Abstract
In this article, a new approach is used to introduce an additional parameter to a continuous class of distributions. The new class is referred to as a new extended-F family of distributions. The new extended-Weibull distribution, as a special submodel of this family, is discussed. General expressions for some mathematical properties of the proposed family are derived, and maximum likelihood estimators of the model parameters are obtained. Furthermore, a simulation study is provided to evaluate the validity of the maximum likelihood estimators. Finally, the flexibility of the proposed method is illustrated via two applications to real data, and the comparison is made with the Weibull and some of its well-known extensions such as Marshall–Olkin Weibull, alpha power-transformed Weibull, and Kumaraswamy Weibull distributions.
1. Introduction
In many practical situations, classical distributions do not provide adequate fits to real data. For example, when modeling data with a monotonic hazard rate function (hrf), one frequently uses the Rayleigh, exponential, or Weibull distributions. Among these models, the Weibull distribution is the most prominent one for modeling real phenomena of nature. Unfortunately, the Weibull model is inappropriate in modeling data having nonmonotonic hrf such as unimodal, modified unimodal, or bathtub-shaped.
To address the abovementioned problems, the researchers have shown an increased interest in developing more flexible distributions. This has been performed via extending the classical distributions by introducing additional parameter(s) to the baseline model. In this regard, numerous generalized families of distributions have been proposed and studied over the last three decades for modeling data in many applied areas such as economics, engineering, biological studies, environmental sciences, medical sciences, and finance. Some well-known families include odd exponentiated half-logistic-G [1], Topp Leone odd Lindley-G [2], Marshall–Olkin alpha power-G [3], transmuted transmuted-G [4], generalized Burr XII power series class [5], Weibull Marshall–Olkin [6], new exponentiated TX [7], Weibull-G Poisson [8], odd Dagum-G [9], arcsine exponentiated-X [10], and odd log-logistic Lindley-G [11] families, among many others.
Recently, Ahmad et al. [12] proposed a new method of introducing an additional parameter to extend the existing distributions, called the extended alpha power-transformed (EAPT) family of distributions. The cumulative distribution function (cdf) of the EAPT family is defined bywhere is an additional shape parameter and is the cdf of the baseline model depending on the vector of parameters .
Ahmad et al. [13] proposed another new method to define new lifetime distributions, called new extended alpha power-transformed (NEAPT) family that is defined by the cdf:
In this article, we further propose a new method to provide flexible lifetime distributions called a new extended-F (NE-F)family with additional shape parameter . The proposed NE-F family is specified by the following cdf:where and . Here, in (3), the baseline cdf is weighted by the quantity . Clearly, when , the cdf (3) reduces to the baseline model. The probability density function (pdf) and hrf corresponding to (3) are specified by
The key motivations for using the NE-F family of distributions in practice are as follows:(i)A very simple and convenient method of adding an additional parameter to modify the existing distributions(ii)To improve the characteristics and flexibility of the existing distributions(iii)To introduce the extended version of the baseline distribution whose cdf, survival function (sf), and hrf have closed forms(iv)To compare the goodness of fit with other well-known models having the same, as well as higher, number of parameters(v)To provide better fits than the other competing modified models
This paper is unfolded as follows. Section 2 offers a special submodel of the new family called the new extended-Weibull (NE-W) distribution. Mathematical properties of the NE-F family are derived in Section 3. The maximum likelihood estimators (MLEs) of the model parameters are obtained in Section 4. A Monte Carlo simulation study is provided in the same section. Two practical applications are discussed in Section 5. Finally, Section 6 concludes the article.
2. The NE-W Distribution
Consider the distribution and density functions of the Weibull random variable given by and . Then, the cdf of the NE-W distribution is given by
The pdf corresponding to (5) is given by
Plots for the pdf and hrf of the NE-W for selected parametric values are shown in Figures 1 and 2, respectively.
