Improved MPPT algorithm: Artificial neural network trained by an enhanced Gauss-Newton method

Fayrouz Dkhichi; Fayrouz Dkhichi

doi:10.3934/electreng.2023020

AIMS Electronics and Electrical Engineering

2023, Volume 7, Issue 4: 380-405. doi: 10.3934/electreng.2023020

Previous Article Next Article

Research article Topical Sections

Improved MPPT algorithm: Artificial neural network trained by an enhanced Gauss-Newton method

Fayrouz Dkhichi ^,

Laboratory of Networks, Computer Science, Telecommunications & Multimedia, Electrical Engineering Department, Superior School of Technology, Hassan II University of Casablanca, Morocco

Received: 26 August 2023 Revised: 09 November 2023 Accepted: 23 November 2023 Published: 07 December 2023

A novel approach defined by the artificial neural network (ANN) model trained by the improved Gauss-Newton in conjunction with a simulated annealing technique is used to control a step-up converter. To elucidate the superiority of this innovative method and to show its high precision and speed in achieving the right value of the Maximum Power Point (MPP), a set of three comparative Maximum Power Point Tracker (MPPT) methods (Perturbation and observation, ANN and ANN associated with perturbation and observation) are exanimated judiciously. The behavior of these methods is observed and tested for a fixed temperature and irradiance. As a result, the proposed approach quickly tracks the right MPP = 18.59 W in just 0.04382 s. On the other hand, the outstanding ability of the suggested method is demonstrated by varying the irradiance values (200 W/m², 300 W/m², 700 W/m², 1000 W/m², 800 W/m² and 400 W/m²) and by varying the temperature values (15℃, 35℃, 45℃ and 5℃). Therefore, the ANN trained by Gauss-Newton in conjunction with simulated annealing shows a high robustness and achieves the correct value of MPP for each value of irradiance with an efficiency 99.54% and for each value of temperature with an efficiency 99.98%; the three other methods sometimes struggle to achieve the right MPP for certain irradiance values and often remains stuck in its surroundings.

Keywords:

Citation: Fayrouz Dkhichi. Improved MPPT algorithm: Artificial neural network trained by an enhanced Gauss-Newton method[J]. AIMS Electronics and Electrical Engineering, 2023, 7(4): 380-405. doi: 10.3934/electreng.2023020

Related Papers:

[1]	Fathy H. Riad, Eslam Hussam, Ahmed M. Gemeay, Ramy A. Aldallal, Ahmed Z.Afify . Classical and Bayesian inference of the weighted-exponential distribution with an application to insurance data. Mathematical Biosciences and Engineering, 2022, 19(7): 6551-6581. doi: 10.3934/mbe.2022309
[2]	S. H. Sathish Indika, Norou Diawara, Hueiwang Anna Jeng, Bridget D. Giles, Dilini S. K. Gamage . Modeling the spread of COVID-19 in spatio-temporal context. Mathematical Biosciences and Engineering, 2023, 20(6): 10552-10569. doi: 10.3934/mbe.2023466
[3]	Kai Wang, Zhenzhen Lu, Xiaomeng Wang, Hui Li, Huling Li, Dandan Lin, Yongli Cai, Xing Feng, Yateng Song, Zhiwei Feng, Weidong Ji, Xiaoyan Wang, Yi Yin, Lei Wang, Zhihang Peng . Current trends and future prediction of novel coronavirus disease (COVID-19) epidemic in China: a dynamical modeling analysis. Mathematical Biosciences and Engineering, 2020, 17(4): 3052-3061. doi: 10.3934/mbe.2020173
[4]	Sarah R. Al-Dawsari, Khalaf S. Sultan . Modeling of daily confirmed Saudi COVID-19 cases using inverted exponential regression. Mathematical Biosciences and Engineering, 2021, 18(3): 2303-2330. doi: 10.3934/mbe.2021117
[5]	Manal M. Yousef, Rehab Alsultan, Said G. Nassr . Parametric inference on partially accelerated life testing for the inverted Kumaraswamy distribution based on Type-II progressive censoring data. Mathematical Biosciences and Engineering, 2023, 20(2): 1674-1694. doi: 10.3934/mbe.2023076
[6]	Mohamed S. Eliwa, Buthaynah T. Alhumaidan, Raghad N. Alqefari . A discrete mixed distribution: Statistical and reliability properties with applications to model COVID-19 data in various countries. Mathematical Biosciences and Engineering, 2023, 20(5): 7859-7881. doi: 10.3934/mbe.2023340
[7]	Francisco Julian Ariza-Hernandez, Juan Carlos Najera-Tinoco, Martin Patricio Arciga-Alejandre, Eduardo Castañeda-Saucedo, Jorge Sanchez-Ortiz . Bayesian inverse problem for a fractional diffusion model of cell migration. Mathematical Biosciences and Engineering, 2024, 21(4): 5826-5837. doi: 10.3934/mbe.2024257
[8]	Wael S. Abu El Azm, Ramy Aldallal, Hassan M. Aljohani, Said G. Nassr . Estimations of competing lifetime data from inverse Weibull distribution under adaptive progressively hybrid censored. Mathematical Biosciences and Engineering, 2022, 19(6): 6252-6275. doi: 10.3934/mbe.2022292
[9]	Walid Emam, Khalaf S. Sultan . Bayesian and maximum likelihood estimations of the Dagum parameters under combined-unified hybrid censoring. Mathematical Biosciences and Engineering, 2021, 18(3): 2930-2951. doi: 10.3934/mbe.2021148
[10]	Xiaomei Feng, Jing Chen, Kai Wang, Lei Wang, Fengqin Zhang, Zhen Jin, Lan Zou, Xia Wang . Phase-adjusted estimation of the COVID-19 outbreak in South Korea under multi-source data and adjustment measures: a modelling study. Mathematical Biosciences and Engineering, 2020, 17(4): 3637-3648. doi: 10.3934/mbe.2020205

Abstract

1. Introduction

Data modeling has become extremely complicated in recent years as a result of the massive amount of data collected from many sectors, mainly in engineering, medicine, ecology, and renewable energy. The most popular option for analyzing count data sets is the Poisson distribution. The Poisson distribution has the drawback of being unable to represent overdispersed data sets. Overdispersion happens when the variation exceeds the mean. For count data sets, many researchers have presented mixed-Poisson distributions such as Poisson inverse Gaussian by ^[1], Conway–Maxwell–Poisson ^[2], Generalized Poisson Lindley ^[3], Poisson Weibull ^[4], Poisson Ishita ^[5], Poisson quasi-Lindley ^[6], Poisson Xgamma ^[7,8], Poisson XLindley ^[9], Poisson Moment Exponential ^[10], among authors. Even though there are several discrete models in the literature, there is still plenty of room to suggest a new discretized model that is acceptable under a variety of scenarios.

Let X be a random variable having Ramos and Louzada distribution ^[11] with the probability density function (PDF) given by

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

(1)

where $\tau$ is the scale parameter.

In this study, a new one-parameter discrete distribution for modeling count observations is introduced by compounding the Poisson distribution with Ramous-Louzada (RL) distribution. The resulting model is called the Poisson Ramous-Louzada (PRL) distribution. The major reason for the selection of the RL distribution as a compounding distribution is because of its simple form, which is needed to compute the statistical properties of the proposed distribution and estimate the unknown parameter. The proposed model may be used to model count datasets, which are frequently seen in real-world data modeling. To build a mixed Poisson model, it is assumed that the Poisson model's parameter is a random variable (RV) with a continuous distribution, and the count variable is drawn from the Poisson distribution conditional on the random parameter. As a result, the count variable's marginal distribution is a mixed Poisson distribution.

The remainder of the paper is structured as follows: The new model is described in Section 2 and gives graphical representations of PMF, and HRF. Section 3 deduces several mathematical characteristics. Section 4 estimates the PRL parameter using the following classical estimation methods, maximum likelihood estimation (MLE), Anderson Darling (AD), Cramer von Mises (CVM), ordinary least-squares (OLS) and weighted least squares (WLS), and a simulation study is also given. Section 5 additionally discusses the Bayesian model formulation for the suggested distribution. Section 6 examines three real-world data sets to demonstrate the versatility of the PRL distribution. Section 6 also includes a Bayesian study of real-world data sets using Markov chain Monte Carlo methods. Section 7 concludes with some recommendations.

2. The Structure of the new model

A random variable $X$ is said to follow the Poisson Ramos-Louzada distribution if it possesses the following stochastic representation

$\left(X|\theta \right)\sim Poisson\left(g\left(\theta \right)\right)$

$\left(\theta |\tau \right)\sim RL\left(\tau \right)$

We call the marginal distribution of X the Poisson Ramos-Louzada distribution. The model is denoted by $PRL\left(\tau \right)$ .

