# A New Version of Weighted Weibull Distribution: Modelling to COVID-19 Data

Authors: Amani Abdullah Alahmadi; Mohammed Alqawba; Waleed Almutiry; A. W. Shawki; Sharifah Alrajhi; Sanaa Al-Marzouki; Mohammed Elgarhy
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this study, we will look at a new flexible model known as the new double-weighted Weibull distribution. The new Weibull double-weighted distribution model is highly versatile because numerous submodels are included. The proposed model is very flexible because its density function has many shapes; it can be right skewness, decreasing, and unimodal. Also, the hazard rate function can be increasing, decreasing, up-side-down, and J-shaped. Diverse features of the novel are computed. These qualities include moments, incomplete moments, and Lorenz and Bonferroni curves and quantiles, as well as entropy and order statistics. The maximum likelihood approach is used to estimate the model's parameters. In order to evaluate the accuracy and performance of maximum likelihood estimators, simulation data are presented. The utility and adaptability of the proposed model are demonstrated by utilizing three significant datasets: daily fatalities confirmed cases of COVID-19 in Egypt and Georgia and relief times of twenty patients using an analgesic.


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## Body

## 1. Introduction

In 1951, Swedish scientist Walled Weibull created the Weibull (Wei) distribution. The Wei distribution is a frequently used distribution for modelling lifetime data in dependability where the hazard rate function is monotone. However, when the true hazard shape is unimodal or bathtub, the two-parameter Wei distribution is inadequate in many applications, such as lifetime analysis. To deal with bathtub-shaped failure rates, many generalizations of the Wei distribution have been proposed in the statistical literature.The probability density function (pdf) and cumulative distribution function (cdf) have the following shape:(1)gx=αβxβ−1e−αxβ,x>0,α,β>0,(2)Gx=1−e−αxβ,x>0,where β is a positive shape parameter and α is a positive scale parameter.Weighted (W) distributions may be used to increase comprehension of standard distributions as well as provide techniques for extending distributions for additional flexibility in fitting a dataset. A W distribution can be obtained in a variety of ways. In 1934, Fisher proposed the concept of W distribution. Rao [1] and Patil and Rao [2] discuss applications of a W distribution to biased samples in many disciplines such as medicine, ecology, dependability, and branching processes (1978). Patil and Rao [2] suggested a double-weighted (DW) family of distributions, which has the following characteristics:(3)fwx=w1xgxw2xWD,where w1x,w2x>0 and WD=∫0∞w1xgxw2xdx is a normalizing constant that forces fwx to integrate to 1. Table 1 shows the literature for authors who use the double-weighted family.Table 1
Some literature of double-weighted distributions.
ModelAuthorsDW exponential distributionAl-khadim and hantoosh [3]DW Rayleigh distributionRishwan [4]DW inverse Wei distributionAl-khadim and hantoosh [5]The motivation and limitations of our study to introduce a new double-weighted Weibull (NDWW) is a novel double-weighted distribution. The mathematical features of the proposed model are described in detail in the expectation that it may find wider applications in dependability, engineering, and other study areas. It includes a number of interesting features and allows for greater flexibility in incorporating filtered and uncensored survival data into real-world applications. To illustrate our point, we use three real-life datasets, which correspond to the daily death toll from COVID-19 in Egypt and Georgia and the relief times of twenty patients taking analgesics. NDWW offers a better fit than other statistical distributions, according to a study. Also, our new model serves the data which are positive data.The remainder of this study can be divided into the following categories. Section2 defines the NDWW. Section 3 derives some of the distribution's broad statistical features. The maximum likelihood method is used to estimate the distribution's parameters in Section 4. Section 5 conducts a simulation analysis to determine the model parameters of one distribution. Section 6 delves into an illustrated goal based on real-world facts. The study closes with some last thoughts.


