Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

Syed T. R. Rizvi; Sana Ghafoor; Aly R. Seadawy; Ahmed H. Arnous; Hakim AL Garalleh; Nehad Ali Shah; Syed T. R. Rizvi; Sana Ghafoor; Aly R. Seadawy; Ahmed H. Arnous; Hakim AL Garalleh; Nehad Ali Shah

doi:10.3934/math.20241027

AIMS Mathematics

2024, Volume 9, Issue 8: 21144-21176. doi: 10.3934/math.20241027

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Research article Special Issues

Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

1.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
2.
Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, 41411, Kingdom of Saudi Arabia
3.
Department of Engineering Mathematics and Physics, Higher Institute of Engineering, El-Shorouk Academy-11837, Egypt
4.
Department of Mathematical Science, College of Engineering, University of Business and Technology, Jeddah 21361, Saudi Arabia
5.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, 602105, Tamilnadu, India

Received: 29 February 2024 Revised: 01 June 2024 Accepted: 05 June 2024 Published: 01 July 2024
MSC : 35A09, 35A24, 35C08

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

Keywords:

Citation: Syed T. R. Rizvi, Sana Ghafoor, Aly R. Seadawy, Ahmed H. Arnous, Hakim AL Garalleh, Nehad Ali Shah. Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model[J]. AIMS Mathematics, 2024, 9(8): 21144-21176. doi: 10.3934/math.20241027

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Abstract

1. Introduction

The count data sets emerge in various fields like the yearly number of destructive earthquakes, number of patients of a specific disease in a hospital ward, failure of machines, number of patients due to coronavirus, number of monthly traffic accidents, hourly bacterial growth, and so on. Various discrete probability models have been utilized to model these kinds of data sets. Poisson and negative binomial distributions are frequently for modeling count observations. On the other hand, in the advanced scientific eon, the data generated from different fields is getting complex day by day, however, existing discrete models do not provide an efficient fit.

Discretization of continuous distribution can be applied by using different approaches (survival discretization-mixed-Poisson-infinite series). The most widely used technique is the survival discretization approach by [1]. One of the important virtues of this methodology is that the generated discrete model retains the same functional form of the survival function as that of its continuous counterpart. Due to this feature, many survival characteristics of the distribution remain unchanged. The discretization approach to any continuous model depends on the domain of the random variable $X$ . In literature, several discrete distributions are introduced and studied based on the discretization survival function. See for example, discrete Rayleigh [2], discrete a mixture of gamma and exponential [3], discrete Burr and Pareto [4], discrete Rayleigh generator [5], discrete Lindley [6], discrete Burr-XII [7], discrete Burr III [8], two-parameter discrete Lindley [9], discrete log-logistic [10], discrete Gompertz [11], discrete alpha power inverse Lomax [12], discrete Poisson and Ailamujia [13], discrete Half-Logistic [14], discrete Marshall-Olkin Weibull [15], discrete Gompertz-G family [16], discrete Burr-Hatke [17], three-parameter discrete Lindley [18], new discrete Lindley [19], exponentiated discrete Lindley [20], discrete generalized Lindley [21], discrete Gumble [22], discrete inverted Topp-Leone [23], and references cited therein.

Although various distributions are available in literate to analyze count observations, there is still a need to introduce a more flexible and suitable distribution under different conditions. The fundamental purpose of this paper is to propose discrete Ramos-Louzada distribution, which is a one-parameter lifetime distribution introduced by [24]. The proposed one-parameter distribution herein has distinctive properties which makes it among the best choice for modeling over-dispersed and positively skewed data with leptokurtic-shaped. A continuous random variable $X$ is said to have Ramos-Louzada distribution if its probability density function (pdf) can be written as

$g\left(x;\lambda \right) = \frac{1}{{\lambda }^{2}\left(\lambda -1\right)}\left({\lambda }^{2}-2\lambda +x\right){e}^{-\frac{x}{\lambda }}; x\ge 0, \lambda \ge 2,$

(1)

where λ is the shape parameter. The corresponding survival function (sf) to Eq (1) can be formulated as

$G\left(x;\lambda \right) = \frac{{\lambda }^{2}-\lambda +x}{\lambda \left(\lambda -1\right)}{e}^{-\frac{x}{\lambda }}; x\ge 0, \lambda \ge 2.$

(2)

In this article, the discrete version of Ramos and Louzada distribution is proposed and studied in detail. The following are some interesting features of the proposed distribution: Its statistical and reliability characteristics can be expressed as closed forms. Its failure rate is showing an increasing pattern. The suggested distribution evaluated time and count data sets more effectively than competing distributions. As a result, we feel that the proposed model is the greatest option for attracting a wider range of applications and industries.

