key: cord-0896974-8kyvhz10
authors: Riad, Fathy H.; Alruwaili, Bader; Gemeay, Ahmed M.; Hussam, Eslam
title: Statistical modeling for COVID 19 virus spread in Kingdom of Saudi Arabia and Netherlands
date: 2022-03-18
journal: Alexandria Engineering Journal
DOI: 10.1016/j.aej.2022.03.015
sha: f99e6136170e8d1e61f931b1a85010bf861cb2cc
doc_id: 896974
cord_uid: 8kyvhz10

The aim of this work is to develop a new outstanding lifetime distribution, dubbed the Power Bilal (PB) distribution. Both of the pdf and cdf of the Power Bilal distribution have a simple forms. The suggested distribution’s moments, incomplete moments as well as the quantile function are deduced and acquired in explicit forms as a result of its simple forms. Seven estimation methods for estimating the Power Bilal distribution parameters are mentioned, and numerical simulations are used to compare the proposed approaches using partial and overall ranks. According to the results of this work, the Anderson-Darling estimators are advised to estimation the parameters of the Power Bilal distribution. We shows the importance and flexibility of the Power Bilal distribution by comparing it to other existing competing distributions using two different real data-sets from COVID-19 mortality rates of two countries.

One of statistics' primary objectives is to develop a useful statistical models for natural and real life events represented by known statistical probability distributions. Where the probability distributions are being used to modeling the uncertain and risky potential life phenomenon of interest. Numerous probability distributions have been developed as a result of the complexity and difficulty of modeling natural life phenomena using conventional distributions. Sometimes, the known and available probability distributions continue to be unable to accurately shows and represents the data for certain natural phenomena. These modifications and expansions result in the expansions and modification of generalised probability distributions.

By adding of a few new or extra parameters to well-known probability distributions enhanced their appropriateness for natural phenomena data and increased their precision in presenting the distribution's tail shape. There are some useful ways to extend and enhance the flexibility to the classical statistical distributions by adding additional parameter to the distribution such as the power (P) transformation. For example, Let X and Y are defined as two random variables. The power-transformation say X = Y 1 λ have been adopted by many researchers to construct power distributions.

The two-parameter (PB) distribution, which has a number of interesting properties, is acquired in this article by referring to the distributions above. The implemented PB distribution does have a more flexible PDF since it can be skew to right as positive skewed, skew to the left as negative skewed, or symmetric, which allows for additional tail flexibility. It is able to simulate diminishing, increasing, bathtub, and reverse-J hazard rates. Additionally, the suggested distribution has a precise closed-form cdf and is quite easy to handle. These advantages gives the distribution an appealing prospect for applications in a variety of fields, including bioengineering life testing, durability, and econometric data.

Recently, many authors were interested in introducing new lifetime distributions for fitting real lifetime data. Among them, e. g., [25, 10, 29, 4, 2, 12, 28, 24, 26] . On the other hand, it is well known that, the order statistics can deal and apply with the applications and properties of the random variables and of functions related them, see [13] , [6] , [9] for reference.

The critical question is whether we require this distribution. To address this question, we summarise the PB distribution's significance: (i) The PB distribution's statistical functions have simple and closed-form expressions. (ii) the PB distribution's properties are deduced explicitly without the use of any special and specific mathematical functions; and (iii) the proposed PB distribution offers an extra flexibility than current distributions in terms of the shape of the hazard rate function. This artical is presented as the following: In Section 2 we introduces the proposed distribution PB, along with its pdf and cdf functions, also the graphical plot of the pdf and HRF are included in this section. Some statistical properties for the PB distributions are established in Section 3. Seven classical estimation methods were presented in Section 4. The simulation study along with its numerical results was done in Section 5. Here comes the data analysis in Section 6. Last but not least the concluded observations extracted form this research article was found in Section 7.