## 2. The New Double-Weighted Weibull

In this section, we introduce a new four-parameter model which is called the NDWW model by using new two transformationsw1x=xθandw2x=G¯cx, and by inserting (1) and (2) in (3), then the pdf and cdf of the NDWW are given by(4)fwx=βαcβ+1θ/β+1Γθ/β+1xθ+β−1e−αcβ+1xβ,x>0,α,β,θ,c>0,and(5)Fx=γθ/β+1,αcβ+1xβΓθ/β+1,x>0,α,β,θ,c>0,respectively; where the gamma function is defined by(6)Γa=∫0∞ta−1e−tdt,a>0.and incomplete gamma function is defined by(7)γa,x=∫0xta−1e−tdt,a>0,x>0.On the basis ofX, the survival function Sx, hazard function hx, and reversed hazard function ax are given(8)Sx=1−γθ/β+1,αcβ+1xβΓθ/β+1,x>0,hx=βαcβ+1θ/β+1xθ+β−1e−αcβ+1xβΓθ/β+1−γθ/β+1,αcβ+1xβ,x>0,and(9)ax=βαcβ+1θ/β+1xθ+β−1e−αcβ+1xβγθ/β+1,αcβ+1xβ.Figure1 shows the pdf and hazard rate function graphs for the NDWW distribution.Figure 1
pdf and hazard rate of the NDWW model.We may deduce from Figure1 that the pdf of the NDWW distribution can be unimodal, declining, inverted J shaped, and right skewed. In addition, as shown in Figure 1, the hazard rate function of the NDWW distribution might be rising, decreasing, or J-shaped.It should be noted thatα and c remain scale parameters, whereas β and θ are shape parameters. In Table 2, certain distributions appear as special instances of the NDWW distribution.Table 2
Submodels of the NDWW distribution.
ΘβαcDistributionAuthor————NDWWNEWk+λ——0Weighted Weibull distribution (WWW)NEWk+3——0Volume biased weighted Weibull distribution (VBWW)Ahmed et al. [6]k+2——0Area biased weighted Weibull distribution (ABWW)Perveen et al. [7]k+1——0Size biased weighted Weibull distribution (SBWW)Ahmad [6]Kβ——0Weighted Weibull distribution (WW)Dey et al. [8]—1—0Weighted exponential distribution (WE)Ahmed et al. [6]11—0Length biased (LB) exponential distribution (LBE)Dara and ahmad [9].—21/(2k)0Weighted Rayleigh distribution (WR)Ajami and jahanshahi [10]121/(2k)0LB Rayleigh distribution (LBR)NEW221/(2k)0Area biased Rayleigh distribution (ABR)NEW01—0One parameter exponential distribution (E)Sea ahmed et al. [6]02—0One parameter Rayleigh distribution (R)Sea ajami and jahanshahi [10]0——0Weibull distribution (W)Sea dey et al. [8]—1—0LB gamma distribution (LBG)Ahmed et al. [6]—110One-parameter gamma distribution (G)Sea Ahmed et al. [6]βk−1—λβ0Generalized gamma distribution (GG)Khodabin and Ahmadabadi [11]k-2211Chi distribution2(k-1)2λ20Generalized normal distribution (GN)Saralees [12]−12σ21Half-normal distributionCooray and ananda [13]βk+1—λβ0LB weighted weibull distribution (LBWW)Kishore and tanusree [14]k-22—0Generalized Rayleigh distributionSee das and roy [15]22—0LB Rayleigh distribution (LBR)Bashir and rasul [16]2(k-1)2kc2—Weighted generalized Rayleigh distribution (WGR)Das and roy [15]2k-12kc2/c2+1—LB weighted generalized Rayleigh distribution (LBWGR)Das and roy [15]βk+1—1/λβ0LB weighted Weibull distribution (LBWW)Das and roy [17]11λ−k0LB weighted exponential distribution (LBWE)Al-kadim and hussein [18]22—1/k−1LB weighted Rayleigh distribution (LBWR)Al-kadim and hussein [18]12λ/20Maxwell distribution (M)See dar et al. [19]12λ−k/20Weighted Maxwell distribution (WM)Joshi and modi [20]k+12λ/20Weighted Maxwell distribution (WM1)Dar et al. [19]22λ/20LB Maxwell distribution (LBM)NEW32λ/20Area biased Maxwell distribution (ABM1)NEW2λ−12λ/k0Nakagami distribution (N)See mudasir and ahmed [21]2λ+σ−22λ/k0Weighted Nakagami distribution (WN)Mudasir and ahmed [21]2λ2λ/k0Area biased Nakagami distribution (ABN)NEW2λ−12λ/k0LB Nakagami distribution (LBN)Mudasir and ahmad [21]λ+11—1Weighted Ailamujia distribution (WA)11—1Ailamujiadistribution (A)See Jan et al. (2017)21—1LB Ailamujia distribution (LBA)NEW31—1Area biased Ailamujia distribution (ABA)NEW


## 3. Structural Properties

The quantile function, mean residual life function, moment generating function (MGF), moments, Lorenz and Bonferroni curves, mean, and variance for the NDWW distribution are shown in this section.