The rest of the study is organized as follows: In Section 2, we introduce a new distribution using survival discretization methodology. Different mathematical properties are derived in Section 3. Parameter estimation and simulation study are presented in Section 4. Four data sets are utilized to show the flexibility of the proposed model in Section 5. Finally, Section 6 provides some conclusions.

2. Synthesis of discrete Ramos-Louzada distribution

Let $Y$ be a continuous random variable with sf $G\left(y;\boldsymbol{\eta }\right),$ then the pmf of the discrete random variable $X = \left\lfloor Y \right\rfloor$ can be expressed as

$Pr\left(X = x;\boldsymbol{\eta }\right) = G\left(x;\boldsymbol{\eta }\right)-G\left(x+1;\boldsymbol{\eta }\right); x\in {\mathbb{Z}}_{+},$

where $\left\lfloor \cdot \right\rfloor$ denotes the floor function, which returns the highest integer value smaller or equal than its argument, and $\boldsymbol{\eta }$ is a parameter vector $1\times k$ . If the random variable $Y$ have Ramos-Louzada (RL) distribution, then the pmf of discrete RL (DRL) distribution can be written as

$\mathrm{P}\mathrm{r}\left(X = x;\lambda \right) = p\left(x;\lambda \right) = \frac{{e}^{-\frac{x}{\lambda }}}{\lambda \left(\lambda -1\right)}\left[({\lambda }^{2}-\lambda +x)(1-{e}^{-\frac{1}{\lambda }}){- e}^{-\frac{1}{\lambda }}\right]; x = \mathrm{0, 1}, 2, \dots ,$

(3)

where $\lambda \ge 2$ is the shape parameter. The pmf $p\left(x+1;\lambda \right)$ can be expressed as a weight from the pmf $p\left(x;\lambda \right)$ as follows

$p\left(x+1;\lambda \right) = \frac{{e}^{-\frac{1}{\lambda }}\left[\left({\lambda }^{2}-\lambda +x+1\right)\left(1-{e}^{-\frac{1}{\lambda }}\right){-e}^{-\frac{1}{\lambda }}\right]}{\left[\left({\lambda }^{2}-\lambda +x\right)\left(1-{e}^{-\frac{1}{\lambda }}\right){-e}^{-\frac{1}{\lambda }}\right]}p\left(x;\lambda \right).$

Figure 1 illustrates some pmf plots of the DRL models based on different values of the model parameter $\lambda .$ It is found that the pmf can be used a positively skewed data with a uni-modal shape.

Figure 1. The pmf plots of the DRL distribution.

DownLoad: Full-Size Img PowerPoint

Based on $F\left(x;\lambda \right) = \mathrm{P}\mathrm{r}\left(X\le x;\lambda \right) = 1-G\left(x;\lambda \right)+\mathrm{P}\mathrm{r}\left(X = x;\lambda \right)$ , the cumulative distribution function (cdf) of the DRL distribution can be formulated as

$F\left(x;\lambda \right) = 1-\frac{{\lambda }^{2}-\lambda +x+1}{\lambda \left(\lambda -1\right)}{e}^{-\frac{x+1}{\lambda }}; x = \mathrm{0, 1}, 2, \dots .$

(4)

The corresponding sf to Eq (4) can be expressed as

$S\left(x;\lambda \right) = \frac{{\lambda }^{2}-\lambda +x}{\lambda \left(\lambda -1\right)}{e}^{-\frac{x}{\lambda }}; x = \mathrm{0, 1}, 2, \dots .$

(5)

The hazard rate function (hrf) of the DRL model is given by

$h\left(x;\lambda \right) = \frac{\left({\lambda }^{2}-\lambda +x\right)\left(1-{e}^{-\frac{1}{\lambda }}\right)-{e}^{-\frac{1}{\lambda }}}{\left({\lambda }^{2}-\lambda +x\right)}; x = \mathrm{0, 1}, 2, \dots ,$

(6)

where $h\left(x;\lambda \right) = \frac{\mathrm{P}\mathrm{r}\left(X = x;\lambda \right)}{1-F\left(x-1;\lambda \right)}.$ Mathematically, the shape of the hrf of the DRL model is always increasing, which makes it an effective statistical tool for modeling data, especially in the engineering and medical fields. Figure 2 shows the hrf plots of the new discrete model based on various values of the model parameter.

Figure 2. The hrf plots of the DRL distribution.

