Abstract
Nowadays, researchers in applied sectors are highly motivated to propose and study new generalizations of the existing distributions to provide the best fit to data. To provide a close fit to data in numerous sectors, a series of new distributions have been proposed. In this study, we propose a new family called the new generalized- (for short, “NG-”) family of distributions. Based on the NG- method, a novel modification of the Weibull model called the new generalized-Weibull (for short, “NG-Weibull”) distribution is studied. The heavy-tailed characteristics of the NG- distributions are derived. The maximum likelihood estimators of the NG- distributions are also obtained. Based on the special case (i.e., NG-Weibull) of the NG- family, a simulation study is provided. The practical performance of the new NG-Weibull model is assessed by analyzing the COVID-19 data set. The fitting results of the NG-Weibull model are compared with three other competing models. Based on certain statistical measures, it is observed that the NG-Weibull model is the best competitive model.
1. Introduction
In the practice of applied statistics, statisticians are highly motivated and attracted to introduce new probability distributions. This motivation has gained popularity in the recent decade. In this credit, the development and introduction of new families of distribution have received much attention. The new families are developed by introducing new parameters [1–4].
Theoretically and practically, some distributions do not have enough flexibility to counter the complex forms of the data sets. For example, the classical models do not provide a reasonable fit to the extreme value data [5]. This is the main attracting reason to go for the development of new statistical distributions. The development of the modified forms of the existing distributions helps to improve the fitting power of the existing distributions [6].
Recently, Chipepa et al. [7] introduced a new generalized family of distributions called the OGHLW-G (odd generalized half-logistic Weibull-G) family of distributions. They have obtained some structural properties of the OGHLW-G family. They have also discussed some special models of the OGHLW-G family to show the importance of the new models. Chipepa et al. [8] also proposed the Burr III-Topp-Leone-G family of distributions for analyzing (i) carbon fibres data, (ii) fracture toughness of silicon nitride data, and (iii) turbocharger failure times data.
Ahmad and Hussain [9] introduced a NEF-Weibull (new extended flexible-Weibull) distribution to model the lifetime data with bathtub-shaped failure rates. Some other modified and extended distributions include the beta-Weibull (B-Weibull) distribution of Famoye et al. [10]; the flexible Weibull (FWEx) distribution of Bebbington et al. [11]; the generalized modified Weibull (GM-Weibull) distribution proposed by Carrasco et al. [12]; the Kumaraswamy Weibull (K-Weibull) distribution proposed by Cordeiro et al. [13]; the beta modified Weibull (BM-Weibull) distribution of Silva et al. [14]; the exponentiated modified Weibull extension (EMWE) distribution introduced by Sarhan and Apaloo [15]; the Gumbel-Lomax (GL) distribution suggested by Tahir et al. [16]; and the Marshall–Olkin inverse Lomax distribution (MO-ILD) studied by Maxwell et al. [17].
Another prominent approach for extending and updating the existing distributions is called the new exponential- family of Ahmad et al. [5]. They introduced the new exponential- family by combining the exponential distribution with the T- approach of Alzaatreh et al. [18].
Eliwa et al. [19] further contributed to the literature by introducing the exponentiated form of the Chen-G family of distributions. For other recent contributions to the development of new approaches for updating the existing distributions, we refer to (i) the truncated burr- family [20], (ii) the Fréchet Topp Leone- family [21], (iii) the transmuted alpha power- family [22], (iv) the Marshall–Olkin Zubair- family [23], () a new family of continuous (NFC) distributions [24], and (vi) the Topp-Leone Odd Burr III- (TLOBIII-) family [25].
In this article, we propose a NG- family of distributions to obtain the modified and updated versions of the existing distributions. The NG- method can be implemented to obtain new flexible distributions for modeling data in various sectors such as reliability, survival analysis, healthcare, biomedical engineering, and lifetime studies. The next section is devoted to introduce the proposed NG- family of distributions.
2. A New Generalized- Family of Distributions: the Proposed Method
In this section, the probability density function (PDF), cumulative distribution function (CDF), survival function (SF), hazard function (HF), and cumulative HF (CHF) for the NG- family are computed.
Definition: A random variable has a NG- family, if it is CDF is given by
The function defined in (1) is a valid DF, if and only if, . Corresponding to , the PDF is given by
Furthermore, in link to and , the , HF , and CHF are given by
respectively.
In this paper, we implement the NG- family approach to introduce an updated version of the Weibull distribution called the NG-Weibull (new generalized Weibull) distribution. In the next section, we provide the expressions for the PDF, CDF, SF, and HF of the NG-Weibull distribution. Furthermore, the plots for the PDF, CDF, SF, and HF of the NG-Weibull distribution are also provided.
3. A NG-Weibull Distribution: a Special Model
With parameters and , consider the CDF of the Weibull distribution is given by
The respective PDF , SF , and HF are, respectively, given bywhere .
Using (4) in (1), we get the CDF of the NG-Weibull distribution, given by
The PDF of the NG-Weibull distribution is given by
Some possible behaviors for of the NG-Weibull are presented in Figure 1. These plots are obtained for (i) (red curve), (ii) (blue curve), (iii) (gold curve), (iv) (green curve), and (iv) (black curve).
Furthermore, the SF and HF of the NG-Weibull distribution are given by
respectively.
For , and , the plots of CDF and SF of the NG-Weibull distribution are provided in Figure 2.
Furthermore, some possible behaviors for of the NG-Weibull are presented in Figure 3. These plots are obtained for (i) (red curve), (ii) (green curve), and (iii) (blue curve).