### 3.1. Quantile Function

The quantile function is(10)Qu=1αcβ+1γ−1θβ+1,uΓθβ+1β.


### 3.2. Moments

IfX has the pdf (4), then rth moment is obtained as follows:(11)μr`=∫0∞xrfxdx=βαcβ+1θ/β+1Γθ/β+1∫0∞xr+θ+β−1e−αcβ+1xβdx.then(12)μr`=EXr=Γ1+r/β+θ/βαcβ+1r/βΓ1+θ/β.The first fourrth moments can be obtained by putting r = 1, 2, 3, and 4 as(13)μ1`=Γ1+1/β+θ/βαcβ+11/βΓ1+θ/β,μ2`=Γ1+2/β+θ/βαcβ+12/βΓ1+θ/β,μ3`=Γ1+3/β+θ/βαcβ+13/βΓ1+θ/β,and(14)μ4`=Γ1+4/β+θ/βαcβ+14/βΓ1+θ/β.The variance can be calculated as(15)varx=μ2`−μ1`2=Γ1+2/β+θ/βαcβ+12/βΓ1+θ/β−Γ1+1/β+θ/βαcβ+11/βΓ1+θ/β2.By definition of moment generating function ofX and using (4), we have(16)Μxt=∑r=0∞trr!μr`=∑r=0∞trr!Γ1+r/β+θ/βαcβ+1r/βΓ1+θ/β.


### 3.3. Conditional Moments

Thesth conditional moment, say τst, is defined by(17)τst=∫t∞xsfxdx.Using (3), then τst will be as follows:(18)τst=βαcβ+1θ/β+1Γθ/β+1∫t∞xs+θ+β−1e−αcβ+1xβdx.Then,(19)τst=Γ1+s+θ/β,αcβ+1xβαcβ+1s/βΓ1+θ/β.


### 3.4. Incomplete Moments

Thesth incomplete moment, say φst, is defined by(20)φst=∫0txsfxdx.Using (3), then, φst will be as follows:(21)φst=βαcβ+1θ/β+1Γθ/β+1∫0txs+θ+β−1e−αcβ+1xβdx.Then,(22)φst=γ1+s+θ/β,αcβ+1xβαcβ+1s/βΓ1+θ/β.


### 3.5. Mean Residual Life Function

The mean residual life function may be calculated by using the following formula:(23)μx=EX−x|X>x=∫x∞yfxdyF¯x−x,In the NDWW distribution, we obtain the MRL function as follows:(24)μx=Γ1+1+θ/β,αcβ+1xβ/αcβ+11/βΓ1+θ/β1−γθ/β+1,αcβ+1xβ/Γθ/β+1−x.


### 3.6. Lorenz and Bonferroni Curves

The Lorenz (Lo(x)) and Bonferroni (Bo(x)) curves are inequality metrics that are widely employed in income and wealth distributions [22]. They are gained in the following order:(25)Lox=∫0txfxdxΕX=γ1+1+θ/β,αcβ+1xβΓ1+1+θ/β,and(26)Box=LoxFx=γ1+1+θ/β,αcβ+1xβΓ1+1+θ/βγ1+θ/β,αcβ+1xβ.


### 3.7. Order Statistics

LetX1: n < X2:n < … < Xn: n be the matching ordered random sample from an n-person population. The sth order statistic's pdf is defined as(27)fs:nx=n!s−1!n−s!fxFxs−11−Fxn−s.By entering (4) and (5) into (27), the pdf of the sth order statistic for the NDWW distribution may be derived:(28)fs:nx=n!s−1!n−s!βαcβ+1θ/β+1Γθ/β+1xθ+β−1e−αcβ+1xβγθ/β+1,αcβ+1xβs−1×1−γθ/β+1,αcβ+1xβn−s.The smallest pdf of order statistics can be obtained ats = 1 as(29)f1:nx=nβαcβ+1θ/β+1Γθ/β+1xθ+β−1e−αcβ+1xβ1−γθβ+1,αcβ+1xβn−1.The greatest pdf of order statistics can be obtained ats = n as(30)fn:nx=nβαcβ+1θ/β+1Γθ/β+1xθ+β−1e−αcβ+1xβγθβ+1,αcβ+1xβn−1.


## 3.1. Quantile Function

The quantile function is(10)Qu=1αcβ+1γ−1θβ+1,uΓθβ+1β.


## 3.2. Moments

IfX has the pdf (4), then rth moment is obtained as follows:(11)μr`=∫0∞xrfxdx=βαcβ+1θ/β+1Γθ/β+1∫0∞xr+θ+β−1e−αcβ+1xβdx.then(12)μr`=EXr=Γ1+r/β+θ/βαcβ+1r/βΓ1+θ/β.The first fourrth moments can be obtained by putting r = 1, 2, 3, and 4 as(13)μ1`=Γ1+1/β+θ/βαcβ+11/βΓ1+θ/β,μ2`=Γ1+2/β+θ/βαcβ+12/βΓ1+θ/β,μ3`=Γ1+3/β+θ/βαcβ+13/βΓ1+θ/β,and(14)μ4`=Γ1+4/β+θ/βαcβ+14/βΓ1+θ/β.The variance can be calculated as(15)varx=μ2`−μ1`2=Γ1+2/β+θ/βαcβ+12/βΓ1+θ/β−Γ1+1/β+θ/βαcβ+11/βΓ1+θ/β2.By definition of moment generating function ofX and using (4), we have(16)Μxt=∑r=0∞trr!μr`=∑r=0∞trr!Γ1+r/β+θ/βαcβ+1r/βΓ1+θ/β.


## 3.3. Conditional Moments

Thesth conditional moment, say τst, is defined by(17)τst=∫t∞xsfxdx.Using (3), then τst will be as follows:(18)τst=βαcβ+1θ/β+1Γθ/β+1∫t∞xs+θ+β−1e−αcβ+1xβdx.Then,(19)τst=Γ1+s+θ/β,αcβ+1xβαcβ+1s/βΓ1+θ/β.


## 3.4. Incomplete Moments

Thesth incomplete moment, say φst, is defined by(20)φst=∫0txsfxdx.Using (3), then, φst will be as follows:(21)φst=βαcβ+1θ/β+1Γθ/β+1∫0txs+θ+β−1e−αcβ+1xβdx.Then,(22)φst=γ1+s+θ/β,αcβ+1xβαcβ+1s/βΓ1+θ/β.


## 3.5. Mean Residual Life Function

The mean residual life function may be calculated by using the following formula:(23)μx=EX−x|X>x=∫x∞yfxdyF¯x−x,In the NDWW distribution, we obtain the MRL function as follows:(24)μx=Γ1+1+θ/β,αcβ+1xβ/αcβ+11/βΓ1+θ/β1−γθ/β+1,αcβ+1xβ/Γθ/β+1−x.


## 3.6. Lorenz and Bonferroni Curves

The Lorenz (Lo(x)) and Bonferroni (Bo(x)) curves are inequality metrics that are widely employed in income and wealth distributions [22]. They are gained in the following order:(25)Lox=∫0txfxdxΕX=γ1+1+θ/β,αcβ+1xβΓ1+1+θ/β,and(26)Box=LoxFx=γ1+1+θ/β,αcβ+1xβΓ1+1+θ/βγ1+θ/β,αcβ+1xβ.


## 3.7. Order Statistics

LetX1: n < X2:n < … < Xn: n be the matching ordered random sample from an n-person population. The sth order statistic's pdf is defined as(27)fs:nx=n!s−1!n−s!fxFxs−11−Fxn−s.By entering (4) and (5) into (27), the pdf of the sth order statistic for the NDWW distribution may be derived:(28)fs:nx=n!s−1!n−s!βαcβ+1θ/β+1Γθ/β+1xθ+β−1e−αcβ+1xβγθ/β+1,αcβ+1xβs−1×1−γθ/β+1,αcβ+1xβn−s.The smallest pdf of order statistics can be obtained ats = 1 as(29)f1:nx=nβαcβ+1θ/β+1Γθ/β+1xθ+β−1e−αcβ+1xβ1−γθβ+1,αcβ+1xβn−1.The greatest pdf of order statistics can be obtained ats = n as(30)fn:nx=nβαcβ+1θ/β+1Γθ/β+1xθ+β−1e−αcβ+1xβγθβ+1,αcβ+1xβn−1.