Theorem 1: The PMF of PRL distribution is given by

$P\left(X = x, \tau \right) = \frac{{\left(1+\frac{1}{\tau }\right)}^{-x}\left(x-1+\tau (\tau -1)\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}};x = \mathrm{0, 1}, \mathrm{2, 3}, \dots \&\tau \ge 2$

Proof: The PMF of the new probability model can be obtained as

$g\left(x|\theta \right) = \frac{{e}^{-\theta }{\theta }^{x}}{x!};x = \mathrm{0, 1}, \mathrm{2, 3}, \dots \&\theta > 0$

when its parameter $\theta$ follows RL distribution

$f\left(\theta ;\tau \right) = \frac{\left({\tau }^{2}-2\tau +\theta \right)}{{\tau }^{2}\left(\tau -1\right)}{e}^{-\left(\frac{\theta }{\tau }\right)}$

We have

$\begin{array}{l} P\left(X = x, \tau \right) = \underset{0}{\overset{\infty }{\int }}g\left(x|\theta \right)f\left(\theta ;\tau \right)d\theta \\ \ \ \ \ \ \ \ = \frac{1}{x!{\tau }^{2}\left(\tau -1\right)}\underset{0}{\overset{\infty }{\int }}{e}^{-\theta }{\theta }^{x}\left({\tau }^{2}-2\tau +\theta \right){e}^{-\left(\frac{\theta }{\tau }\right)}d\theta \\ \ \ \ \ \ \ \ = \frac{1}{x!{\tau }^{2}\left(\tau -1\right)}\left(\left({\tau }^{2}-2\tau \right)\underset{0}{\overset{\infty }{\int }}{e}^{-\theta }{\theta }^{x}{e}^{-\left(\frac{\theta }{\tau }\right)}d\theta +\underset{0}{\overset{\infty }{\int }}{e}^{-\theta }{\theta }^{x+1}{e}^{-\left(\frac{\theta }{\tau }\right)}d\theta \right)\\ \ \ \ \ \ \ \ = \frac{1}{x!{\tau }^{2}\left(\tau -1\right)}\left(\left({\tau }^{2}-2\tau \right){\left(1+\frac{1}{\tau }\right)}^{-x-1}\mathrm{\Gamma }\left(1+x\right)+{\left(1+\frac{1}{\tau }\right)}^{-2-x}\mathrm{\Gamma }\left(2+x\right)\right) \\ \ \ \ \ \ \ \ P\left(X = x, \tau \right) = \frac{{\left(1+\frac{1}{\tau }\right)}^{-x}\left(x-1+\tau (\tau -1)\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}};x = \mathrm{0, 1}, \mathrm{2, 3}, \dots \&\tau \ge 2. \end{array}$

(2)

The PMF behavior of the Poisson Ramos-Louzada distribution for various parameter values is shown in Figure 1.

Figure 1. PMF visualization plots for the PRL distribution.

DownLoad: Full-Size Img PowerPoint

As can be seen, the PMF has a positively skewed and can be used to discuss the count data that is positively skewed. The corresponding CDF of the discrete Poisson Ramos-Louzada distribution is given as

$\begin{array}{l} F\left(X = x\right) = {p}_{r}\left(X\le x\right) = 1-\sum\limits_{v = x+1}^{\infty }P\left(v\right)\\ \ \ \ \ \ \ \ \ \ \ \ \ = 1-\frac{{\left(1+\frac{1}{\tau }\right)}^{-x}\tau \left(x+{\tau }^{2}\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}};x = \mathrm{0, 1}, 2, \dots ;\tau \ge 2. \end{array}$

(3)

The corresponding survival function is

$S\left(x;\tau \right) = \frac{{\left(1+\frac{1}{\tau }\right)}^{-x}\tau \left(x+{\tau }^{2}\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}},$

(4)

The hazard rate function (HRF), and reversed hazard rate function can be expressed as

$h\left(x;\tau \right) = \frac{x+\tau \left(\tau -1\right)-1}{\tau \left(x+{\tau }^{2}\right)},$

(5)

and

$r\left(x;\tau \right) = \frac{1-x+\tau -{\tau }^{2}}{x\tau +{\tau }^{3}-{\left(1+\frac{1}{\tau }\right)}^{x}\left(\tau -1\right){\left(1+\tau \right)}^{2}}.$

(6)

The graphs below depict the behavior of the HRF of the discrete PRL distribution for various parameter values.

Figure 2. HRF visualization plots for the PRL distribution.

DownLoad: Full-Size Img PowerPoint

3. Statistical properties of PRL distribution

This section has examined some statistical measures of the PRL distribution. Moments, the moment generating function (MGF), and the probability generation function are among them (pgf).

3.1. Moments of PRL distribution

Assume $X$ is a PRL random variable, the r^th factorial moments can be derived as

${\mu }_{\left(r\right)}^{'} = E\left[E\left({X}^{\left(r\right)}|\theta \right)\right], \ \ {\rm{where}} \ \ {X}^{\left(r\right)} = X\left(X-1\right)\left(X-2\right)\dots \left(X-r+1\right)$

$= \frac{1}{{\tau }^{2}\left(\tau -1\right)}\underset{0}{\overset{\infty }{\int }}\left[\sum\limits_{x = 0}^{\infty }{x}^{\left(r\right)}\frac{{e}^{-\theta }{\theta }^{x}}{x!}\right]\left({\tau }^{2}-2\tau +\theta \right){e}^{-\left(\frac{\theta }{\tau }\right)}d\theta$

$= \frac{1}{{\tau }^{2}\left(\tau -1\right)}\underset{0}{\overset{\infty }{\int }}\left[{\theta }^{r}\sum\limits_{x = r}^{\infty }\frac{{e}^{-\theta }{\theta }^{x-r}}{\left(x-r\right)!}\right]\left({\tau }^{2}-2\tau +\theta \right){e}^{-\left(\frac{\theta }{\tau }\right)}d\theta$

Taking $x+r$ in place of $x$ within the bracket, we get

$\begin{array}{c} {\mu }_{\left(r\right)}^{'} = \frac{1}{{\tau }^{2}\left(\tau -1\right)}\underset{0}{\overset{\infty }{\int }}\left[{\theta }^{r}\sum\limits_{x = 0}^{\infty }\frac{{e}^{-\theta }{\theta }^{x}}{x!}\right]\left({\tau }^{2}-2\tau +\theta \right){e}^{-\left(\frac{\theta }{\tau }\right)}d\theta\\ = \frac{1}{{\tau }^{2}\left(\tau -1\right)}\underset{0}{\overset{\infty }{\int }}{\theta }^{r}\left({\tau }^{2}-2\tau +\theta \right){e}^{-\left(\frac{\theta }{\tau }\right)}d\theta \\ = \frac{{\tau }^{r}\left(-1+r+\tau \right)\mathrm{\Gamma }\left(1+r\right)}{\tau -1}. \end{array}$

(7)

The first four factorial moments can be expressed as

${\mu }_{\left(1\right)}^{'} = \frac{{\tau }^{2}}{\tau -1},$

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

${\mu }_{\left(3\right)}^{'} = \frac{6{\tau }^{3}\left(2+\tau \right)}{\tau -1},$

and

${\mu }_{\left(4\right)}^{'} = \frac{24{\tau }^{4}\left(3+\tau \right)}{\tau -1}.$

The first four moments about the mean of the PRL distribution are obtained.