The NG-Weibull distribution is an updated version of the Weibull distribution. It has certain advantages over some other updated versions of the Weibull distribution.(i)The NG-Weibull distribution has a closed-form CDF, which makes it easier to generate random numbers for the simulation study.(ii)The NG-Weibull is capable of capturing the bathtub shape of the HF. Statistical models with a bathtub shape of the HF are very useful for modeling the healthcare engineering data sets.(iii)The NG-Weibull distribution provides a close fit to the COVID-19 data set. Therefore, the implementation of the NG-Weibull distribution could be a useful choice for modeling data in health and other related sectors.
Besides the abovementioned advantages, the NG-Weibull distribution also has certain limitations. For example,(i)The NG-Weibull distribution is a continuous model, and it is employed to analyze the mortality rates of COVID- 19 infected people. Therefore, it could not be implemented to analyze the data sets that are not continuous in nature, such as (i) the number of COVID-19 daily registered cases, (ii) the number of COVID-19 confirmed deaths, and (iii) the number of COVID-19 recovered cases.(ii)Since the PDF of the NG- distributions has a complicated form, more computational work would be required to derive the distributional properties.
4. The HT Characteristics
The statistical distributions that possess HT (heavy-tailed) behavior are very useful for dealing with extreme value phenomena. In this section, we prove mathematically the HT characteristics of the NG- distributions.
4.1. The Regularly Varying Tail Behavior
This subsection is devoted to proving the RVTB (regularly varying tail behavior) of the NG- distributions. According to Karamata’s theorem [26], in terms of SF , we have
Theorem If is the SF of the regular varying (RVa) distribution, then is also an RVa distribution.
Proof. Let assume is finite but nonzero . Using (3), we haveUsing in (9), we getOn simplification, we getSince, is a CDF. Therefore, we haveHence, from (11), we getwhich is finite but nonzero ; thus, is an RVa distribution.
Now, we prove the HT behavior of the special case of the NG- family. Using (4) in (11), we get
4.2. A Supportive Application of the RVTB
Consider the distribution of has the power-law behavior (PLB), then we have
By incorporating Karamata’s theorem, we can write aswhere stands for the slowly varying function. From (3), we have
Using in (17), we get
Since . Therefore, we can write (18), as follows:where
If is a slowly varying function, then the result obtained in (19) is true. According to Resnick [27], for all , we have to prove
By incorporating (19), we get
Applying to both sides of (22), we get
5. Estimation and Simulation Study
In this section, the procedure of the maximum likelihood estimation for the model parameters of the NG- distributions is implemented. Furthermore, a simulation study for different values of the model parameters is discussed.
5.1. Maximum Likelihood Estimation
In this subsection, the computation of the maximum likelihood estimators (MLEs) for the parameters of NG- is provided. Consider be a set of observed values of a sample randomly selected from NG- distributions with parameters , and . In link to the PDF of the NG- distributions given by (2), the LF (likelihood function) is
Corresponding to , the log LF (LLF) is given by
On behalf of the parameters of the NG- distributions, the partial derivative of the LLF are, respectively, given by
The MLEs of the parameters can be obtained by solving
5.2. Simulation Study
In this subsection, the performance of MLEs of is evaluated by conducting a simulation study. The process is carried out as follows:(i)A sequence of RS (random sample), say of sizes, , and 1000 are obtained from the NG-Weibull distribution.(ii)Two statistical quantities (MSE and bias) are considered to evaluate , and . These quantities are, respectively, given by(iii)Two different cases are considered for the four parameters of the NG-Weibull distribution.(iv)One thousand repetitions are done to compute the bias and the MSE for , and .
The numerical results are provided in Tables 1–3. From the numerical illustration presented in Tables 1–3, it can be observed that as sample size increases (i) the values of the MLEs get closer and closer to the parameters , and (ii) the values of the MSE and bias of , and tend to zero.
6. Analyzing the COVID-19 Data Set
This section is devoted to demonstrating the usefulness of the NG- method by applying its special model (NG-Weibull distribution) to analyze the COVID-19 data set. The data set represents the mortality rates of COVID-19 infected person in Canada. Corresponding to the COVID-19 data set, the fitting results of the NG-Weibull distribution are compared with three other competing models. These models are given by(i)Weibull distribution where .(ii)K-Weibull distribution where .(iii)Alpha power transformed Weibull (APT-Weibull) distribution where .
Some commonly used analytical goodness of fit measures are considered to show which distribution better fits the data. These measures are given by
Corresponding to the COVID-19 data set, Table 4 gives the values of the MLEs with standard errors enclosed in parentheses. However, Table 5 gives the analytical measures of the competing models for the COVID-19 data set.
Based on the results in Table 4, it is clear that the implementation of the NG-Weibull model is a suitable choice to use for modeling the COVID-19 data set. Furthermore, for supporting the numerical results obtained in Table 4, the estimated PDF, CDF, SF, and PP (probability-probability) plots of the NG-Weibull model are presented in Figure 4. The plots in Figure 4 visually confirm the close-fitting of the NG-Weibull distribution.
7. Concluding Remarks
In this paper, a new family of distributions called the NG- family was introduced. The NG- method was introduced using the exponential distribution in combination with the T- approach. The purpose of introducing the NG- family was to update the distributional flexibility of the classical/modified distributions for modeling data in applied sectors. Based on the NG- distributions approach, an updated/modified form of the Weibull model called the NG-Weibull was studied in depth. The HT characteristics of the NG-Weibull distribution were proved empirically. Based on the NG-Weibull distribution, a simulation study was also conducted. The simulation study showed that the values of MLEs are quite steady and are close to the true parameters’ values as the sample gets increased. Finally, to establish the applicability of the NG-Weibull distribution, a data set related to the mortality rates of COVID-19 infected people was analyzed. The performance of the NG-Weibull distribution was compared with three other competitors. Based on the different six criteria, it is shown that the NG-Weibull distribution outperforms the competitive models [5].
Data Availability
The data are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest.