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.
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: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:where and is a normalizing constant that forces to integrate to 1. Table 1 shows the literature for authors who use the double-weighted family.
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. Section 2 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 transformations , and by inserting (1) and (2) in (3), then the pdf and cdf of the NDWW are given byandrespectively; where the gamma function is defined by
and incomplete gamma function is defined by
On the basis of X, the survival function , hazard function , and reversed hazard function are givenand
Figure 1 shows the pdf and hazard rate function graphs for the NDWW distribution.
We may deduce from Figure 1 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 that α 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.
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
3.2. Moments
If has the pdf (4), then moment is obtained as follows:then
The first four rth moments can be obtained by putting r = 1, 2, 3, and 4 asand
The variance can be calculated as
By definition of moment generating function of X and using (4), we have
3.3. Conditional Moments
The sth conditional moment, say , is defined by
Using (3), then will be as follows:
Then,
3.4. Incomplete Moments
The sth incomplete moment, say , is defined by
Using (3), then, will be as follows:
Then,
3.5. Mean Residual Life Function
The mean residual life function may be calculated by using the following formula:
In the NDWW distribution, we obtain the MRL function as follows:
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:and
3.7. Order Statistics
Let X1: n < X2:n < … < Xn: n be the matching ordered random sample from an n-person population. The order statistic's pdf is defined as
By entering (4) and (5) into (27), the pdf of the order statistic for the NDWW distribution may be derived:
The smallest pdf of order statistics can be obtained at s = 1 as
The greatest pdf of order statistics can be obtained at s = n as
4. Maximum Likelihood Estimation
If n 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:
The first partial derivatives of the parameters arewhere 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 , , , and , respectively.
5. Numerical Results
In this section, we evaluate the ML estimators' performance in terms of sample size n. 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 sizes n = 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)Tables 3–5 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.
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).
Tables 6 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).
The 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.
The 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].
We 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 Figures 2–7 for the purpose visual comparison. According to these plots, we observe that the NDWW distribution is better than other distributions.
(a)
(b)
(a)
(b)
(a)
(b)
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].
Abbreviations
| Wei: | Weibull |
| pdf: | The probability density function |
| cdf: | Cumulative distribution function |
| W: | Weighted |
| DW: | Double weighted |
| NDWW: | New double-weighted Weibull |
| MGF: | Moment generating function |
| LL: | Log likelihood |
| MSEs: | Mean square errors |
| MLEs: | Maximum likelihood estimates |
| SErs: | Standard errors. |
Data Availability
The data used to support the findings of the study can be obtained from the corresponding author upon request.
Conflicts of Interest
There are no conflicts of interest in this paper's publishing.
Acknowledgments
The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.