${\mu }_{2} = \frac{{\tau }^{2}\left({\tau }^{2}+\tau -3\right)}{{\left(\tau -1\right)}^{2}},$

(8)

${\mu }_{3} = \frac{{\tau }^{2}\left(2{\tau }^{4}+3{\tau }^{3}-14{\tau }^{2}+4\tau +7\right)}{{\left(\tau -1\right)}^{3}},$

(9)

${\mu }_{4} = \frac{{\tau }^{2}(9{\tau }^{6}+18{\tau }^{5}-92{\tau }^{4}+41{\tau }^{3}+77{\tau }^{2}-41\tau -15)}{{\left(\tau -1\right)}^{4}},$

(10)

Using Eqs (8)–(10), the Index of Dispersion (ID), coefficient of skewness (CS), and coefficient of Kurtosis (CK) can be derived in closed forms,

$ID\left(X\right) = \frac{Var\left(X\right)}{Mean\left(X\right)} = \frac{{\tau }^{2}+\tau -3}{\tau -1},$

(11)

$CS\left(X\right) = \frac{{\mu }_{3}}{{\left({\mu }_{2}\right)}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}} = \frac{{\tau }^{2}\left(7+4\tau -14{\tau }^{2}+3{\tau }^{3}+2{\tau }^{4}\right)}{{\left(\tau -1\right)}^{3}{\left(\frac{{\tau }^{2}\left(-3+\tau +{\tau }^{2}\right)}{{\left(-1+\tau \right)}^{2}}\right)}^{3/2}},$

(12)

and

$CK\left(X\right) = \frac{9{\tau }^{6}+18{\tau }^{5}-92{\tau }^{4}+41{\tau }^{3}+77{\tau }^{2}-41\tau -15}{{\tau }^{2}{\left({\tau }^{2}+\tau -3\right)}^{2}}.$

(13)

The moment-generating function of RV X can be expressed as

$\begin{array}{l} {M}_{X}\left(s\right) = \sum\limits_{x = 0}^{\infty }{e}^{xs}P\left(X = x, \tau \right)\\ \ \ \ \ \ \ \ \ \ = \frac{\tau \left(\tau -{e}^{s}\left(\tau -2\right)-1\right)-1}{\left(\tau -1\right){\left(1+\tau -{e}^{s}\tau \right)}^{2}}. \end{array}$

(14)

The probability-generating function of PRL distribution can be derived as

$\begin{array}{l} {P}_{X}\left(t\right) = \sum\limits_{x = 0}^{\infty }{t}^{x}P\left(X = x, \tau \right)\\ \ \ \ \ \ \ \ = \frac{-1-\tau +2t\tau +{\tau }^{2}-t{\tau }^{2}}{(-1+\tau ){(-1-\tau +t\tau )}^{2}}. \end{array}$

(15)

Table 1 displays some computational statistics of the PRL distribution for sundry parameter values.

Table 1. Some computational statistics of PRL distribution.

τ	${\bf E}\left({\bf X}\right)$	${\bf V}{\bf a}{\bf r}\left({\bf X}\right)$	${\bf C}{\bf S}\left({\bf X}\right)$	${\bf C}{\bf K}\left({\bf X}\right)$	${\bf I}{\bf D}\left({\bf X}\right)$	${\bf C}{\bf V}\left({\bf X}\right)$
2	4.00000	12.0000	1.44338	6.08333	3.00000	0.86603
3	4.50000	20.2500	1.67901	7.05761	4.50000	1.00000
4	5.33333	30.2222	1.79405	7.66025	5.66667	1.03078
5	6.25000	42.1875	1.85607	8.02222	6.75000	1.03923
6	7.20000	56.1600	1.89348	8.25493	7.80000	1.04083
7	8.16667	72.1389	1.91786	8.41326	8.83333	1.04002
8	9.14286	90.1224	1.93468	8.52586	9.85714	1.03833
9	10.1250	110.1094	1.94678	8.60883	10.87500	1.03638
10	11.1111	132.0988	1.95579	8.67174	11.88889	1.03441
15	16.0714	272.0663	1.97871	8.83698	16.92857	1.02632
20	21.0526	462.0499	1.98750	8.90278	21.94737	1.02103

| Show Table

DownLoad: CSV

4. Parameter estimation

In this section, the parameter of PRL distribution is examined using some classical estimation approaches. The considered estimation approaches are maximum likelihood, Anderson-Darling, Cramer von Mises, least squares, and weighted least squares.

4.1. Maximum likelihood estimation

Let ${X}_{1}, {X}_{2}, {X}_{3}, \dots {X}_{n}$ be a random sample of failure times from PRL distribution, and the likelihood function for the parameter $\tau$ can be written as

$L\left(\tau |x\right) = \prod _{i = 1}^{n}\frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{i}}\left({x}_{i}-1+\tau (\tau -1)\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}},$

(16)

and log-likelihood function is specified by

$l\left(\tau |x\right) = \sum\nolimits_{i = 1}^{n}\mathrm{log}{\left(1+\frac{1}{\tau }\right)}^{-{x}_{i}}+\sum\nolimits_{i = 1}^{n}\mathrm{log}\left({x}_{i}-1+\tau (\tau -1)\right)-n\mathrm{log}\left(\tau -1\right)-n\mathrm{log}{\left(1+\tau \right)}^{2}.$

(17)

We get the following equation by deriving Eq (17) with regard to parameter $\tau$ :

$\frac{\partial l}{\partial \tau } = \sum\nolimits_{i = 1}^{n}\frac{{x}_{i}}{\left(1+\frac{1}{\tau }\right){\tau }^{2}}+\sum\nolimits_{i = 1}^{n}\frac{2\tau -1}{{x}_{i}+\tau \left(\tau -1\right)-1}-\frac{n}{\left(\tau -1\right)}-\frac{2n}{\left(\tau +1\right)}.$

(18)

The ML estimate is obtained by equating the above equation to zero and solving it for parameter $\tau$ . However, the ensuing expression has not a closed-form result and the required results can be obtained using iterative procedures.

4.2. Anderson darling estimation

The Anderson-Darling (AD) estimator $\widehat{\tau }$ of parameter $\tau$ can be defined by minimizing the following expression

$\mathrm{A}\mathrm{D}\left(\tau \right) = -n-\frac{1}{n}\sum\limits_{i = 1}^{n}\left(2i-1\right)\left[\mathrm{log}\left(F\left({x}_{\left(i:n\right)}|\tau \right)\right)+\mathrm{log}\left(1-F\left({x}_{\left(i:n\right)}|\tau \right)\right)\right],$

$\mathrm{A}\mathrm{D}\left(\tau \right) = -n-\frac{1}{n}\sum\nolimits_{i = 1}^{n}\left(2i-1\right)\left[\mathrm{log}\left(1-\frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{\left(i:n\right)}}\tau \left({x}_{\left(i:n\right)}+{\tau }^{2}\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}}\right)+\mathrm{log}\left(\frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{\left(i:n\right)}}\tau \left({x}_{\left(i:n\right)}+{\tau }^{2}\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}}\right)\right],$

Alternatively, the estimator can also be obtained by solving the following nonlinear equation

$\sum\limits_{i = 1}^{n}\text{‍}\left(2i-1\right)\left[\frac{{\rm{ \mathsf{ϕ} }}\left({x}_{\left(i:n\right)}|\tau \right)}{F\left({x}_{\left(i:n\right)}|\tau \right)}-\frac{{\rm{ \mathsf{ϕ} }}\left({x}_{\left(n+1-i:n\right)}|\tau \right)}{1-F\left({x}_{\left(n+1-i:n\right)}|\tau \right)}\right] = 0$

where ${\rm{ \mathsf{ϕ} }}\left({x}_{i:n}|\tau \right) = \frac{d}{d\tau }F\left({x}_{\left(i:n\right)}|\tau \right)$ and it reduces to

${\rm{ \mathsf{ϕ} }}\left({x}_{i:n}|\tau \right) = \frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{\left(i:n\right)}}\left(-{{x}_{\left(i:n\right)}}^{2}\left(\tau -1\right)-\left(\tau -3\right){\tau }^{2}-{x}_{\left(i:n\right)}\left(-1+\tau -3{\tau }^{2}+{\tau }^{3}\right)\right)}{{(\tau -1)}^{2}{(1+\tau )}^{3}}$

(19)

4.3. Ordinary least squares estimation

The ordinary least-square (OLS) estimator of the PRL model parameter can be obtained by minimizing

$\mathrm{L}\mathrm{S}\mathrm{E}\left(\tau \right) = \sum\limits_{i = 1}^{n}{\left[F\left({x}_{\left(i:n\right)}|\tau \right)-\frac{i}{n+1}\right]}^{2},$

with respect to the parameter $\tau$ . Moreover, the LSE of $\tau$ is also obtained by solving

$\sum\limits_{i = 1}^{m}\text{‍}\left[1-\frac{i}{1+n}-\frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{\left(i:n\right)}}\tau \left({x}_{\left(i:n\right)}+{\tau }^{2}\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}}\right]{\rm{ \mathsf{ϕ} }}\left({x}_{i:n}|\tau \right) = 0,$

4.4. Weighted least-square estimation

The WLS estimate (WLSE) of $\tau$ , say $\widehat{\tau }$ , can be determined by minimizing

$\mathrm{W}\mathrm{L}\mathrm{S}\mathrm{E}\left(\tau \right) = \sum\limits_{i = 1}^{n}{\frac{{\left(n+1\right)}^{2}\left(n+2\right)}{i\left(n-i+1\right)}\left[F\left({x}_{\left(i:n\right)}|\tau \right)-\frac{i}{n+1}\right]}^{2},$

with respect to $\tau$ . The WLSE of $\tau$ can also be obtained by solving

$\sum\limits_{i = 1}^{n}\text{‍}\frac{{\left(1+n\right)}^{2}\left(2+n\right)}{i\left(n-i+1\right)}\left[1-\frac{i}{1+n}-\frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{\left(i:n\right)}}\tau \left({x}_{\left(i:n\right)}+{\tau }^{2}\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}}\right]{\rm{ \mathsf{ϕ} }}\left({x}_{i:n}|\tau \right) = 0,$

In which ${\rm{ \mathsf{ϕ} }}\left({x}_{i:n}|\tau \right)$ is presented in (19).