DownLoad: Full-Size Img PowerPoint

The reversed hazard rate function (rhrf) and the second rate of failure are given as

$\stackrel{˘}{r} = \frac{{e}^{-\frac{x}{\lambda }}\left[({\lambda }^{2}-\lambda +x)(1-{e}^{-\frac{1}{\lambda }}){-e}^{-\frac{1}{\lambda }}\right]}{\lambda \left(\lambda -1\right)-\left({\lambda }^{2}-\lambda +x+1\right){e}^{-\frac{x+1}{\lambda }}}; x = \mathrm{0, 1}, 2, \dots$

(7)

and

${r}^{*}\left(x\right) = log \left[\frac{\left({\lambda }^{2}-\lambda +x\right){e}^{\frac{1}{\lambda }}}{{\lambda }^{2}-\lambda +x+1}\right]; x = \mathrm{0, 1}, 2, \dots ,$

(8)

where $\stackrel{˘}{r} = \frac{\mathrm{P}\mathrm{r}\left(X = x;\lambda \right)}{F\left(x;\lambda \right)}$ and ${r}^{*}\left(x\right) = \mathrm{log}\left[\frac{S(x;\lambda )}{S(x+1;\lambda )}\right]$ . Mathematically, and after simple algebra steps, it is found that the shape of the rhrf of the DRL model is decreasing only. Figure 3 shows some rhrf plots of the proposed model based on specific parameter values.

Figure 3. The rhrf plots of the DRL distribution.

DownLoad: Full-Size Img PowerPoint

3. Distributional properties

In this Section, the probability generating function (pgf) as well as its rth moment are investigated. Assume the random variable $X$ have a DRL model, then the pgf can be expressed as

${W}_{X}\left(z\right) = \sum\limits_{x = 0}^{\infty }{z}^{x}\mathrm{P}\mathrm{r}\left(X = x;\lambda \right) = 1+\left(z-1\right)\sum\limits_{x = 1}^{\infty }{z}^{x-1}S\left(x;\lambda \right) \\ = 1+{e}^{-\frac{1}{\lambda }}\left(z-1\right)\left[\frac{1}{\left(1-z{e}^{-\frac{1}{\lambda }}\right)}+\frac{1}{\lambda \left(\lambda -1\right){\left(1-z{e}^{-\frac{1}{\lambda }}\right)}^{2}}\right].$

(9)

On replacing z by ${e}^{z}$ in Eq (9), the moment generating function (mgf) can be written as

${M}_{X}\left(z\right) = 1+{e}^{-\frac{1}{\lambda }}\left({e}^{z}-1\right)\left[\frac{1}{\left(1-{e}^{z}{e}^{-\frac{1}{\lambda }}\right)}+\frac{1}{\lambda \left(\lambda -1\right){\left(1-{e}^{z}{e}^{-\frac{1}{\lambda }}\right)}^{2}}\right].$

(10)

The first four moments around the origin $\left({\mu \text{'}}_{1}, {\mu \text{'}}_{2}, {\mu \text{'}}_{3}, {\mu \text{'}}_{4}\right)$ can be written as

${\mu \text{'}}_{1} = \frac{{\lambda -\lambda }^{2}+\left({\lambda }^{2}-\lambda +1\right){e}^{\frac{1}{\lambda }}}{\lambda \left(\lambda -1\right){\left({e}^{\frac{1}{\lambda }}-1\right)}^{2}},$

${\mu \text{'}}_{2} = \frac{\left(\lambda \left(\lambda -1\right)+1\right){e}^{\frac{2}{\lambda }}+{3e}^{\frac{1}{\lambda }}-\lambda \left(\lambda -1\right)}{\lambda \left(\lambda -1\right){\left({e}^{\frac{1}{\lambda }}-1\right)}^{3}},$

${{\mu }^{\text{'}}}_{3} = \frac{\left[\lambda \left(\lambda -1\right)+1\right]{e}^{\frac{3}{\lambda }}+\left[3\lambda \left(\lambda -1\right)+10\right]{e}^{\frac{2}{\lambda }}-\left[3\lambda \left(\lambda -1\right)-7\right]{e}^{\frac{1}{\lambda }}-\lambda \left(\lambda -1\right)}{\lambda \left(\lambda -1\right){\left({e}^{\frac{1}{\lambda }}-1\right)}^{4}}$

and

${\mu \text{'}}_{4} = \frac{\left[\lambda \left(\lambda -1\right)+1\right]{e}^{\frac{4}{\lambda }}+\left[10\lambda \left(\lambda -1\right)+25\right]{e}^{\frac{3}{\lambda }}+67{e}^{\frac{2}{\lambda }}+\left[10\lambda \left(\lambda -1\right)-3\right]{e}^{\frac{1}{\lambda }}-\lambda \left(\lambda -1\right)}{\lambda \left(\lambda -1\right){\left({e}^{\frac{1}{\lambda }}-1\right)}^{5}}.$

Based on the rth moments, the variance can be expressed as

${\sigma }^{2} = \frac{\left[{\lambda }^{4}-2{\lambda }^{3}+2{\lambda }^{2}-\lambda \right]{e}^{\frac{3}{\lambda }}-\left[2{\lambda }^{4}-4{\lambda }^{3}+2{\lambda }^{2}-1\right]{e}^{\frac{2}{\lambda }}+\left[{\lambda }^{4}-2{\lambda }^{3}+\lambda \right]{e}^{\frac{1}{\lambda }}}{{\lambda }^{2}{\left(\lambda -1\right)}^{2}{\left({e}^{\frac{1}{\lambda }}-1\right)}^{4}}.$

(11)

The dispersion index (di) is defined by variance to mean ratio. The di indicates that the reported model is suitable for under-, equi- or over-dispersed data sets. Using the derived moments, the coefficients skewness and kurtosis can be listed in closed forms. Some numerical computations for mean, variance, di, skewness, and kurtosis based on DRL parameters are listed in Table 1.