In this part of the paper we introduced the PB distribution by adding power parameter λ to enhance the efficiency of the distribution. This is done as follows: The pdf and cdf of Bilal (B) distribution [3] introduced respectively as follows:

(1)

For the PB distributions, suppose x defined as continue random variable follows the BD, than the PDF and CDF of PB distribution are defined as follows

where α > 0 is define as the scale parameters and λ > 0 is define as the shape parameter. The survival function (SF) and hazard rate function (HRF) of PB distributions are defined as follows:

Figures 1 and 2 shows the plots of the pdf and HRF of the PB distribution, respectively. By plotting the pdf of the PB distribution, we can see that its shapes can be symmetric, skewed to left or right sides, J-shaped or inverse J-shaped. The HRF of PB distributions can be upside-down, increasing, decreasing, increasing-constant or J-shaped. 

Here we introduce some properties for the Power Bilal distribution PB, such as the moments, (MGF) moment generating function, and the CF characteristic function. Also, we will introduce (MRL) the mean residual life, and (MIT) the mean period of inactivity.

Here we introduce the moments and the incomplete moments for the PB distributions. The moments about the origin r th is donated as follow: Figure 3 : Shows the plots for the means, variance, skewness, and kurtosis of PB distribution in three dimensions for different values for α and λ .

The r th incomplete moments of the PB distribution is defined as following:

The MGF of PB model is defined by

by putting t = jt in MGF, we get the CF of PB model.

The MRL (define also as the expected additional life length for a unit) denotes the extra life span age. Where the MRL value for the PB distribution is introduced as follows:

where IC 1 (t) is first incomplete moments. The MIT is defined as the time of waiting that elapsed after the failure of an item with condition that this failure has happened in (0, t).The MIT of PB distribution is introduced as following:

Let X is a random variable, then the entropy is introduced as a measure of the randomness amount of information in distribution. For PB distribution the continuous Rényi, Tsallis entropies are given as follows:

As r approaches 1, the Rényi entropy reduces to the Shannon entropy, HX(1). The Shannon entropy of the PB distribution has following value:

By considering that Φ(z) = d dz log(Γ(z)) where Υ is known as Euler Mascheroni constant.

Let X 1 , . . . , X n are defined as random samples follows the distribution of PB , and X 1:n ,. . . , X n:n are the corresponding order statistics, then pdf and cdf , respectively, of the i th order statistics introduced as follows:

x λ α is hyper geometric function.

Putting i = 1, we obtain pdf and cdf of the minimum order statistics (W n). Where the limit distribution for W n is equal to following form please (see, Theorem (2.1.5) in Galambos [18] )

n .

Putting i = n generates the pdf and cdf of maximum order statistics (Z n ). It has the form (as demonstrated in Theorem (2.1.1) in Galambos). [18] )

Every distribution has number of parameters that determine its behaviour. It is very important to get estimation for the parameters of any distributions either by classical approaches or Bayesian methods. In this part of the paper we will discuss seven estimations approaches of the PB parameters namely:Maximum likelihood estimation (MLE), Anderson-Darling (ADE), Right-tail Anderson-Darling (RADE), Cramér-von Mises (CVME), Ordinary Least-Squares (OLSE), Weighted Least-Squares (WLSE) and Maximum Product of Spacing (MPSE). For more details about these estimation method, see [15] and [14] .

Here we used the most likely approach is to estimate the values of the unknown parameters of the PB model. The PB distribution's log-likelihood reduces to the following:

log (e − 1)

x λ i α

where Θ Θ Θ = (α, λ) . From previous equation, we get

Solving the previous equations mathematically is complicated, so they equations are solved by numerical method. We can use the Newton Raphson Method for solving these equation numerically.

Suppose that a random sample ordered shown as follows, x (1) , . . . , x (n) from F (x) of the PB distributions. By minimising the ADE of the PB parameters can be estimated as below:

The ADE can be derived by solving the next non-linear equation

The RADEs of the PB parameters are getting by minimizing

with respect to α, λ. The RADE can be found by solving the following

where ∆ 1 (·|α, λ) and ∆ 2 (·|α, λ) are defined in Equations (7) and (8). 8 

The CVME is obtained depend on the difference between the cdf estimation and the empirical cdf . Where the CVME of the PB parameters can be get by minimizing

with respect to α and λ. The CVME is calculated by solving the next non-linear equation

where ∆ 1 (·|α, λ) and ∆ 2 (·|α, λ) are defined in Equations (7) and (8).

The OLSE of the parameters of the PB model can be computed by minimizing the next function with respect to α and λ,

Further, the OLSE can be calculated by solving the next non-linear equation

where ∆ 1 (·|α, λ) and ∆ 2 (·|α, λ) are defined in Equations (7) and (8).

The WLSE of the parameters of the PB model can be calculated by minimizing the next function

with respect to α and λ. Also, the WLSE can be calculated by solving the next non-linear equation

and ∆ 1 (·|α, λ) and ∆ 2 (·|α, λ) are defined in Equations (7) and (8).

The MPS method is considered as a good alternative to the MLE method because it approximates the Kullback-Leibler information measure.

The MPSE for α M P SE , λ M P SE can be found by maximizing the geometric mean of the spacing

, with respect to α and λ, or by another way by maximizing the logarithm of the geometric mean of the sample spacings

Then the MPSE of the PB parameters can be calculated by solving next the nonlinear equations

and ∆ 1 (·|α, λ) and ∆ 2 (·|α, λ) are defined in Equations (7) and (8).

Here, we conducted a simulation study of the proposed distribution in order to evaluate the estimation methods' efficiency. Now, we try comparing seven estimators using simulation results: WLSE, OLSE, MLE, MPSE, CVME, ADE, and RADE. We will estimate the (|Bias( Θ Θ Θ)|), using the following equation.

The numerical simulations can be used to establish a guideline for selecting the most appropriate estimation technique for the PB parameters. The R software (version 4.0.3) R is utilized to generate 1000 random samples from the PB distributions for sample sizes n = 20, 70, 150, 400 and 600, along with different parametric values for the distribution.

The numerical simulation for various estimators and seven parameter combinations are presented, including the absolute value of bias, MSE, and MRE are tabulated in Tables 1-4 . Furthermore, these tables show each estimator's rank in regards to the other estimators in each row; superscripts indicate the indicators, and the Ranks denotes the sum is the partial sum of the ranks for all column in a sample size-dependent sample. The table refRanks PB summarises the estimators' partial and overall rankings. 

We used the results conducted from Table 5 , and after analyzing the results we conducted the following findings 1. The behaviour of the PB parameter estimates obtained using the seven estimators is quite reliable. Furthermore, the bias decreases as n increases, implying that all these estimates are asymptotically unbiased. Besides that, the MSE and MRE values decrease as n increases, proving the estimators' consistency.

2. All estimator methods showed consistency, except the MLE estimator method, which showed consistency for all parameter combinations. Table 5 we can clearly deduce that the MLEs approaches is the best method to estimate the parameters of the distribution. Table 5 and for the parameter combinations, we comes up with that the MLE method outperformed all the others estimator methods (overall score of 35). Hence, based on the results, we can consider the MLE method as the best.

Here we discusses the flexibility of PB distribution by fitting two real data sets from for the COVID-19 infections in two different countries, and asses its performance. Also we compared the proposed distribution with other competing distributions. The discrimination criteria including minus maximized log-likelihood (− ), AIC method, CAIC method, BIC method, Hannan information criterion HQIC method, Cramér-Von Mises W method, Anderson-Darling A method, and Kolmogorov-Smirnov (K-S) statistics with its corresponding p-value, are utilized to compare between the fitted competitive distributions. The two analyzed data sets are used to present the flexibility of PB model as compared with others distributions such as Bilal (B) [3] , exponential (E), linear exponential (LE) [27] , Nadarajah-Haghighi exponential (NHE) [22] , Weibull (W), Lindley (L), modified Kies-exponential (MKE) [8] , Burr-Hatke (BH) [20] and Frechet (F) distributions. Tables 6 and 8 report parameters estimates for the given data sets by using MLE estimation method (as were recommended in simulations section). Figure 4 , 6 discusses the plots of the fitted PB pdf , cdf , SF, and P-P plots for the second data sets. Further, Figure 5 , 7 provide the plots of the probability-probability (PP) of the PB model and other fitted models for the two data sets by different estimation methods, respectively. Figure 8 , 10, discuss the maximization of the estimates, and prove that the estimates are global maximum. Figure 9 , and 10 discusses the existence of the estimates and proves that they are unique.