4.5. Cramer Von-Misses estimator

The Cramer von Mises (CVM) is a minimum distance-based estimator. The CVM of the PRL distribution can be obtained by minimizing

$\mathrm{C}\mathrm{V}\mathrm{M}\left(\tau \right) = \frac{1}{12n}+\sum\limits_{i = 1}^{n}{\left[\mathrm{log}\left(F\left({x}_{\left(i:n\right)}|\tau \right)\right)-\frac{2i-1}{2n}\right]}^{2},$

with respect to the parameter $\tau$ .

The CVME of $\tau$ is also obtained by solving

$\sum\limits_{i = 1}^{n}\text{‍}\left[1-\frac{2i-1}{2n}-\frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{\left(i:n\right)}}\tau \left({x}_{\left(i:n\right)}+{\tau }^{2}\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}}\right]{\rm{ \mathsf{ϕ} }}\left({x}_{i:n}|\tau \right) = 0.$

4.6. Simulation

In this section, we performed a simulation study to evaluate the accuracy of all considered estimators. In the simulation run, we generate 10,000 samples of size n = 10, 25, 50,100,200, and 300 from PRL distribution and then calculate the average estimates (AE), absolute bias (AB), mean relative error (MRE) and mean square error (MSE). For this purpose, we consider the six sets of values of parameter $\tau$ . The simulation results are presented in Tables 2–7.

Table 2. Parameter Estimates based on simulated samples for the parameter

$\tau = 2.1$ .

Measures	$\mathit{\boldsymbol{n}}$	MLE	OLSE	WLSE	ADE	CVME
AE	10	2.4683	3.0734	2.7620	3.0486	2.7156
	25	2.3209	2.4309	2.1472	2.5131	2.3049
	50	2.2357	2.1799	2.1001	2.2501	2.1488
	100	2.1808	2.1098	2.1000	2.1307	2.1045
	200	2.1416	2.1002	2.1000	2.1013	2.1004
	300	2.1274	2.1000	2.1000	2.1002	2.1000
AB	10	0.3683	0.9734	0.6620	0.9486	0.6156
	25	0.2209	0.3309	0.0472	0.4131	0.2049
	50	0.1357	0.0799	0.0001	0.1501	0.0488
	100	0.0808	0.0098	0.0000	0.0307	0.0045
	200	0.0416	0.0002	0.0000	0.0013	0.0004
	300	0.0274	0.0000	0.0000	0.0002	0.0000
MRE	10	0.1754	0.4635	0.3152	0.4517	0.2931
	25	0.1052	0.1576	0.0225	0.1967	0.0976
	50	0.0646	0.0381	0.0001	0.0715	0.0233
	100	0.0385	0.0046	0.0000	0.0146	0.0021
	200	0.0198	0.0001	0.0000	0.0006	0.0002
	300	0.0130	0.0000	0.0000	0.0001	0.0000
MSE	10	0.7743	3.3333	2.0870	3.4064	2.3300
	25	0.2951	0.8431	0.0912	1.1611	0.5520
	50	0.1391	0.1614	0.0001	0.3712	0.0985
	100	0.0627	0.0157	0.0000	0.0651	0.0068
	200	0.0271	0.0003	0.0000	0.0025	0.0006
	300	0.0169	0.0000	0.0000	0.0006	0.0000

| Show Table

DownLoad: CSV

Table 3. Parameter Estimates based on simulated samples for the parameter

$\tau = 3.0$ .

Measures	$\mathit{\boldsymbol{n}}$	MLE	OLSE	WLSE	ADE	CVME
AE	10	3.1396	4.3862	4.3030	4.1871	4.2023
	25	3.0268	4.0219	3.9440	3.8787	3.9351
	50	2.9950	3.9277	3.9411	3.7879	3.8614
	100	2.9948	3.9041	4.0457	3.7762	3.8804
	200	2.9964	3.9231	4.2116	3.8175	3.9187
	300	2.9970	3.9378	4.3245	3.8419	3.9348
AB	10	0.1396	1.3862	1.3030	1.1871	1.2023
	25	0.0268	1.0219	0.9440	0.8787	0.9351
	50	0.0050	0.9277	0.9411	0.7879	0.8614
	100	0.0052	0.9041	1.0457	0.7762	0.8804
	200	0.0036	0.9231	1.2116	0.8175	0.9187
	300	0.0030	0.9378	1.3245	0.8419	0.9348
MRE	10	0.0465	0.5763	0.5672	0.5519	0.5599
	25	0.0089	0.4499	0.4713	0.4301	0.4473
	50	0.0017	0.3944	0.4427	0.3736	0.3907
	100	0.0017	0.3580	0.4347	0.3302	0.3570
	200	0.0012	0.3318	0.4495	0.3062	0.3321
	300	0.0010	0.3237	0.4659	0.2977	0.3217
MSE	10	1.5854	5.3423	4.9727	4.6497	4.8640
	25	0.7594	2.7425	2.7410	2.4596	2.6652
	50	0.4343	1.9387	2.1702	1.7096	1.8726
	100	0.2391	1.4819	1.9558	1.2643	1.4561
	200	0.1248	1.1972	1.9742	1.0271	1.1933
	300	0.0832	1.0952	2.0686	0.9387	1.0861

| Show Table

DownLoad: CSV

Table 4. Parameter Estimates based on simulated samples for the parameter

$\tau = 4.0$ .

Measures	$\mathit{\boldsymbol{n}}$	MLE	OLSE	WLSE	ADE	CVME
AE	10	3.9375	5.2163	5.2168	5.1234	5.0765
	25	3.9262	4.9401	4.9726	4.8907	4.8748
	50	3.9620	4.9088	5.0659	4.8573	4.8859
	100	3.9713	4.8963	5.1787	4.8401	4.8697
	200	3.9834	4.8858	5.3031	4.8439	4.8789
	300	3.9988	4.8862	5.3711	4.8506	4.8756
AB	10	0.0625	1.2163	1.2168	1.1234	1.0765
	25	0.0738	0.9401	0.9726	0.8907	0.8748
	50	0.0380	0.9088	1.0659	0.8573	0.8859
	100	0.0287	0.8963	1.1787	0.8401	0.8697
	200	0.0166	0.8858	1.3031	0.8439	0.8789
	300	0.0012	0.8862	1.3711	0.8506	0.8756
MRE	10	0.0156	0.4834	0.4807	0.4714	0.4854
	25	0.0185	0.3538	0.3585	0.3405	0.3536
	50	0.0095	0.2878	0.3167	0.2742	0.2889
	100	0.0072	0.2466	0.3054	0.2321	0.2424
	200	0.0042	0.2264	0.3260	0.2157	0.2256
	300	0.0003	0.2229	0.3428	0.2138	0.2205
MSE	10	2.9043	6.2750	6.0663	5.8402	6.1414
	25	1.3856	3.1215	3.0986	2.8793	3.0677
	50	0.7547	1.9790	2.2677	1.8032	2.0032
	100	0.3717	1.3758	1.8782	1.2184	1.3353
	200	0.1822	1.0527	1.9039	0.9610	1.0504
	300	0.1202	0.9624	2.0149	0.8867	0.9453

| Show Table

DownLoad: CSV

Table 5. Parameter Estimates based on simulated samples for the parameter

$\tau = 5.0$ .