3. Properties of the NE-F Family
In this section, we derive some general properties of the NE-F family including the linear representation, moments, moment-generating function (mgf), and order statistics.
3.1. Linear Representation
Using the exponential series, the cdf of the NE-F family reduces to
Applying the binomial expasion, we have
By differentiating (8), the pdf of the NE-F family reduces towhereand refers to the exponentiated-F (Ex-F) family pdf with power parameter . Hence, the pdf of the NE-F family is expressed as a linear combination of Ex-F densities. Equation (9) can be used to obtain several mathematical properties of the NE-F family from those properties of the Ex-F class.
Henceforth, let refer to a random variable having the Ex-F distribution with parameter . Some mathematical properties of can be expressed from those of .
3.2. Moments and Generating Function
The th moment of follows simply from (9) as
The s th incomplete moment of is expressed from (9) as
The first incomplete moment of can be obtained from (12) aswhere is the first incomplete moment of the Ex-F class.
The mgf of can be derived from equation (9) aswhere is the mgf of and . Hence, follows from the Ex-F generating function.
3.3. Order Statistics
The order statistics are very important in many fields of statistical theory and its practice. Let be a random sample from the NE-F family. The pdf of th order statistic, denoted by , , takes the following form:
Using the pdf and cdf of the NE-F family, we can write
Applying both exponential and binomial series, we get
Combining (16) and (17), the pdf of the th order statistic reduces towhere, as before, , for ,
Equation (18) refers to the pdf of the NE-F order statistics as a linear mixture of Ex-F densities and can be used to derive some mathematical properties of the th order statistic from the properties of .
4. Maximum Likelihood Estimation and Monte Carlo Simulations
This section covers the estimation of the NE-W parameters. Section 4.1 offers estimation of the NE-W parameters using the maximum likelihood method. Section 4.2 explores the performance of the maximum likelihood estimators (MLEs) in terms of biases and mean squared errors (MSEs) by means of a Monte Carlo simulation study.
4.1. Maximum Likelihood Estimation
Here, we consider the estimation of the unknown parameters of the NE-W model from complete samples via the maximum likelihood approach. Let be the observed values of a random sample of this distribution with parameter vector . The log-likelihood function for , say , takes the following form:
The log-likelihood function can be maximized either directly by using the R (AdequecyModel package), SAS (PROC NLMIXED), or the Ox program (subroutine MaxBFGS) or by solving the nonlinear likelihood equations. The partial derivatives of the log-likelihood function are given by
4.2. Monte Carlo Simulations
This section provides a comprehensive simulation study to explore the behavior of the MLEs. The NE-F family is easily simulated by inverting (3) as follows: If U has a uniform U (0,1) distribution, then the nonlinear equation is as follows:where t is the solution of . Expression (22) can be used to simulate any special subcase of the NE-F family. Particularly, the quantile function of the NE-W distribution has the following form:where refers to the negative branch of the Lambert W-function that is known as a product log function in Mathematica software and Wolfram Language.
Now, we consider the NE-W distribution to assess the behavior of the MLEs in estimating the NE-W parameters. We simulate the NE-W distribution for two combinations of parameters using Monte Carlo simulations with replications. We calculate the MSEs and the biases of the parameter estimates using the optim()R-function with an argument method = “L-BFGS-B.” We generate samples of sizes from the NE-W distribution with and . The reason for choosing the initial values of the model parameters is to check the performance of the MLEs, for example, to see whether (i) the simulated values of the model parameters approach the true value or not, (ii) the MSEs decrease or not, and (iii) the biases approach zero or not. The estimated biases and MSEs of are defined, respectively, by
The numerical results for the abovementioned measures are displayed in Tables 1 and 2. It is noted, from these tables, that the estimated biases decrease when the sample size increases. Furthermore, the estimated MSEs decay toward zero as increases. This fact reveals the consistency property of the MLEs.