## 4. Maximum Likelihood Estimation

Ifn independent items are tested and the lifetime distribution of each component is supplied by (3), then the log likelihood (LL) function based on the sample that was seen x = (x1, x2, K, xn) is defined:(31)lnL=nlnβ+nθβ+1lnα+nθβ+1lncβ+1−nlnΓθβ+1+θ+β−1∑i=1nlnxi−αcβ+1∑i=1nxiβ.The first partial derivatives of the parameters are(32)∂lnL∂θ=nβlnα+nβlncβ+1−nβψ1+θβ+∑i=1nlnxi=0,(33)∂lnL∂α=nαθβ+1−cβ+1∑i=1nxiβ=0,(34)∂lnL∂c=nβcβ−1θβ+1ψcβ+1−αβcβ−1∑i=1nxiβ=0,(35)∂lnL∂β=nβ−nθβ2lnαcβ+1+ncβlncθ/β+1cβ+1+nθβ2ψ1+θβ+∑i=1nlnxi−αcβ+1∑i=1nxiβlnxi−αcβlnc∑i=1nxiβ=0,where ψx=Γx/Γ¯x is the digamma function.Because the LL equations (32-35) do not have analytic solutions, we will utilize iterative techniques to compute these equations' numerical solutions to provide maximum LL estimators of θ,α, c, and β, say θ^MLE, α^MLE, c^MLE, and β^MLE, respectively.


## 5. Numerical Results

In this section, we evaluate the ML estimators' performance in terms of sample sizen. A numerical evaluation of the performance of ML estimators for the NDWW distribution is performed. Estimates are X1, X2, K, and Xn and are evaluated using the Mathematica program based on the following quantities for each sample size: empirical mean square errors (MSEs). The following are the numerical procedures:(i)
A random sample of sizesn = 100, 500, and 1000 is taken into account; these random samples are produced from the NDWW distribution using the inversion approach.(ii)
Six sets of parameters are taken into account.(iii)
The NDWW model's ML estimates (MLEs) are assessed for each parameter value and sample size.(iv)
Repeat this process 10000 times to obtain the means and MSEs of the MLE for various parameter values in both models and for each sample size.(v)
Tables3–5 present empirical findings. These tables show that the estimates are fairly consistent and near to the real value of the parameters as sample sizes grow.Table 3
MLE and MSE of NDWW distribution for set 1and set 2.
nParametersInitMLEMSEInitMLEMSE100α0.5000.9330.3400.5000.5340.262β1.5001.0040.2800.5000.5040.003θ1.0000.8040.9151.5001.2150.255c0.5000.3860.4380.3000.4690.048500α0.5000.8870.3240.5000.4940.077β1.5001.3360.2090.5000.5140.003θ1.0000.8550.8871.5001.3670.124c0.5000.4110.2660.3000.4440.0371000α0.5000.5570.0970.5000.5030.076β1.5001.4570.1000.5000.5170.002θ1.0000.9200.4241.5001.5370.095c0.5000.5200.1420.3000.3380.032Table 4
MLE and MSE of NDWW distribution for set 3 and set 4.
nParametersInitMLEMSEInitMLEMSE100α0.1000.4750.1911.3001.0871.346β0.7000.4390.0721.1001.0190.187θ0.7000.9370.3791.1001.4860.726c0.3000.3600.0212.8002.6830.205500α0.1000.2400.1571.3001.1800.789β0.7000.5420.0701.1001.0710.123θ0.7000.8460.3331.1001.4440.687c0.3000.3560.0192.8002.6980.1361000α0.1000.1370.06061.3001.3370.597β0.7000.6600.0621.1001.0770.056θ0.7000.7240.1841.1001.2910.504c0.3000.3100.0182.8002.7530.081Table 5
MLE and MSE of NDWW distribution for set 5 and set 6.
nParametersInitMLEMSEInitMLEMSE100α1.8002.1190.8740.5000.5340.262β0.8000.7500.1240.5000.5140.003θ0.8001.1420.98411.5001.2150.755c3.0003.6300.9480.3000.4690.048500α1.8002.0180.7670.5000.4940.177β0.8000.7640.0560.5000.5140.003θ0.8001.0820.5911.5001.3670.524c3.0003.2720.4560.3000.4040.0371000α1.8001.86130.5110.5000.5030.076β0.8000.7740.0410.5000.5070.001θ0.8000.8530.4171.5001.5370.395c3.0003.1440.3940.3000.3380.032