Measures	$\mathit{\boldsymbol{n}}$	MLE	OLSE	WLSE	ADE	CVME
AE	10	4.8793	6.2159	6.2417	6.1934	6.0518
	25	4.9283	5.9406	6.0539	5.9254	5.8979
	50	4.9480	5.9061	6.0826	5.8767	5.8738
	100	4.9839	5.8739	6.1814	5.8623	5.8668
	200	4.9940	5.8508	6.2729	5.8430	5.8573
	300	4.9858	5.8588	6.3355	5.8436	5.8443
AB	10	0.1207	1.2159	1.2417	1.1934	1.0518
	25	0.0717	0.9406	1.0539	0.9254	0.8979
	50	0.0520	0.9061	1.0826	0.8767	0.8738
	100	0.0161	0.8739	1.1814	0.8623	0.8668
	200	0.0060	0.8508	1.2729	0.8430	0.8573
	300	0.0142	0.8588	1.3355	0.8436	0.8443
MRE	10	0.0241	0.4372	0.4343	0.4289	0.4335
	25	0.0143	0.2971	0.3010	0.2896	0.2975
	50	0.0104	0.2376	0.2514	0.2249	0.2334
	100	0.0032	0.1979	0.2415	0.1921	0.1968
	200	0.0012	0.1769	0.2549	0.1739	0.1775
	300	0.0028	0.1737	0.2671	0.1705	0.1714
MSE	10	4.5278	8.0561	7.8298	7.6115	7.7488
	25	1.9727	3.5841	3.6415	3.3645	3.5746
	50	0.9619	2.1944	2.3526	1.9809	2.1332
	100	0.4770	1.4498	1.9140	1.3619	1.4272
	200	0.2355	1.0642	1.8717	1.0091	1.0642
	300	0.1600	0.9522	1.9495	0.9070	0.9279

| Show Table

DownLoad: CSV

Table 6. Parameter Estimates based on simulated samples for the parameter

$\tau = 7.0$ .

Measures	$\mathit{\boldsymbol{n}}$	MLE	OLSE	WLSE	ADE	CVME
AE	10	6.8852	8.3751	8.3080	8.2719	8.0785
	25	6.9508	7.9654	8.0939	7.9661	7.9015
	50	6.9772	7.9105	8.0665	7.8996	7.8877
	100	6.9808	7.8671	8.1001	7.8491	7.8347
	200	6.9902	7.8377	8.2073	7.8376	7.8279
	300	6.9994	7.8330	8.2567	7.8323	7.8247
AB	10	0.1148	1.3751	1.3080	1.2719	1.0785
	25	0.0492	0.9654	1.0939	0.9661	0.9015
	50	0.0228	0.9105	1.0665	0.8996	0.8877
	100	0.0192	0.8671	1.1001	0.8491	0.8347
	200	0.0098	0.8377	1.2073	0.8376	0.8279
	300	0.0006	0.8330	1.2567	0.8323	0.8247
MRE	10	0.0164	0.3971	0.3834	0.3728	0.3801
	25	0.0070	0.2536	0.2487	0.2396	0.2494
	50	0.0033	0.1922	0.1930	0.1835	0.1908
	100	0.0027	0.1539	0.1695	0.1468	0.1502
	200	0.0014	0.1307	0.1741	0.1286	0.1298
	300	0.0001	0.1240	0.1797	0.1228	0.1229
MSE	10	8.0853	13.221	12.275	11.590	11.818
	25	3.1995	5.2182	4.9697	4.6070	5.0214
	50	1.5333	2.9054	2.8651	2.6487	2.8488
	100	0.7477	1.7950	2.0349	1.6308	1.7225
	200	0.3819	1.2139	1.8654	1.1544	1.1984
	300	0.2538	1.0264	1.8495	0.9997	1.0176

| Show Table

DownLoad: CSV

Table 7. Parameter Estimates based on simulated samples for the parameter

$\tau = 15.0$ .

Measures	$\mathit{\boldsymbol{n}}$	MLE	OLSE	WLSE	ADE	CVME
AE	10	14.870	16.725	16.755	16.639	16.487
	25	14.975	16.192	16.194	16.166	16.071
	50	15.003	15.979	16.045	16.015	15.948
	100	15.014	15.881	16.023	15.876	15.818
	200	14.992	15.818	16.066	15.853	15.801
	300	15.014	15.810	16.081	15.826	15.813
AB	10	0.1302	1.7254	1.7551	1.6394	1.4871
	25	0.0254	1.1917	1.1941	1.1658	1.0705
	50	0.0026	0.9791	1.0445	1.0145	0.9477
	100	0.0142	0.8806	1.0226	0.8755	0.8180
	200	0.0084	0.8184	1.0661	0.8532	0.8008
	300	0.0135	0.8103	1.0806	0.8263	0.8125
MRE	10	0.0087	0.3328	0.3258	0.3120	0.3262
	25	0.0017	0.2138	0.2007	0.1975	0.2060
	50	0.0002	0.1496	0.1444	0.1445	0.1491
	100	0.0009	0.1114	0.1076	0.1043	0.1085
	200	0.0006	0.0832	0.0879	0.0814	0.0824
	300	0.0009	0.0733	0.0817	0.0710	0.0729
MSE	10	27.483	42.816	40.605	37.786	40.567
	25	11.128	16.789	14.977	14.517	15.753
	50	5.4177	8.2091	7.5370	7.6602	8.2052
	100	2.7917	4.5139	4.1585	3.9290	4.2486
	200	1.3835	2.4799	2.6592	2.3612	2.4262
	300	0.8964	1.8851	2.2170	1.7610	1.8652

| Show Table

DownLoad: CSV

4.7. Bayesian analysis

The Bayesian parameter estimation technique is an alternate to classical maximum likelihood estimation. In Bayesian estimation, a prior distribution must be defined for each unknown parameter. Consider a set of data $x = {x}_{1}, {x}_{2}, \dots, {x}_{n}$ taken from discrete PRL distribution and the likelihood function is provided by

$L\left(\tau |x\right) = \prod _{i = 1}^{n}\frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{i}}\left({x}_{i}-1+\tau (\tau -1)\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}}.$

(20)

The Bayesian model is constructed by stating the prior distribution for the model parameter and then multiplying it with the likelihood function for the provided data using the Bayes theorem to generate the posterior distribution function. The prior distribution of parameter $\tau$ is denoted as $p\left(\tau \right)$ .

$p\left(\tau |x\right)\propto L\left(\tau |x\right)p\left(\tau \right).$

For the proposed distribution, the gamma distribution is considered a prior distribution with known hyperparameters such as $\tau \sim Gamma\left(\alpha, \beta \right)$ . The posterior expression, up to proportionality, may be found by multiplying the likelihood by the prior, and this can be represented as

$p\left(\tau |x\right)\propto \frac{{\beta }^{\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{\tau }^{\alpha -1}\mathrm{exp}\left(-\tau \beta \right)\prod\limits_{i = 1}^{n}\frac{{\left(1+\frac{1}{\tau }\right)}^{-{x}_{i}}\left({x}_{i}-1+\tau (\tau -1)\right)}{\left(\tau -1\right){\left(1+\tau \right)}^{2}}$

The posterior density is not mathematically tractable; for inference purposes, we will utilize the Markov Chain Monte Carlo (MCMC) approach to mimic posterior samples, allowing for easy sample-based conclusions.

In the present study, we explore the application of MCMC algorithms implemented in the package MCMCpack of the R program to simulate samples from the joint posterior distribution. For this purpose, we generated 1006000 samples of the joint posterior distribution of interest. The effects of the initial values in the iterative process are eliminated after a burn-in phase of 6000 simulated samples. To achieve approximately independent samples, a thinning interval of size 300 was utilized. The parameter Bayes estimates were gained by taking the expected value of generated samples. Traceplots and the Geweke diagnostic were used to monitor the convergence of the simulated sequences. The asymptotic standard error of the difference divided by the difference between the two means of non-overlapping parts of a simulated Markov chain is the basis of the Geweke convergence diagnostic. We may say that a chain has reached convergence if its corresponding absolute z score is smaller than 1.96 since this z score asymptotically follows a typical normal distribution. The construction of interesting posterior summaries was done using the R software package MCMCpack.

5. Application

This section is ardent to prove the usefulness of the discrete Poisson Ramos-Louzada distribution in the modeling of three datasets. We compare the fits of the proposed distribution with some renowned one-parameter discrete distributions, discrete Raleigh ^[12], Poisson, discrete Pareto ^[13] and discrete Burr-Hatke ^[14], discrete Inverted Topp-Leone ^[15]. The Kolmogorov-Smirnov (KS) test, Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) are used to compare the fitted models. We also illustrate the estimation procedures based on censored samples proposed in the previous section with three examples from the literature.

5.1. Data Ⅰ (Failure times of electronic components)

A sample of the failure time of 15 electronic components in an acceleration life test ^[16]. The observations are 1, 5, 6, 11, 12, 19, 20, 22, 23, 31, 37, 46, 54, 60, and 66. The mean and variance of the first dataset are 27.533 and 431.94 respectively. The dispersion index value is 15.689 which indicates that the dataset is overdispersed. We determine the MLEs, standard errors (SE), and model selection measures (AIC, BIC, and KS) for the first dataset using the R software's maxLik package. These results are shown in Table 8 along with the model selection measures.

Table 8. ML Estimates and goodness-of-fit for the first dataset.

