A New Extended-F Family: Properties and Applications to Lifetime Data

In this article, a new approach is used to introduce an additional parameter to a continuous class of distributions. (e new class is referred to as a new extended-F family of distributions. (e 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.


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.
is 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. e cumulative distribution function (cdf ) of the EAPT family is defined by where α is an additional shape parameter and F(x; ξ) is the cdf of the baseline model depending on the vector of parameters ξ ∈ R.
Ahmad et al. [13] proposed another new method to define new lifetime distributions, called new extended alpha powertransformed (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 θ. e proposed NE-F family is specified by the following cdf: Here, in (3), the baseline cdf is weighted by the quantity e θF(x;ξ) . Clearly, when θ � 1, the cdf (3) reduces to the baseline model. e probability density function (pdf ) and hrf corresponding to (3) are specified by e 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 wellknown models having the same, as well as higher, number of parameters (v) To provide better fits than the other competing modified models is 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. e 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.

The NE-W Distribution
Consider the distribution and density functions of the Weibull random variable given by F(x) � 1 − e − cx α , x ≥ 0, α, c > 0 and f(x) � αcx α− 1 e − cx α . en, the cdf of the NE-W distribution is given by e 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.

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.

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 to where and h k+1 (x) � (k + 1)f(x; ξ) F(x; ξ) k refers to the exponentiated-F (Ex-F) family pdf with power parameter (k + 1) > 0. 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 Y k+1 refer to a random variable having the Ex-F distribution with parameter (k + 1). Some mathematical properties of X can be expressed from those of Y k+1 .

Moments and Generating Function.
e r th moment of X follows simply from (9) as e s th incomplete moment of X is expressed from (9) as 2 Journal of Mathematics e first incomplete moment of X can be obtained from (12) as where J k+1 (t) � t − ∞ xh k+1 (x)dx is the first incomplete moment of the Ex-F class. e mgf of X can be derived from equation (9) as where M k+1 (t) is the mgf of Y k+1 and τ(t, k) � 1 0 exp[tQ G (u)]u k du. Hence, M(t) follows from the Ex-F generating function.

Order Statistics.
e order statistics are very important in many fields of statistical theory and its practice. Let X 1 , . . . , X n be a random sample from the NE-F family. e   pdf of rth order statistic, denoted by X r: n , r � 1, . . . , n, 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 r th order statistic reduces to where, as before, 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 rth order statistic from the properties of Y s .

Maximum Likelihood Estimation and Monte Carlo Simulations
is 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.

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 x 1 , x 2 , . . . , x n be the observed values of a random sample of this distribution with parameter vector Θ � (α, θ, c) T . e log-likelihood function for Θ, say ℓ � ℓ(Θ), takes the following form: e 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. e partial derivatives of the log-likelihood function are given by

Monte Carlo Simulations.
is section provides a comprehensive simulation study to explore the behavior of the MLEs. e 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 log t + θ(1 − t) − log(u) � 0. 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 W(s) refers to the negative branch of the Lambert Wfunction 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 1, 000 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 N � 1000 samples of sizes n � 25, 50, 75, . . . , 1000 from the NE-W distribution with set 1: α � 1.4, θ � 0.8, c � 1.2 and set 2: α � 0.9, θ � 1.2, c � 1.5. e 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. e estimated biases and MSEs of Θ � (α, θ, c) T are defined, respectively, by e 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 n increases. Furthermore, the estimated MSEs decay toward zero as n increases. is fact reveals the consistency property of the MLEs. (1) Weibull distribution:

The Applicability of the NE-W Distribution
(2) MOW distribution [14]: [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. erefore, 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.

Data 1: Infected Guinea Pigs Data.
e first data set consists of 72 observations, and it reported in Bjerkedal [17]. e data represent the Guinea pigs infected with virulent tubercle bacilli. e NE-W and other competitors are applied to this data set. It is observed that the proposed model provides better fit than other competitors. e values of the model parameters are presented in Table 3. e discrimination measures of the fitted models are provided in Table 4. e analytical measures of the NE-W model and other    competitive models are provided in Table 5. e estimated pdf and cdf are sketched in Figure 3, whereas the probabilityprobability (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.

Data 2: Survival Times of Head and Neck Cancer
Patients.
e 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. e NE-W and other selected distributions are applied to analyze this data set. e values of the model parameters are presented in Table 6. e 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. e estimated pdf and cdf are sketched in Figure 5, which shows that the proposed distribution fits the estimated pdf and cdf plots very closely. e 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.

Concluding Remarks
is 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. e 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. e 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. e 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 e references of the data sets used in this paper are provided within the main body of the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.   Journal of Mathematics