## 6. Applications

In this section, we apply the NDWW distribution to illustrate our point; we use three real-life datasets, which correspond to the daily death toll from COVID-19 in Egypt and Georgia and the relief times of twenty patients taking analgesics. To compare the NDWW distribution to other fitted distributions, five, four, three, two, and one parameters are used. The NDWW distribution is compared to the (DWE), (LBE), (W), new modified Weibull distribution (NMW), weighted generalized gamma distribution (WGG), weighted transmuted Weibull distribution (WTW), transmuted Weibull distribution (TW), and weighted Weibull distribution (WWW).The first dataset proposes a concrete application with an actual dataset to assess the interest in the NDWW distribution. The considered data, called the daily deaths, confirmed cases of COVID-19 in Egypt from 19 March to 15 June 2020. The dataset was obtained from the following electronic address: https://www.care.gov.eg/(1, 1, 2, 4, 5, 1, 1, 3, 6, 6, 4, 1, 5, 6, 6, 8, 5, 7, 7, 9, 9, 15, 17, 11, 13, 5, 14, 5, 13, 9, 19, 15, 11, 14, 12, 11, 7, 13, 10, 20, 22, 21, 12, 14, 9, 14, 7, 16, 17, 13, 21, 11, 11, 8, 11, 12, 15, 21, 20, 18, 15, 14, 21, 16, 11, 28, 29, 19, 14, 19, 29, 34, 34, 46, 46, 47, 36, 38, 40, 32, 39, 34, 35, 36, 35, 45, 62, 91, 97).Tables6 and 7 provide the ML estimates of the distribution parameters together with their standard errors (SErs). The analytical measures include the Akaike information criterion (Z1), the correct Akaike information criterion (Z2), the Bayesian information criterion (Z3), and the Hannan-Quinn information criterion (Z4), the Anderson Darling statistic (Z5), the Cramér-von Mises statistic (Z6), the Kolmogorov–Smirnov test (Z7), and the p value (Z8).Table 6
MLEs and (SErs) for the dataset (1) COVID-19.
Distributionα^β^θ^c^λ^NDWW1.1940.690.1462.107—(0.966)(0.295)(0.00016)(0.00005)—WTW0.0000650.8350.1140.176—(0.158)(0.112)(0.863)(0.119)—NMW1.1950.0000560.0280.021.006(0.094)(0.0001)(0.009)(5.051)(26.457)WGG0.7160.690.5241.1581.522(2789)(0.295)(2079)(4232)(1434)DWE——0.10761.943———(0.008)(153.721)—LBE——0.107————(0.008)——Table 7
Numerical values of Z1, Z2, Z3, Z4, Z5, Z6, Z7, and Z8 for the dataset (1) COVID-19.
DistributionZ1Z2Z3Z4Z5Z6Z7Z8NDWW700.22700.696700.017704.2320.621550.09470.075110.69687WTW700.507700.984700.305704.520.631190.099250.081510.59543NMW704.86705.583704.607709.8761.495640.1035050.098230.35689WGG702.22702.943701.967707.2350.621930.094810.075110.69687DWE703.964704.103703.863705.971.556290.2614940.119590.15677LBE701.995702.041701.945702.9991.53510.257940.119580.1568The second dataset is taken from the work of Gross and Clark [23]. The values of Z1, Z2, Z3, Z4, Z5, Z6, Z7, and Z8 are given in Tables 8 and 9.Table 8
MLEs and (SErs) for the dataset (2).
Distributionα^β^θ^c^NDWW19.9940.4735.86124.27(18.594)(0.398)(18.786)(136.949)TW0.0633.0960.711-(0.033)(0.47)(0.316)-WTW0.0182.0620.1020.451(7.738)(0.211)(0.632)(0.074)W-2.7870.122--(0.427)(0.056)-DWE--0.00021.579--(0.573)(0.496)WWW5.530.74510.6436.395(1.799)(0.448)(16.104)(1.799)Table 9
Numerical values of Z1, Z2, Z3, Z4, Z5, Z6, Z7, and Z8 for the dataset (2).