Table 1. Mean, variance, di, skewness, and kurtosis of the DRL distribution.

	$\lambda$
Measure	2.0	2.5	3.0	3.5	4.0	4.5	5.0	5.5	8.0
Mean	3.500	3.678	4.014	4.414	4.847	5.299	5.762	6.234	8.652
Variance	8.079	11.80	15.71	20.03	24.82	30.10	35.87	42.13	80.90
di	2.308	3.207	3.913	4.538	5.121	5.680	6.224	6.757	9.351
Skewness	1.395	1.519	1.629	1.708	1.764	1.806	1.837	1.862	1.928
Kurtosis	5.940	6.353	6.821	7.199	7.495	7.727	7.910	8.057	8.484

| Show Table

DownLoad: CSV

According to Table 1, it is noted that the DRL model can be used effectively to model overdispersion data as di is greater than one, which makes it a proper probability tool to discuss actuarial data. Moreover, the new discrete probabilistic model can be utilized to analyze positively skewed data with leptokurtic-shaped.

4. Estimation methods

In this section, six estimation methods are used to estimate the unknown parameter of DRL distribution. The considered estimation methods are maximum likelihood estimation (mle), method of moments (mom), least-squares estimation (lse), Anderson-Darling estimation (ade), Cramer von-Misses estimation (cvme), and maximum product of spacing estimator (mpse).

4.1. Maximum likelihood estimation

Assume a random sample ${x}_{1}, {x}_{2}, \dots , {x}_{n}$ from the DRL model, then the log-likelihood function can be expressed as

$L\left(\lambda |x\right) = -\frac{1}{\lambda }\sum _{i = 1}^{n}{x}_{i}+\sum _{i = 1}^{n}\mathrm{ln}\left[\left({\lambda }^{2}-\lambda +{x}_{i}\right)\left(1-{e}^{-\frac{1}{\lambda }}\right){-e}^{-\frac{1}{\lambda }}\right]-n\mathrm{ln}\lambda -n\mathrm{ln}\left(\lambda -1\right).$

(12)

Differentiating the Eq (12) with respect to the parameter $\lambda$ , we get the non-linear equation as follows

$\frac{\partial L\left(\lambda |x\right)}{\partial \lambda } = \frac{1}{{\lambda }^{2}}\sum _{i = 1}^{n}{x}_{i}+\sum _{i = 1}^{n}\frac{\left[\left(2\lambda -1\right)\left(1-{e}^{-\frac{1}{\lambda }}\right)-\left(1-\frac{1}{\lambda }+\frac{1}{{\lambda }^{2}}+\frac{{x}_{i}}{{\lambda }^{2}}\right){e}^{-\frac{1}{\lambda }}\right]}{\left[\left({\lambda }^{2}-\lambda +{x}_{i}\right)\left(1-{e}^{-\frac{1}{\lambda }}\right){-e}^{-\frac{1}{\lambda }}\right]}-\frac{n}{\lambda }-\frac{n}{\lambda -1},$

(13)

the exact solution of Eq (13) is not easy, so we will maximize it by using optimization approaches, for example, the Newton-Raphson approach using R software.

4.2. Method of moment estimation

Based on the mom definition, we must equate the sample mean to the corresponding population mean, and then solve the non-linear equation for the parameter

$\frac{{\lambda -\lambda }^{2}+\left({\lambda }^{2}-\lambda +1\right){e}^{\frac{1}{\lambda }}}{\lambda \left(\lambda -1\right){\left({e}^{\frac{1}{\lambda }}-1\right)}^{2}} = \frac{1}{n}\sum _{i = 1}^{n}{x}_{i}.$

(14)

To solve Eq (14), the uniroot function should be utilized.

