**1. Introduction**

Count data modeling is a challenging task in many areas, including, but not limited to, public health, medicine, epidemiology, applied science, sociology, and agriculture. In many situations, the life length of a device cannot be measured on a continuous scale and the survival function is assumed to be a function of a count random variable instead of being a function of a continuous-time random variable. Therefore, discrete distributions are somewhat meaningful to model lifetime data in situations where output may be of a discrete nature. The traditional discrete distributions have limited applicability as models for reliability, failure times, aggregate loss, etc., especially with the count data with overdispersion in which the variance is greater than the mean. This has led to the development of some discrete distributions based on popular continuous models in reliability analysis, actuarial sciences survival analysis, etc. The discretization of continuous distributions has produced many discrete distributions in the last few decades in the statistical literature. However, the quest for a quintessential model remains the crux of the matter in the diverse scientific paradigm.

One of the many approaches to define new models is the discretization of distributions. Until recently, the majority of discrete lifetime distributions have been proposed in the statistical literature by discretizing the survival function *S*(*x*) of continuous lifetime distributions (see the work of authors, for example, in references [1–12]).

The probability mass function (pmf) *P*(*X* = *x*) is defined as follows

*P*(*X* = *x*) = *S*(*x*) − *S*(*x* + 1) *x* = 0, 1, 2, . . .

Away from this method, Afify [12] have introduced and studied a new discrete Lindley distribution by constructing a mixture of discrete analogs to the continuous components used in creating the continuous Lindley distribution.

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In this paper, we propose and study a new probability mass function (pmf), denoted by *px*, by compounding the binomial and the NDL distributions. The basic principle of this method is stated as if *N*(input) and *X*(output) are two random variables denoting the number of particles entering and leaving an attenuator, then the probability functions *p*(*n*) and *f*(*x*) of these two random variables are connected by the binomial decay transformation

$$P(X = x) = \sum\_{n=x}^{\infty} \binom{n}{x} p^x (1-p)^{n-x} p(n); \ \ge = 0, 1, \dots, \infty \tag{1}$$

where 0 ≤ *p* ≤ 1 is the attenuating coefficient which is discussed by Hu et al. [7]. They considered *p*(*n*) as a Poisson distribution with the parameter *λ* > 0, and then they showed that Pr(*X* = *x*) is the Poisson distribution with the parameter *λp*. For clarity, attenuators are electrical devices built to lower the amount of voltage flowing through them without severely compromising the signal's integrity. They serve as a safeguard against systems being exposed to signals with power levels that are too high to be decoded. Déniz [13] introduced uniform Poisson distribution using the idea of Hu et al. [7] by interchanging in Equation (1) the binomial distribution and the discrete uniform distribution and maintaining *P*(*n*) as the Poisson distribution. Some new discrete distributions also are proposed in the literature using the methodology of [7]. Akdo ˘gan et al. [14] proposed uniform-geometric distribution and Co¸skun et al. [15] constructed binomial–discrete Lindley distribution.

The rest of the paper is arranged as follows: Section 2 defines the natural discrete Lindley distribution and proposes the new binomial–natural discrete Lindley distribution with important properties, subsequently. In Section 3, various parameter estimation and simulation studies are given. Section 4 concerns the real data illustration of the findings. In Section 5, some conclusions are provided.

#### **2. Natural Discrete Lindley Distribution**

Recently, Al-Babtain et al. [16] proposed and studied a new natural discrete analog of the continuous Lindley distribution as a mixture of geometric and negative binomial distributions. The new distribution is called natural discrete Lindley (NDL) distribution and it has many interesting properties that make it superior to many other discrete distributions, particularly in analyzing over-dispersed count data. The NDL can be applied in the collective risk models and is competitive with the Poisson distribution to fit automobileclaim-frequency data. Let *N* be a non-negative random variable obtained as a finite mixture of geometric (*p*) and negative binomial (2, *p*) with mixing probabilities *<sup>p</sup> <sup>p</sup>*+<sup>1</sup> and <sup>1</sup> *<sup>p</sup>*+<sup>1</sup> , respectively, then the probability mass function of the NDL distribution is defined as

$$P(N=n) = \frac{p^2}{p+1}(2+n)(1-p)^n \quad ; \quad n = 0, 1, 2, \dots \text{ and } \ p \in (0, 1) \tag{2}$$