DistributionZ1Z2Z3Z4Z5Z6Z7Z8NDWW42.45745.12339.66143.2340.516940.087750.156260.71323TW45.49546.99543.39846.0780.943570.15720.172660.59002WTW46.64649.31343.8547.4230.870240.144410.183350.51201W45.17345.87943.77545.5621.068150.1760.211960.33003DWE49.77550.4848.37750.1632.551330.217070.252590.15575WWW43.05445.72140.25843.8320.57210.096840.164940.64802The third dataset proposes a concrete application with an actual dataset to assess the interest in the NDWW distribution. The considered data, called the daily deaths, confirmed cases of COVID-19 in Georgia from 1 Jan to 31 March 2021. The dataset was obtained from the following electronic address: https://github.com/CSSEGISandData/COVID-19/(23, 44, 31, 25, 18, 20, 28, 34, 22, 23, 23, 24, 26, 21, 26, 23, 17, 25, 15, 14, 11, 24, 16, 17, 16, 25, 12, 19, 21, 11, 19, 16, 14, 13, 19, 18, 11, 14, 15, 8, 5, 10, 15, 7, 9, 11, 14, 13, 9, 7, 19, 10, 12, 10, 6, 12, 10, 14, 11, 10, 12, 9, 12, 10, 4, 9, 15, 10, 12, 9, 12, 9, 5, 2, 8, 7, 9, 9, 8, 9, 10, 4, 8, 10, 6, 8, 5, 14, 8, 5). The values of Z1, Z2, Z3, Z4, Z5, Z6, Z7, and Z8 are given in Tables 10 and 11. Many studies proposed COVID-19 datasets, such as, the works of Alanazi et al. [24], Abdul Latif et al. [25], and Kaur et al. [26].Table 10
MLEs and (SErs) for the dataset (3) COVID-19.
Distributionα^β^θ^c^λ^NDWW12.760.2766.7725.318-(12.545)(0.267)(21.414)(177.294)-WTW2.861.7930.5890.0062-(1.887)(0.252)(0.665)(0.0066)-NMW1.8890.00290.00250.000091.889(0.283)(0.00288)(0.00229)(0.27)(0.28)WGG0.00740.6863.7581.5192.86(18.343)(0.00079)(188.427)(30.49)3(134.32)Table 11
Numerical values of Z1, Z2, Z3, Z4, Z5, Z6, Z7, and Z8 for the dataset (3) COVID-19.
DistributionZ1Z2Z3Z4Z5Z6Z7Z8NDWW600.834601.305600.651604.8670.41680.064540.06920.78194WTW601.129601.6600.946605.1610.457960.081090.083560.5561NMW610.64611.355610.412615.6811.224550.214580.109330.23224WGG603.278603.993603.05608.3190.45250.078630.07960.61856We observe the NDWW distribution gives the best fit among the other selected models for the three datasets. The fitted pdfs and cdfs plots and pp plots of mentioned distributions for the first, second, and third data are represented in Figures2–7 for the purpose visual comparison. According to these plots, we observe that the NDWW distribution is better than other distributions.Figure 2
The fitted pdfs (a) and cdfs (b) for the dataset (1).
(a)(b)Figure 3
The pp plots for the dataset (1).Figure 4
The fitted pdfs (a) and cdfs (b) for the dataset (2).
(a)(b)Figure 5
The pp plots for the dataset (2).Figure 6
The fitted pdfs (a) and cdfs (b) for the dataset (3).
(a)(b)Figure 7
The pp plots for the dataset (3).


## 7. Conclusions

NDWW (new double-weighted Weibull distribution) is a new four-parameter double-weighted Weibull distribution that is suggested. As a result, more data may be examined using the model. Moments, moment generating function (MGF), and incomplete moments are addressed. Quantile function, mean residual life, AND Lorenz and Bonferroni curves are also examined. The maximum likelihood approach is used to estimate the parameters of the models. The behavior of MLE estimations is examined using simulation. The suggested model's utility is demonstrated by three real-world datasets: daily fatalities confirmed instances of COVID-19 in Egypt and Georgia and relief times of twenty patients getting an analgesic. In the future, we plan to use Neutrosophic statistics, as many authors use this new branch, such as, Mahapatra et al. [27] and Mahapatra et al. [28].



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*Source: 3994361-2022-04-21.xml*