4.3. Least squares estimation

To estimate the parameter minimizing the sum of squares of residuals, a standard approach like the lse should be used. For the estimation of the parameter of DRL distribution, the lse can be obtained by minimizing

$\mathrm{l}\mathrm{s}\mathrm{e}\left(\lambda \right) = \sum _{i = 1}^{n}{\left[1-\frac{\left({\lambda }^{2}-\lambda +{\mathrm{x}}_{i:n}\right)}{\lambda \left(\lambda -1\right)}{e}^{-\frac{{\mathrm{x}}_{i:n}+1}{\lambda }}-\frac{i}{n+2}\right]}^{2},$

(15)

with respect to the parameter $\lambda .$

4.4. Anderson-Darling estimation

The ade of the parameter can be derived by minimizing the following equation

$\mathrm{a}\mathrm{d}\mathrm{e}\left(\lambda \right) = -n-\frac{1}{n}\sum _{i = 1}^{n}{\left({2}_{i}-1\right)\left[\mathrm{log}\left(1-\frac{\left({\lambda }^{2}-\lambda +{\mathrm{x}}_{i:n}\right)}{\lambda \left(\lambda -1\right)}{e}^{-\frac{{\mathrm{x}}_{i:n}+1}{\lambda }}\right)+\mathrm{log}\left(\frac{\left({\lambda }^{2}-\lambda +{\mathrm{x}}_{i:n}\right)}{\lambda \left(\lambda -1\right)}{e}^{-\frac{{\mathrm{x}}_{i:n}+1}{\lambda }}\right)\right]}^{2},$

(16)

with respect to the parameter $\lambda .$

4.5. Cramer von-Misses estimation

The cvme is an estimation method. This method is derived as the difference between the empirical cdf and fitted cdf where

$\mathrm{c}\mathrm{v}\mathrm{m}\mathrm{e}\left(\lambda \right) = \frac{1}{12n}+\sum _{i = 1}^{n}{\left[1-\frac{\left({\lambda }^{2}-\lambda +{\mathrm{x}}_{i:n}\right)}{\lambda \left(\lambda -1\right)}{e}^{-\frac{{\mathrm{x}}_{i:n+1}}{\lambda }}-\frac{2i-1}{2n}\right]}^{2}.$

(17)

4.6. Maximum product of spacing estimator

For $u = \mathrm{1, 2}, 3, \dots , h+1$ , assume ${D}_{u}\left(\lambda \right) = F\left({x}_{\left(u\right)}|\lambda \right)-F\left({x}_{\left(u-1\right)}|\lambda \right),$ be the uniform spacings of a random sample from the DRL model, where $F\left({x}_{\left(0\right)}|\lambda \right) = 0$ , $F\left({x}_{(h+1)}|\lambda \right) = 1$ and ${\sum }_{r = 1}^{h+1}{D}_{u}\left(\lambda \right) = 1$ . The mpse of the parameter $\lambda$ , say $\widehat{\lambda }$ , can be estimated by maximizing the geometric mean of the spacings

$\mathrm{m}\mathrm{p}\mathrm{s}\mathrm{e}\left(\lambda \right) = {\left[\prod _{u = 1}^{h+1}{D}_{u}\left(\lambda \right)\right]}^{\frac{1}{h+1}},$

(18)

with respect to the parameter $\lambda$ .

5. Simulation

In this section, we discussed the results of the simulation study to compare the estimation performance of the proposed estimators based on the DRL model. The performance of considered estimators is evaluated via absolute biases and mean square errors. We simulate $\mathrm{10,000}$ random samples from the DRL distribution using the following sample sizes $n = 10, 20, 50,100,200$ , and $500$ . The results calculated for parameter values $\lambda = \left(2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 8.0, 10.0\right)$ using R Software. The absolute bias and mse for the parameter $\lambda$ are presented in Tables 2 and 3.

Table 2. Simulation results of the DRL distribution for different parameter values.