One of the most important features of this distribution is that it has a single parameter and it has attractive properties, which makes it suitable for applications not only in insurance settings but also in other fields where over-dispersions are observed. For more details about this distribution, see Al-Babtain et al. [16]. Given the usefulness of NDL, the discrete analogue due to NDL known as the binomial NDL (BNDL) seems to be naturally interesting to explore.

#### *2.1. The Proposed Discrete Analog*

The probability mass function (1) can be expressed as

$$P(X=x) = \sum\_{n=x}^{\infty} P(X=x|N=n)P(N=n)\_n$$

where *P*( *X*|*N* = *n*) has the binomial *b*(*n*, *p*) distribution. Suppose that *N* is the random variable from NDL with parameter *p* given in (2); then, the probability mass function of the discrete random variable *X* is obtained as

$$\begin{split} p\_x(\mathbf{x};p) &= P(\mathbf{X}=\mathbf{x}) = \sum\_{n=x}^{\infty} P(\mathbf{X}=\mathbf{x}|N=n) P(N=n) = \sum\_{n=x}^{\infty} \binom{n}{x} p^x (1-p)^{n-x} \frac{p^2}{p+1} (2+n)(1-p)^n \\ &= \sum\_{n=x-1}^{\infty} \binom{n}{x} p^x (1-p)^{n-x} \frac{p^2}{p+1} (2+n)(1-p)^n = \sum\_{k=0}^{\infty} \binom{x+k}{x} p^x (1-p)^k \frac{p^2}{p+1} (2+x+k)(1-p)^{x+k} \\ &= \frac{p^2}{p+1} \sum\_{k=0}^{\infty} \binom{x+k}{x} p^x (2+x+k)(1-p)^{x+2k} = \frac{(1-p)^x (1+x+2p-p^2)}{(p+1)(2-p)^{x+2}}; \quad \mathbf{x} \\ &= 0, 1, 2, \dots \text{ and } p \in (0,1) \end{split} \tag{3}$$

If *X* has the pmf (3), then it is called a binomial natural discrete Lindley (BNDL) random variable and it is denoted by *X* ∼ BNDL(*p*). For *n* = 0, this means that no particles enter into the attenuator and it will be termed as failure. Consequently, the corresponding cumulative distribution function (cdf) of BNDL distribution is given by

$$F(\mathbf{x}; p) = P(X \le \mathbf{x}) = \sum\_{t=0}^{\mathbf{x}} p\_{\mathbf{x}}(t) = \sum\_{t=0}^{\mathbf{x}} \frac{(1-p)^t (1+t+2p-p^2)}{(p+1)(2-p)^{t+2}} = 1 - \frac{(1-p)^{\mathbf{x}+1} (3+\mathbf{x}+p-p^2)}{(p+1)(2-p)^{\mathbf{x}+2}}.\tag{4}$$

Figure 1 shows the probability mass function (pmf) plots of the proposed distribution for various values of parameter *p*. Thus, the pmf is always a decreasing function, and the new discrete random variable tends to take small values when *p* increases. The stochastic process tends to happen very quickly once the parameter value grows, which is implied quite strongly by the model's behavior. Therefore, the BNDL model is a logical substitute for the traditional exponential distribution to characterize such phenomena. Additionally, the flexibility of the proposed BNDL can be tested for varied count data sources. For example, this model may be helpful for simulating aggregate losses that are typically limited to actuarial data by maximizing the overall garment fit for a particular number of sizes and accommodation rate, crucial to assessing the goodness of the scaling system. Furthermore, it may be helpful to overcome the problem of over-dispersed data in social sciences, as in anthropology where civilizations grew near the existence of a consistent water source, which is necessary for human survival. Figure 2 complements the results of Figure 1.