For some specific data, selection of the modes is one ways of the scientific study in choosing the predictive model from a group of proposed models. There are a numbers of statistical methods that can be used to determine the fitness of competing the statistical distributions, and the common criteria are the (AIC) method and the (BIC)method. A model with the lowest values can be selection as the best model for the data sets. These are some methods formulas: The AIC is given by

The CAIC is given by

The BIC is given by

The HQIC is given by

Where is defined the log-likelihood function value's under the MLE, also k is the number of parameters in the proposed model, and n is defined the sample size.We used the AIC and BIC tests to show that the distribution persented is the most right fit for the data. To make the comparison between other distributions, we have to do this comparison based on some criteria. These are some names and references of criterion criterione: AIC criterion, see (Akaike [5] ), BIC criterion, see (Schwarz [23] ), HQIC criterion , see (Hannan and Quinn [19] ) and CAIC criterion, see (Bozdogan [17] ), where all these criteria were used to compare the goodness of fit of the proposed model with other competing distributions.

The first data that we used in this paper represents the COVID-19 data of the Netherlands for a period of thirty days during the period from 31 March to 30 April 2020, where this data formed of rough mortality rate. For more details on this data see and other Covid-19 data please see ( [7] , [11] , [21] , [16] ). 

The second data that we used in this paper represents the COVID-19 data of Saudi Arabia for a period of 283 days during the period from 20 April 2020 to 17 January 2021, where this data represents the daily mortality rates and it is reported on [1]. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q Figure 6 : The fitted PB pdf , cdf , SF, and P-P plots for the Data set II. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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Figure 8, 10 discuss that the estimated parameters make maximization for the Log-likelihood function. We estimate the roots using Mathematica 12, by the aid of NMaximize Function, Which always finds global maximum, not local maximum. Also we assured this results by graphing the log-likelihood function, as shown we can find the red dot donates that the estimates is at max point thought the whole curve. Figure 9 ,and 11 shows that the estimates not only maximum but also unique. As we can see the graph of the derivative equation is decreasing function and the curve intersects the X-axis in one point which is the estimate, so we can conclude that this roots (estimates) are unique. 2. Additionally, the P-value for the KS statistic is greatest when the lifetime is PB distribution. This leads us to deduce that PB better fits the two real sets.

3. Estimated pdf and cdf of model plots shown in Figures 4, 6 imply that the proposed distribution is an excellent choice for fitting and modeling the above COVID-19 data.

4. Figure 8 , 10 confirms that the log-likelihood function has a global maximum roots for the model parameters.

5. Figure 9 ,11 show that as the log-likelihood function has global maximum roots the MLEs are unique.

As it is clear that the log-likelihood function is a decreasing and intersects the x-axis at only one point. Therefore, we can say that the log-likelihood function has unique roots, and they are global maximum at the same time.

In this work introduced a new lifetime distribution named the PB distribution. We investigated its statistical properties. Numerous classical estimation methods were considered to obtain point estimates for the unknown PB parameters α, λ. To compare the performance of various estimation methods, a simulation analysis was done by using the R package. Two different COVID-19 data were used to indicate the superiority of the proposed distribution. It was determined that PB fit the data effectively than most other competing distributions. Figure  9 ,11 shows the existence of MLEs as the log-likelihood function cross the x-axis at one point. As it is clear that the log-likelihood function is a decreasing function and intersects x-axis at one point. Furthermore, Figure 8 , 10 shows that the log-likelihood function has global maximum roots. Therefore, we can say that the log-likelihood function has unique roots, and they are global maximum at the same time. Also we graphed the P-P plot of the estimated pdf and cdf . For the future work, we can use the proposed model to modelling different real data sets in numbers of area such reliability engineering, survival analysis and so on. Also we can extend the PB distribution to introduce the bivariate of PB distributions and apply it to modelling real data set,,.

The authors extend their appreciation to the Deanship of Scientific Research at Jouf University for funding this work through research grant No (DSR-2021-03-0109) 22 

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