Para.	n	Bias						mse
$\lambda$		mle	mom	ade	cvme	lse	mpse	mle	mom	ade	cvme	lse	mpse
2.0	10	0.099	0.072	0.878	0.606	0.956	0.801	0.762	1.496	2.897	2.059	3.034	1.797
	20	0.034	0.035	0.486	0.259	0.444	0.435	0.307	1.103	1.408	0.749	1.212	0.626
	50	0.017	0.160	0.122	0.037	0.065	0.172	0.055	0.811	0.304	0.086	0.148	0.111
	100	0.013	0.226	0.015	0.003	0.005	0.086	0.017	0.697	0.033	0.006	0.009	0.026
	200	0.009	0.277	0.000	0.000	0.000	0.050	0.007	0.620	0.000	0.000	0.000	0.008
	500	0.002	0.338	0.000	0.000	0.000	0.028	0.003	0.569	0.000	0.000	0.000	0.002
2.5	10	0.081	0.131	1.109	1.140	1.397	0.958	1.105	2.032	3.694	3.919	4.392	2.609
	20	0.065	0.239	0.837	0.869	1.060	0.622	0.485	1.487	2.368	2.560	2.765	1.173
	50	0.058	0.293	0.595	0.695	0.798	0.320	0.186	1.063	1.436	1.723	1.785	0.391
	100	0.027	0.298	0.475	0.602	0.672	0.195	0.091	0.816	1.082	1.373	1.428	0.165
	200	0.013	0.255	0.374	0.525	0.589	0.098	0.045	0.602	0.864	1.139	1.185	0.065
	500	0.007	0.146	0.273	0.484	0.532	0.047	0.018	0.316	0.674	0.979	1.014	0.022
3.0	10	0.019	0.224	1.085	1.103	1.304	0.996	1.574	2.613	3.976	4.160	4.504	3.186
	20	0.005	0.268	0.877	0.924	1.044	0.639	0.751	1.793	2.525	2.701	2.865	1.512
	50	0.003	0.193	0.738	0.844	0.893	0.357	0.313	1.003	1.540	1.728	1.760	0.537
	100	0.004	0.130	0.754	0.868	0.872	0.222	0.163	0.567	1.160	1.352	1.341	0.242
	200	0.001	0.053	0.775	0.887	0.896	0.141	0.084	0.238	0.920	1.112	1.106	0.113
	500	0.001	0.006	0.811	0.929	0.940	0.070	0.035	0.066	0.769	0.965	0.980	0.042
4.0	10	0.049	0.209	1.105	1.023	1.181	1.100	2.450	3.694	5.254	5.458	5.552	4.850
	20	0.043	0.169	0.896	0.891	0.985	0.748	1.339	2.065	3.000	3.234	3.315	2.269
	50	0.031	0.049	0.835	0.847	0.899	0.400	0.585	0.763	1.608	1.739	1.789	0.815
	100	0.008	0.014	0.849	0.878	0.891	0.255	0.291	0.328	1.161	1.240	1.285	0.377
	200	0.007	0.016	0.838	0.877	0.882	0.156	0.147	0.159	0.911	1.005	1.010	0.174
	500	0.004	0.002	0.831	0.877	0.878	0.066	0.058	0.064	0.773	0.861	0.862	0.062

| Show Table

DownLoad: CSV

Table 3. Simulation results of the DRL distribution for different parameter values.

Para.	n	Bias						mse
$\lambda$		mle	mom	ade	cvme	lse	mpse	mle	mom	ade	cvme	lse	mpse
5.0	10	0.106	0.077	1.058	1.013	1.242	1.164	3.881	4.545	6.333	6.748	7.579	6.421
	20	0.061	0.036	0.955	0.915	1.007	0.824	2.026	2.335	3.527	3.991	4.014	3.056
	50	0.046	0.015	0.864	0.869	0.904	0.447	0.821	0.839	1.793	1.906	1.970	1.070
	100	0.013	0.002	0.857	0.866	0.857	0.270	0.399	0.411	1.244	1.325	1.303	0.513
	200	0.013	0.006	0.833	0.843	0.849	0.161	0.198	0.214	0.946	1.006	1.009	0.238
	500	0.001	0.003	0.832	0.846	0.843	0.082	0.079	0.078	0.792	0.829	0.822	0.086
6.0	10	0.167	0.115	1.192	1.073	1.169	1.260	5.339	5.663	8.785	8.977	9.398	8.321
	20	0.049	0.027	0.946	0.952	0.975	0.890	2.794	2.926	4.236	4.861	4.686	3.860
	50	0.008	0.019	0.873	0.889	0.881	0.499	1.081	1.079	2.081	2.225	2.292	1.417
	100	0.007	0.016	0.859	0.843	0.870	0.294	0.523	0.517	1.392	1.421	1.522	0.631
	200	0.007	0.006	0.845	0.839	0.846	0.181	0.261	0.269	1.030	1.061	1.065	0.308
	500	0.001	0.001	0.832	0.832	0.835	0.085	0.107	0.108	0.823	0.841	0.847	0.112
8.0	10	0.141	0.089	1.294	1.077	1.320	1.622	8.569	8.656	12.77	12.61	13.63	13.41
	20	0.074	0.009	1.050	0.991	1.111	1.110	4.265	4.178	6.399	6.736	7.355	6.250
	50	0.029	0.003	0.883	0.860	0.918	0.577	1.685	1.630	2.726	3.019	3.118	2.151
	100	0.015	0.014	0.867	0.861	0.859	0.372	0.834	0.814	1.787	1.862	1.859	1.029
	200	0.008	0.019	0.837	0.808	0.842	0.219	0.416	0.422	1.202	1.204	1.285	0.493
	500	0.006	0.001	0.826	0.814	0.820	0.097	0.169	0.175	0.877	0.879	0.902	0.176
10.0	10	0.106	0.086	1.365	1.333	1.446	1.868	12.89	12.67	17.66	19.87	19.51	20.26
	20	0.001	0.063	1.143	1.079	1.067	1.308	6.232	6.279	8.767	9.553	9.777	9.056
	50	0.025	0.036	0.899	0.892	0.933	0.686	2.483	2.373	3.774	4.076	4.248	3.246
	100	0.018	0.016	0.868	0.886	0.872	0.457	1.247	1.246	2.255	2.421	2.464	1.489
	200	0.001	0.009	0.852	0.827	0.829	0.267	0.636	0.599	1.467	1.501	1.501	0.712
	500	0.002	0.007	0.823	0.801	0.818	0.119	0.246	0.240	0.967	0.971	1.005	0.255

| Show Table

DownLoad: CSV

Based on the simulation criteria, it is observed that all estimation approaches work quite well in estimating the parameter λ of the DRL distribution.