Model	MLEs (S.E.)	-LogLik.	AIC	BIC	K-S	P-value
PRL	26.455 (7.2429)	64.995	131.99	132.70	0.1770	0.6700
DR	24.382 (3.1481)	66.394	134.79	135.50	0.2160	0.4300
Poisson	27.533 (1.3548)	151.21	304.41	305.12	0.3810	0.0180
DITL	0.4178 (0.1079)	74.491	150.98	151.69	0.3590	0.0310
DPr	0.3284 (0.0848)	77.402	156.80	157.51	0.4060	0.0097
DBH	0.9992 (0.0076)	91.368	184.74	185.44	0.7910	0.0000

| Show Table

DownLoad: CSV

Figure 3. Plots of fitted CDFs versus empirical CDFs for the first dataset.

DownLoad: Full-Size Img PowerPoint

For Bayesian data analysis, the parameter $\tau$ of the PRL distribution was assumed to have an approximate gamma as the prior distribution, that is, $\tau \sim Gamma(0.001, 0.1)$ . depicts posterior samples for the parameter $\tau$ . The evaluation of the MCMC draws across iterations is assessed using traceplot, posterior density, and ACF plot. From the traceplot, it is interesting to note that the samples produced attained acceptable convergence. The ACF plot indicates that the posterior samples are uncorrelated. Furthermore, the z-score of the Geweke test is –0.2498, indicating that the samples have sufficiently converged to a stable distribution. The posterior mean for τ is ${\tau }_{Bayes} = 13.00418$ with a standard deviation of 2.18641, and the corresponding 95% highest density interval is (9.008356, 17.3976). We observe that the ML and Bayesian estimates are quite similar.

Figure 4. Traceplot, Posterior density, and ACF plot based on the first dataset.

DownLoad: Full-Size Img PowerPoint

5.2. Data Ⅱ (COVID-19 Deaths in China)

A sample of 66 patients died due to COVID-19 in China from January 23, 2022, to March 28, 2020. The data are: 8, 16, 15, 24, 26, 26, 38, 43, 46, 45, 57, 64, 65, 73, 73, 86, 89, 97,108, 97,146,121,143,142,105, 98,136,114,118,109, 97,150, 71, 52, 29, 44, 47, 35, 42, 31, 38, 31, 30, 28, 27, 22, 17, 22, 11, 7, 13, 10, 14, 13, 11, 8, 3, 7, 6, 9, 7, 4, 6, 5, 3 and 5. Some descriptive measures (mean, variance, and dispersion index) for this dataset are 47.742, 1924.8, and 38.696. We acquire the ML estimates for the parameter, and model selection metrics (AIC, BIC, and KS) for the second dataset. These results are shown in Table 9.

Table 9. ML Estimates and goodness-of-fit for the second dataset.

Model	MLEs (S.E.)	-LogLik.	AIC	BIC	K-S	P-value
PRL	48.711 (6.1847)	324.51	651.02	653.21	0.0851	0.7300
DR	47.010 (2.8934)	347.23	696.45	698.64	0.2930	0.0000
Poisson	49.743 (0.8682)	1409.8	2821.6	2823.8	0.4970	0.0000
DITL	0.3539 (0.0436)	366.91	735.81	738.00	0.3290	0.0000
DPr	0.2863 (0.0352)	379.07	760.14	762.33	0.3820	0.0000
DBH	0.9997 (0.0019)	461.02	924.04	926.23	0.8120	0.0000

| Show Table

DownLoad: CSV

For Bayesian data analysis, the parameter tau of the PRL distribution was assumed to have a gamma prior distribution. The associated Geweke z-score is –0.08203, which likewise indicates that the samples have sufficiently converged to a stable distribution. The posterior mean for τ is ${\tau }_{Bayes} = 32.0684$ with a standard deviation of 2.89397, and a 95% HDI of (26.20931, 37.44432). The ML and Bayesian estimates are discernibly similar to one another.

Figure 5. Plots of fitted CDFs versus empirical CDFs for the second dataset.

DownLoad: Full-Size Img PowerPoint

Figure 6. Traceplot, Posterior density, and ACF plot based on the second dataset.

DownLoad: Full-Size Img PowerPoint

5.3. Data set Ⅲ (Deaths due to COVID-19 in Pakistan)

The third dataset is also about deaths due to COVID-19 in Pakistan from 18 March 2020 to 30 June 2020. The data are: 1, 6, 6, 4, 4, 4, 1, 20, 5, 2, 3, 15, 17, 7, 8, 25, 8, 25, 11, 25, 16, 16, 12, 11, 20, 31, 42, 32, 23, 17, 19, 38, 50, 21, 14, 37, 23, 47, 31, 24, 9, 64, 39, 30, 36, 46, 32, 50, 34, 32, 34, 30, 28, 35, 57, 78, 88, 60, 78, 67, 82, 68, 97, 67, 65,105, 83,101,107, 88,178,110,136,118,136,153,119, 89,105, 60,148, 59, 73, 83, 49,137 and 91. Some computational measures, mean, variance and index of dispersion for the third dataset are; 50.057, 1758.8, and 35.135. The MLEs and goodness-of-fit measures for this dataset are given in Table 10.

Table 10. ML Estimates and goodness-of-fit for the third dataset.

Model	MLEs (S.E.)	-LogLik.	AIC	BIC	K-S	P-value
PRL	49.020 (5.4201)	428.30	858.61	861.07	0.0676	0.8210
DR	46.339 (2.4841)	452.55	907.10	909.56	0.2473	0.0000
Poisson	50.058 (0.9742)	1713.0	3428.1	3430.5	0.4954	0.0000
DITL	0.3493 (0.0375)	488.14	978.28	980.75	0.3263	0.0000
DPr	0.2835 (0.0304)	503.61	1009.2	1011.7	0.3558	0.0000
DBH	0.9997 (0.0016)	613.80	1229.6	1232.1	0.7876	0.0000

| Show Table

DownLoad: CSV

Figure 7. Plots of fitted CDFs versus empirical CDFs for the third dataset.

DownLoad: Full-Size Img PowerPoint

For the third dataset, the gamma distribution is again considered as the prior distribution, and the posterior samples for the parameter are described in . Furthermore, the Geweke z-score is used as a diagnostic measure and its value is –0.03794, suggesting convergence of the samples to a stable distribution. The posterior mean for the third dataset is ${\tau }_{Bayes} = 46.96159$ with a standard deviation of 4.92385. The corresponding 95% HDI (37.94273, 57.07319). The ML and Bayes estimate is quite similar to each other.

Figure 8. Traceplot, Posterior density, and ACF plot based on the third dataset.