5. The Applicability of the NE-W Distribution
This section explores the applicability of the proposed NE-W model in biological sciences as compared with other competitive distributions including Weibull, Marshall–Olkin Weibull (MOW), alpha power-transformed Weibull (APTW), and Kumaraswamy Weibull (Ku-W) distributions by analyzing two real-life applications. The distribution functions of the competitive models are as follows:(1)Weibull distribution:(2)MOW distribution [14]:(3)Ku-W distribution [15]:(4)APTW distribution [16]:
First, we check whether the considered data set actually comes from the NE-W model or not using the goodness of fit test, based on the Anderson–Darling (AD) test statistic, Cramer–von-Mises (CM) test statistic, and Kolmogorov–Smirnov (KS) statistic with its corresponding p value. Note that the AD, CM, and KS statistics are to be used only to verify the goodness of fit and not as discrimination criteria. Therefore, we consider four discrimination criteria, based on the log-likelihood function evaluated at the maximum likelihood estimates, including the Akaike information (AIC), Bayesian information (BIC), Hannan–Quinn information (HQIC), and corrected Akaike information (CAIC) criteria.
All the required computations have been performed via the optim()R-function with an argument method = “BFGS.” In general, a model with smaller values of these measures indicates better fit to the data. Based on the considered analytical measures, one can observe that the proposed NE-W model provides the best fit to the two analyzed real-life data sets.
5.1. Data 1: Infected Guinea Pigs Data
The first data set consists of 72 observations, and it reported in Bjerkedal [17]. The data represent the Guinea pigs infected with virulent tubercle bacilli. The NE-W and other competitors are applied to this data set. It is observed that the proposed model provides better fit than other competitors. The values of the model parameters are presented in Table 3. The discrimination measures of the fitted models are provided in Table 4. The analytical measures of the NE-W model and other competitive models are provided in Table 5. The estimated pdf and cdf are sketched in Figure 3, whereas the probability-probability (pp) plot and Kaplan–Meier survival plot are provided in Figure 4. Figures 3 and 4 reveal that the NE-W distribution provides the superior fit to the Guinea pigs infected data.
5.2. Data 2: Survival Times of Head and Neck Cancer Patients
The second data set consists of 44 observations as reported in [18], and it represents the survival times of a group of patients suffering from head and neck cancer who are treated using a combination of radiotherapy. The NE-W and other selected distributions are applied to analyze this data set. The values of the model parameters are presented in Table 6. The discrimination measures of the fitted models are provided in Table 7, whereas the analytical measures of the proposed NE-W model and other competitive models are provided in Table 8. The estimated pdf and cdf are sketched in Figure 5, which shows that the proposed distribution fits the estimated pdf and cdf plots very closely. The PP plot and Kaplan–Meier survival plot are presented in Figure 6. One can see, from Tables 7 and 8 and Figures 5 and 6, that the proposed model outclasses other competitors.
6. Concluding Remarks
This article proposed a new method for generating flexible models, called a new extended-F (NE-F) family. One special submodel of the NE-F family, called a new extended-Weibull (NE-W) distribution, is considered. Some general properties are derived for the NE-F family. The model parameters are estimated via the maximum likelihood along with simulation results to explore the performance of these estimators. Two applications to biological science data are analyzed to illustrate, empirically, the flexibility of the proposed NE-W model. The comparison of the NE-W distribution is conducted with some well-known lifetime distributions such as Weibull, Marshall–Olkin Weibull, alpha power-transformed Weibull, and Kumaraswamy Weibull distributions. The practical applications of the proposed model reveal better fit to both analyzed data sets than other competing models.
It is worth mentioning that the results in this paper can be extended in some ways. For example, extreme stability, characterization properties, entropies, and stochastic orders could be considered. Exponentiated and transmuted versions may be studied, several fundamental properties could be explored, and a bivariate extended-F family may also be established.
Data Availability
The references of the data sets used in this paper are provided within the main body of the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by the Department of Statistics, Yazd University, Yazd, Iran.