6. Data analysis

In this section, the importance of the proposed distribution is discussed by using data sets from different areas. We shall compare the fits of the DRL distribution with different competitive distributions such as Poisson (Poi), discrete Pareto (DPr), discrete Rayleigh (DR), discrete inverse Rayleigh (DIR), discrete Burr-Hatke (DBH), discrete Bilal (DBi), discrete Lindley (DL), new discrete Lindley (NDL), and discrete Burr-XII (DBXII) distributions. The fitted probability distributions are compared using some criteria, namely, the negative log-likelihood ( $-L$ ), Akaike information criterion (aic), and Kolmogorov-Smirnov (ks) test with its p-value.

6.1. Data set I: Covid-19 data in Pakistan

The first data set represents the number of deaths due to coronavirus in Pakistan during the period March 18, 2020, to April 30, 2020, which were obtained from the public reports of the National Institute of Health (NIH), Islamabad, Pakistan (https://covid.gov.pk/stats/pakistan). The mean, variance, and di of data set I are 9.4773,102.39, and 10.804, respectively. The mle(s) along with standard error(s) "se(s)" and goodness-of-fit measures for this data are presented in Table 4.

Table 4. The mle(s), se(s), and goodness-of-fit measures for data set I.

Model	$\lambda$		$\delta$		Goodness-of-fit measures
Model	mle	se	mle	se	$-L$	aic	ks	p-value
DRL	8.8686	1.5033	-	-	145.22	292.43	0.156	0.2300
Poi	9.4773	0.4641	-	-	283.94	569.89	0.391	< 0.0001
DPr	0.5021	0.0757	-	-	162.19	326.38	0.401	< 0.0001
DR	9.9883	0.7535	-	-	168.85	339.70	0.339	< 0.0001
DIR	7.4291	1.2625	-	-	166.31	334.61	0.382	< 0.0001
DBH	0.9950	0.0115	-	-	175.37	352.74	0.647	< 0.0001
DBi	11.838	1.2932	-	-	151.29	304.59	0.213	0.0370
DL	0.8313	0.0165	-	-	149.17	300.33	0.184	0.1000
NDL	0.1640	0.0161	-	-	148.44	298.89	0.237	0.0140
DBXII	0.9536	0.0434	11.907	11.305	150.70	305.40	0.302	0.0007

| Show Table

DownLoad: CSV

The results in Table 4 show that the DRL distribution provides a better fit over other competing discrete models since it has the minimum aic, and ks values with the highest p-value. Figure 4 shows the probability-probability (pp) plots for all tested models which prove the empirical results listed in Table 4.

Figure 4. The pp plots for all fitted distributions for data set I.

DownLoad: Full-Size Img PowerPoint

6.2. Data set II: Hydrology data

The second data set was reported in [25], which represents the exceedance of flood peaks in m³/s of the Wheaton River near Carcross in Yukon Territory, Canada based on the discretization concept. The mean, variance, and di of this data are 11.806,152.38, and 12.908, respectively. The mle(s), se(s), and goodness-of-fit measures for data set II are reported in Table 5.

Table 5. The mle(s), se(s), and goodness-of-fit measures for data set II.

Model	$\lambda$		$\delta$		Goodness-of-fit measures
Model	mle	se	mle	se	$-L$	aic	ks	p-value
DRL	11.214	1.4497	-	-	252.71	507.43	0.133	0.1600
Poi	11.805	0.4049	-	-	564.38	1130.8	0.408	< 0.0001
DPr	0.4770	0.0563	-	-	276.82	555.64	0.311	< 0.0001
DR	12.280	0.7239	-	-	300.65	603.29	0.323	< 0.0001
DIR	4.7947	0.6303	-	-	331.46	664.92	0.497	< 0.0001
DBH	0.9966	0.0072	-	-	302.29	606.57	0.572	< 0.0001
DBi	14.621	1.2479	-	-	272.50	546.99	0.257	0.0002
DL	0.8592	0.0109	-	-	264.30	530.59	0.232	0.0009
NDL	0.1373	0.0107	-	-	262.09	526.17	0.271	< 0.0001
DBXII	0.8205	0.0591	2.6112	0.9287	270.50	544.99	0.228	0.0011

| Show Table

DownLoad: CSV

It is observed that the DRL model is the best among all competitive distributions. Figure 5 illustrates the pp plots for all tested distributions which prove the empirical results reported in Table 5.