DownLoad: Full-Size Img PowerPoint

6. Conclusions

In this paper, we introduce a one-parameter discrete distribution by compounding Poisson with the Ramos-Louzada distribution. The proposed distribution is showing unimodal and positively skewed behavior. The failure rate of new distribution is increasing pattern. Some statistical properties derived include the moment-generating function, probability-generating function, factorial moments, dispersion index, skewness and kurtosis. The model parameter is estimated using the maximum likelihood estimation approach and the behavior of the derived estimator is assessed via a simulation study. The usefulness of the proposed distribution is carried out using three real-life datasets. The proposed distribution provides more efficient results than all considered competitive distributions. The Bayesian analysis is also performed by taking the MCMC approximation approach.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	Saccardo RR, Domingues AM, Battistelle RAG, Bezerra BS, Siqueira RM, dos Santos Neto JB (2023) Investment in photovoltaic energy: An attempt to frame Brazil within the 2030 passage target of the Paris agreement. Cleaner Energy Systems 5: 100070. https://doi.org/10.1016/j.cles.2023.100070 doi: 10.1016/j.cles.2023.100070
[2]	Ma Z, Hu L, Mao H, Shao Q, Tian Z, Luo Y, et al. (2023) Shading effect and energy-saving potential of rooftop photovoltaic on the top-floor room. Solar Energy 265: 112099. https://doi.org/10.1016/j.solener.2023.112099 doi: 10.1016/j.solener.2023.112099
[3]	Junior RNY, Ochoa AAV, Leite GDNP, Silva HCN, da Costa JA, Tiba C, et al. (2023) Real-time energy and economic performance of the multi-zone photovoltaic-drive air conditioning system for an office building in a tropical climate. Energ Convers Manage 297: 117713. https://doi.org/10.1016/j.enconman.2023.117713 doi: 10.1016/j.enconman.2023.117713
[4]	Yin Y, Liu J (2023) Collaborative decision-making model for capacity allocation of photovoltaics energy storage system under Energy Internet in China. J Energy Storage 66: 107456. https://doi.org/10.1016/j.est.2023.107456 doi: 10.1016/j.est.2023.107456
[5]	Traiki G, El Magri A, Lajouad R, Bouattane O (2023) Multi-objective control and optimization of a stand-alone photovoltaic power conversion system with battery storage energy management. IFAC Journal of Systems and Control 26: 100227. https://doi.org/10.1016/j.ifacsc.2023.100227 doi: 10.1016/j.ifacsc.2023.100227
[6]	Burhan M, Chua KJE, Ng KC (2016) Sunlight to hydrogen conversion: Design optimization and energy management of concentrated photovoltaic (CPV-Hydrogen) system using micro genetic algorithm. Energy 99: 115-128. https://doi.org/10.1016/j.energy.2016.01.048 doi: 10.1016/j.energy.2016.01.048
[7]	Das D, Panda DP, Tongbram B, Saha J, Chavan V, Chakrabarti S(2018) Optimization of hybrid InAs stranski krastanov and submonolayer quantum dot heterostructures and its effect on photovoltaic energy conversion efficiency in near infrared region. Solar Energy 171: 64-72. https://doi.org/10.1016/j.solener.2018.06.030 doi: 10.1016/j.solener.2018.06.030
[8]	Corrado C, Leow SW, Osborn M, Chan E, Balaban B, Carter SA (2013) Optimization of gain and energy conversion efficiency using front-facing photovoltaic cell luminescent solar concentrator design. Solar Energy Materials and Solar Cells 111: 74-81. https://doi.org/10.1016/j.solmat.2012.12.030 doi: 10.1016/j.solmat.2012.12.030
[9]	Riyadi TWB, Effendy M, Utomo BR, Wijayanta AT (2023) Performance of a photovoltaic-thermoelectric generator panel in combination with various solar tracking systems. Appl Therm Eng 253: 121336. https://doi.org/10.1016/j.applthermaleng.2023.121336 doi: 10.1016/j.applthermaleng.2023.121336
[10]	Baghaz E, Melhaoui M, Yaden F, Hirech K, Kassmi K (2013) Design realization and optimization of the photovoltaic systems equipped with analog and digital MPPT commands. Energy Procedia 42: 270-279. https://doi.org/10.1016/j.egypro.2013.11.027 doi: 10.1016/j.egypro.2013.11.027
[11]	Masmoudi A, Abdelkafi A, Krichen L, Saidi AS (2022) An experimental approach for improving stability in DC bus voltage of a stand-alone photovoltaic generator. Energy 257: 124797. https://doi.org/10.1016/j.energy.2022.124797 doi: 10.1016/j.energy.2022.124797
[12]	Choi WY, Lee CG (2012) Photovoltaic panel integrated power conditioning system using a high efficiency step-up DC–DC converter. Renewable Energy 41: 227-234. https://doi.org/10.1016/j.renene.2011.10.023 doi: 10.1016/j.renene.2011.10.023
[13]	Nakpin A, Khwan S (2016) A Novel High Step-up DC-DC Converter for Photovoltaic Applications. Procedia Computer Science 86: 409-412. https://doi.org/10.1016/j.procs.2016.05.051 doi: 10.1016/j.procs.2016.05.051
[14]	Cha WJ, Kwon JM, Kwon BH (2016) Highly efficient step-up dc–dc converter for photovoltaic micro-inverter. Solar Energy 135: 14-21. https://doi.org/10.1016/j.solener.2016.05.024 doi: 10.1016/j.solener.2016.05.024
[15]	Mustafa Y, Yin H, Lu Y, Ruderman A (2021) Constrained minimization of switched capacitor converter equivalent resistance by adjusting transistor sizes and duty cycles. Microelectronics Journal 112: 105061. https://doi.org/10.1016/j.mejo.2021.105061 doi: 10.1016/j.mejo.2021.105061
[16]	Hafiz M, Ahmed KZ, Islam D, Rashid ABMH (2011) Design and implementation of a 0.8 V input, 84% duty cycle, variable frequency step-up converter. Microelectron J 42: 648-660. https://doi.org/10.1016/j.mejo.2011.03.005 doi: 10.1016/j.mejo.2011.03.005
[17]	Amir A, Amir A, Seng CH, el Khateb A, Selvaraj J, Rahim NA (2018) Application of modified classical numerical methods for DMPPT on Buck and Boost converters. Solar Energy 173: 437-448. https://doi.org/10.1016/j.solener.2018.07.088 doi: 10.1016/j.solener.2018.07.088
[18]	Baba AO, Liu G, Chen X (2020) Classification and Evaluation Review of Maximum Power Point Tracking Methods. Sustainable Futures 2: 100020. https://doi.org/10.1016/j.sftr.2020.100020 doi: 10.1016/j.sftr.2020.100020
[19]	Pandey NK, Pachauri RK, Choudhury S, Sahu RK (2023) Asymmetrical interval Type-2 Fuzzy logic controller based MPPT for PV system under sudden irradiance changes. Materials today proceeding 80: 710-716. https://doi.org/10.1016/j.matpr.2022.11.074 doi: 10.1016/j.matpr.2022.11.074
[20]	Ullah K, Ishaq M, Tchier F, Ahmad H, Ahmad Z (2023) Fuzzy-based maximum power point tracking (MPPT) control system for photovoltaic power generation system. Results in Engineering 20: 101466. https://doi.org/10.1016/j.rineng.2023.101466 doi: 10.1016/j.rineng.2023.101466
[21]	Mohammadinodoushan M, Abbassi R, Jerbi H, Ahmed FW, Rezvani A (2021) A new MPPT design using variable step size perturb and observe method for PV system under partially shaded conditions by modified shuffled frog leaping algorithm- SMC controller. Sustain Energy Techn 45: 101056. https://doi.org/10.1016/j.seta.2021.101056 doi: 10.1016/j.seta.2021.101056
[22]	Loukriz A, Haddadi M, Messalt S (2016) Simulation and experimental design of a new advanced variable step size Incremental Conductance MPPT algorithm for PV system. ISA Transactions 62: 30-38. https://doi.org/10.1016/j.isatra.2015.08.006 doi: 10.1016/j.isatra.2015.08.006
[23]	Goel A, Goel AK, Kumar A (2022) The role of artificial neural network and machine learning in utilizing spatial information. Spat Inf Res 31: 275-285. https://doi.org/10.1007/s41324-022-00494-x doi: 10.1007/s41324-022-00494-x
[24]	Toumi D, Attous DB, Ibrahim A, Tarek B (2022) Maximum power point tracking of photovoltaic array using fuzzy logic control. International Journal of Power Electronics and Drive Systems 13: 2440. https://doi.org/10.11591/ijpeds.v13.i4.pp2440-2449 doi: 10.11591/ijpeds.v13.i4.pp2440-2449
[25]	Manna S, Singh DK, Akella AK (2023) Hybrid two-stage adaptive maximum power point tracking for stand-alone, grid integration, and partial shaded PV system. Int J Adapt Control. https://doi.org/10.1002/acs.3684
[26]	Manna S, Akella AK, Singh DK (2023) Novel Lyapunov-based rapid and ripple-free MPPT using a robust model reference adaptive controller for solar PV system. Prot Contr Mod Pow Syst 8: 1-25. https://doi.org/10.1186/s41601-023-00288-9 doi: 10.1186/s41601-023-00288-9
[27]	Singh DK, Akella AK, Manna S (2023) Adjustable variable step-based MRAC MPPT for solar PV system in highly fluctuating and cloudy atmospheric conditions. Electrical Engineering 105: 3751-3772. https://doi.org/10.1007/s00202-023-01922-3 doi: 10.1007/s00202-023-01922-3
[28]	Ahmed R, Namaane A, M'Sirdi NK (2013) Improvement in Perturb and Observe Method Using State Flow Approach. Energy Procedia 42: 614-623. https://doi.org/10.1016/j.egypro.2013.11.063 doi: 10.1016/j.egypro.2013.11.063
[29]	Argyros IK, Magreñ án AA (2014) Local convergence analysis of proximal Gauss–Newton method for penalized nonlinear least squares problems. Appl Math Comput 241: 401-408. https://doi.org/10.1016/j.amc.2014.04.087 doi: 10.1016/j.amc.2014.04.087
[30]	Dkhichi F, Oukarfi B, Fakkar A, Belbounaguia N (2014) Parameter identification of solar cell model using Levenberg–Marquardt algorithm combined with simulated annealing. Solar Energy 110: 781-788. https://doi.org/10.1016/j.solener.2014.09.033 doi: 10.1016/j.solener.2014.09.033
[31]	Bao T, Li Z, Pu O, Chan RWK, Zhao Z, Pan Y, et al. (2023) Modal analysis of tracking photovoltaic support system. Solar Energy 265: 112088. https://doi.org/10.1016/j.solener.2023.112088 doi: 10.1016/j.solener.2023.112088
[32]	Ouedraogo A, Diallo A, Goro S, Ilboudo WDA, Madougou S, Bathiebo DJ, et al. (2022) Analysis of the solar power plant efficiency installed in the premises of a hospital — Case of the Pediatric Charles De Gaulle of Ouagadougou. Solar Energy 241: 120-129. https://doi.org/10.1016/j.solener.2022.05.051 doi: 10.1016/j.solener.2022.05.051
[33]	Muhammadsharif FF, Hashim S, Hameed SS, Ghoshal SK, Abdullah IK, Macdonald JE, et al. (2019) Brent's algorithm based new computational approach for accurate determination of single-diode model parameters to simulate solar cells and modules. Solar Energy 193: 782-798. https://doi.org/10.1016/j.solener.2019.09.096 doi: 10.1016/j.solener.2019.09.096
[34]	Ndi FE, Perabi SN, Ndjakomo SE, Abessolo GO, Mengata GM(2021) Estimation of single-diode and two diode solar cell parameters by equilibrium optimizer method. Energy Reports 7: 4761-4768. https://doi.org/10.1016/j.egyr.2021.07.025 doi: 10.1016/j.egyr.2021.07.025
[35]	Zhang SMF, Seif JP, Abbott MD, Le AH, Allen TG, Perez-Wurfl I, et al. (2022) Illumination-dependent temperature coefficients of the electrical parameters of modern silicon solar cell architectures. Nano Energy 98: 107221. https://doi.org/10.1016/j.nanoen.2022.107221 doi: 10.1016/j.nanoen.2022.107221
[36]	Park C, Balaji N, Ahn S, Park J, Cho E, Yi J (2020) Effects of tunneling oxide defect density and inter-diffused carrier concentration on carrier selective contact solar cell performance: Illumination and temperature effects. Solar Energy 211: 62-73. https://doi.org/10.1016/j.solener.2020.09.060 doi: 10.1016/j.solener.2020.09.060
[37]	Luo P, Pan J, Hong J, Liang J, Jiang H, Yang D (2023) An ultrahigh synchronous drive step-up converter for PEMFC and its explicit model predictive control: A neural network fitting strategy. Int J Hydrogen Energy. https://doi.org/10.1016/j.ijhydene.2023.08.113
[38]	Li P, Wang Y, Zuo Z (2023) Robust multiple frequency design on voltage-mode control of DC-DC boost converters. J Franklin I 360: 1207-1225. https://doi.org/10.1016/j.jfranklin.2022.12.017 doi: 10.1016/j.jfranklin.2022.12.017
[39]	Morales DS (2010) Maximum Power Point Tracking Algorithms for Photovoltaic Applications, Faculty of Electronics, Communications and Automation. Master of Science in Technology.
[40]	Naidu IES, Srikanth S, Rao A, Venkatanarayana A (2023) A novel mine blast optimization algorithm (MBOA) based MPPT controlling for grid-PV systems. AIMS Electronics and Electrical Engineering 7: 135-155. https://doi.org/10.3934/electreng.2023008 doi: 10.3934/electreng.2023008
[41]	Sivakumar P, Kader AA, Kaliavaradhan Y, Arutchelvi M (2015) Analysis and enhancement of PV efficiency with incremental conductance MPPT technique under non-linear loading conditions. Renew Energ 81: 543-550. https://doi.org/10.1016/j.renene.2015.03.062 doi: 10.1016/j.renene.2015.03.062
[42]	Wu Y, Jakobsson A, Liu L (2023) Super-resolution Direction of Arrival Estimation Using a Minimum Mean-Square Error Framework. Signal Processing 212: 109164. https://doi.org/10.1016/j.sigpro.2023.109164 doi: 10.1016/j.sigpro.2023.109164
[43]	Dkhichi F, Oukarfi B, El Kouari Y, Ouoba D, Fakkar A, Sabiri Z (2016) Performances of Artificial Neural Network combined with Perturb & Observe technique in maximizing the photovoltaic system power. International Renewable and Sustainable Energy Conference (IRSEC), 951-955. https://doi.org/10.1109/IRSEC.2016.7984023
[44]	Kassem AM (2012) MPPT control design and performance improvements of a PV generator powered DC motor-pump system based on artificial neural networks. Electrical Power and Energy Systems 43: 90–98. https://doi.org/10.1016/j.ijepes.2012.04.047 doi: 10.1016/j.ijepes.2012.04.047
[45]	Ansari QH, Uddin M, Yao JC (2024) Convergence of the Gauss-Newton method for convex composite optimization problems under majorant condition on Riemannian manifolds. J Complexity 80: 101788. https://doi.org/10.1016/j.jco.2023.101788 doi: 10.1016/j.jco.2023.101788
[46]	Cyril Lahore (2012) Optimisation de commandes MPPT. Available from : https://dumas.ccsd.cnrs.fr/dumas-01304277.

This article has been cited by:

1.	Fatimah M. Alghamdi, Muhammad Ahsan-ul-Haq, Muhammad Nasir Saddam Hussain, Eslam Hussam, Ehab M. Almetwally, Hassan M. Aljohani, Manahil SidAhmed Mustafa, Etaf Alshawarbeh, M. Yusuf, Discrete Poisson Quasi-XLindley distribution with mathematical properties, regression model, and data analysis, 2024, 17, 16878507, 100874, 10.1016/j.jrras.2024.100874
2.	Osama Abdulaziz Alamri, Classical and Bayesian estimation of discrete poisson Agu-Eghwerido distribution with applications, 2024, 109, 11100168, 768, 10.1016/j.aej.2024.09.063
3.	Amani Alrumayh, Marco Costa, Bernoulli Poisson Moment Exponential Distribution: Mathematical Properties, Regression Model, and Applications, 2024, 2024, 0161-1712, 10.1155/2024/5687958
4.	Seth Borbye, Suleman Nasiru, Kingsley Kuwubasamni Ajongba, Vladimir Mityushev, Poisson XRani Distribution: An Alternative Discrete Distribution for Overdispersed Count Data, 2024, 2024, 0161-1712, 10.1155/2024/5554949
5.	Waheed Babatunde Yahya, Muhammad Adamu Umar, A new poisson-exponential-gamma distribution for modelling count data with applications, 2024, 0033-5177, 10.1007/s11135-024-01894-x
6.	Yingying Qi, Dan Ding, Yusra A. Tashkandy, M.E. Bakr, M.M. Abd El-Raouf, Anoop Kumar, A novel probabilistic model with properties: Its implementation to the vocal music and reliability products, 2024, 107, 11100168, 254, 10.1016/j.aej.2024.07.035
7.	Abdullah Ali H. Ahmadini, Muhammad Ahsan-ul-Haq, Muhammad Nasir Saddam Hussain, A new two-parameter over-dispersed discrete distribution with mathematical properties, estimation, regression model and applications, 2024, 10, 24058440, e36764, 10.1016/j.heliyon.2024.e36764
8.	Safar M. Alghamdi, Muhammad Ahsan-ul-Haq, Olayan Albalawi, Majdah Mohammed Badr, Eslam Hussam, H.E. Semary, M.A. Abdelkawy, Binomial Poisson Ailamujia model with statistical properties and application, 2024, 17, 16878507, 101096, 10.1016/j.jrras.2024.101096
9.	Khlood Al-Harbi, Aisha Fayomi, Hanan Baaqeel, Amany Alsuraihi, A Novel Discrete Linear-Exponential Distribution for Modeling Physical and Medical Data, 2024, 16, 2073-8994, 1123, 10.3390/sym16091123
10.	Tabassum Naz Sindhu, Anum Shafiq, Abdon Atangana, Tahani A. Abushal, Alia A. Alkhathami, Control Charts for Overdispersed Count Data: Exploring the Poisson Chris‐Jerry Distribution in Agriculture and Medicine, 2025, 0748-8017, 10.1002/qre.3745
11.	Abdullah M. Alomair, Muhammad Ahsan-ul-Haq, Analysis of radiation and corn borer data using discrete Poisson Xrama distribution, 2025, 18, 16878507, 101388, 10.1016/j.jrras.2025.101388
12.	Muteb Faraj Alharthi, Samirah Alzubaidi, A novel discrete statistical model with applications on medical and health real data, 2025, 125, 11100168, 42, 10.1016/j.aej.2025.04.026
13.	Abdullah M. Alomair, Muhammad Ahsan-ul-Haq, Modeling radiation and electronic devices data with Poisson-Darna distribution, 2025, 18, 16878507, 101661, 10.1016/j.jrras.2025.101661
14.	Ali M. Mahnashi, Abdullah A. Zaagan, Poisson copoun distribution: An alternative discrete model for count data analysis, 2025, 128, 11100168, 571, 10.1016/j.aej.2025.05.085

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)