(1 + \theta + \theta \mathbf{x})}, \quad \mathbf{x} > 0. \tag{3}$$

From now on, an av *X*~minLLx (*θ*, *λ*, *β*) with a pdf is defined by Equation (2).

The purpose of this research is to present and explore the mathematical configurations of a newly developed three-parameter distribution, the minimum Lindley Lomax model, in the perspective of compounding. The rest of the article is composed of seven main components. The minLLx model's essential mathematical features are examined in Section 2. Specific characterizations of the new distribution are pursued in Section 3. The minLLx model's maximum likelihood estimates and observed information matrix are established in Section 4. In Section 5, a simulation study is carried out. Two applications are provided in Section 6. Eventually, in Section 7, there are some closing remarks.

## **2. Structural Properties**

The standard mathematical characteristics of the newly suggested minLLx distribution, as stipulated by the cdf in Equation (1), are explored in this phase. In each subcategory, we report a few explicit results.

## *2.1. Quantile Function*

Let the *p*th quantile of the minLLx distribution, say *xp*, is demarcated by *F*(*xp*) = *p*, such that 0 < *p* < 1. Then the root of

$$\mathbf{x}\_p = \frac{1}{\lambda} \left\{ \left[ \frac{(1+\theta)(1-p)e^{\theta \mathbf{x}\_p}}{1+\theta+\theta \mathbf{x}\_p} \right]^{-1/\beta} - 1 \right\}. \tag{4}$$

#### *2.2. The Shape of the minLLx Distribution*

Mathematically, the forms of the minLLx distribution's density and hazard functions can be defined. The acute points of the density function are the roots of the following equation:

$$\frac{-\lambda(1+\beta)}{1+\lambda x} + \left\{ \frac{\theta[\lambda \beta + 2\theta(1+\lambda x)]}{\lambda \beta(1+\theta+\theta x) + \theta^2(1+x)(1+\lambda x)} \right\} = 0.$$

Furthermore, the acute points of the hazard function are the roots of the following equation:

$$\left\{ \frac{\theta[\lambda\beta + 2\theta(1 + \lambda x)]}{\lambda\beta(1 + \theta + \theta x) + \theta^2(1 + x)(1 + \lambda x)} \right\} - \frac{\lambda}{1 + \lambda x} - \frac{\theta}{1 + \theta + \theta x} = 0.$$

The density and hazard functions are visualized in Figures 1 and 2, respectively. The density function has a reverse-J and right-skewed shape with different peeks, while hrf can sometimes be a monotonic (increasing or decreasing), non-monotonic (bathtub), or constant in shape. The standard L and Lx statistical distributions can only create two shapes, whereas the minLLx model can produce a wide number of shapes based on the power parameter beta.

**Figure 1.** Possible figures of the minLLx pdf for parameter values chosen at random.

**Figure 2.** Possible figures of the minLLx hrf for parameter values chosen at random.