Figure 5. The pp plots for all fitted distributions for data set II.

DownLoad: Full-Size Img PowerPoint

6.3. Data set III: Forest fire in Greece

The third data set was listed in [26] and represents the number of fires in Greece forest districts for the period from 1st July 1998 to 31 August 1998. The mean, variance, and di measures are 5.2, 32.382, and 6.2272, respectively. The mle(s), se(s), and goodness-of-fit measures for data set II are listed in Table 6.

Table 6. The mle(s), se(s), and goodness-of-fit measures for data set III.

Model	$\lambda$		$\delta$		Goodness-of-fit measures
Model	mle	se	mle	se	$-L$	aic	ks	p-value
DRL	4.3673	0.5510	-	-	301.11	604.21	0.1510	0.0140
Poi	5.2000	0.2174	-	-	434.16	870.32	0.282	< 0.0001
DPr	0.6250	0.0597	-	-	339.05	680.10	0.352	< 0.0001
DR	5.6788	0.2714	-	-	352.72	707.45	0.261	< 0.0001
DIR	3.5198	0.3748	-	-	360.90	723.80	0.413	< 0.0001
DBH	0.9833	0.0136	-	-	352.42	706.85	0.532	< 0.0001
DBi	6.7993	0.4693	-	-	310.75	623.49	0.107	< 0.0001
DL	0.7337	0.0156	-	-	303.88	609.75	0.193	0.0100
NDL	0.2567	0.0152	-	-	302.73	607.47	0.169	0.0037
DBXII	0.7486	0.0459	2.4582	0.4938	325.00	654.01	0.287	< 0.0001

| Show Table

DownLoad: CSV

It is found that the new discrete model is the best among all tested distributions. Figure 6 shows the pp plots for all competitive distributions which prove the empirical results listed in Table 6.

Figure 6. The pp plots for all fitted distributions for data set III.

DownLoad: Full-Size Img PowerPoint

6.4. Data set IV: AG-positive leukemia

The fourth data set represents the time to death (in weeks) of AG-positive leukemia patients [27]. The mean, variance, and di values are 62.471, 2954.3, and 47.29, respectively. The estimates and goodness-of-fit measures for all competitive distributions are listed in Table 7.

Table 7. The mle(s), se(s), and goodness-of-fit measures for data set IV.

Model	$\lambda$		$\delta$		Goodness-of-fit measures
Model	mle	se	mle	se	$-L$	aic	ks	p-value
DRL	61.943	15.273	-	-	87.425	176.85	0.152	0.8300
Poi	62.470	1.9169	-	-	475.26	952.52	0.470	0.0011
DPr	0.2838	0.0688	-	-	98.335	198.67	0.324	0.0560
DR	58.076	7.0429	-	-	96.794	195.59	0.309	0.0770
DIR	25.310	6.543	-	-	128.59	259.18	0.681	< 0.0001
DBH	0.9999	0.0029	-	-	119.81	241.62	0.716	< 0.0001
DBi	75.109	13.164	-	-	92.886	187.77	0.219	0.3900
DL	0.9692	0.0052	-	-	91.858	185.72	0.215	0.4100
NDL	0.0306	0.0052	-	-	91.458	184.92	0.218	0.3900
DBXII	0.9975	0.0008	117.30	45.982	96.151	196.30	0.327	0.0530

| Show Table

DownLoad: CSV

It is noted that the DRL is the best for this data. Figure 7 shows the pp plots for all tested distributions which prove the empirical results mentioned in Table 7.

Figure 7. The pp plots for all fitted distributions for data set IV.

DownLoad: Full-Size Img PowerPoint

7. Conclusions

In this article, a new one-parameter discrete model has been proposed entitled a discrete Ramos-Louzada (DRL) distribution. The new model can be used effectively in modeling asymmetric data with overdispersion phenomena. Some of its statistical properties have been derived. It was found that all its properties can be expressed in closed forms, which makes the new model can be utilized in different analysis, especially, in time series and regression. Various estimation techniques including maximum likelihood, moments, least squares, Anderson's-Darling, Cramer von-Mises, and maximum product of spacing estimator, have been investigated to get the best estimator for the real data. The estimation performance of these estimation techniques has been assessed via a comprehensive simulation study. The flexibility of the proposed discrete model has been tested utilizing four distinctive real data sets in various fields. Finally, we hope that the DRL distribution attracts wider sets of applications in different fields.

Conflict of interest

The authors declare that they have no conflict of interest to report regarding the present study.

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