A Novel Generalized Family of Distributions for Engineering and Life Sciences Data Applications

Department of Statistics, Shaheed Benazir Bhutto Women University, Peshawar, Pakistan Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai 90110, Songkhla, (ailand Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat, Pakistan College of Computer Science, King Khalid University, Abha, Saudi Arabia Department of Information Technology, Faculty of Science and Technology, Suan Sunandha Rajabhat University, Bangkok 10300, (ailand


Introduction
A prominent subject in distribution theory is the advancement of novel methods to expand the existing families of lifetime distributions. For modeling data, many distributions have been extensively used and applied in various fields such as actuarial sciences, biological sciences, demography, social sciences, engineering, and medical sciences. A number of lifetime models are available in the literature to analyze the data. However, in many situations, the classical distributions are not suitable for describing and predicting real-world phenomena. Due to this reason, attempts are made to describe new techniques for creating new distributions with the addition of one more parameter to the baseline model. Generally, an extra parameter is included by means of generators, or present distributions are joined to have more flexible models [1]. e justification of these amendments is to take along more flexibility to the conventional models such as to introduce skewness in symmetric distributions, to obtain reliably better fits than other proposed distributions and to create heavy-tailed distributions for modeling varied real datasets, and to obtain flexible models for fitting all types of hazard rate functions [2]. erefore, several researchers adopted numerous ideas, for example, Azzalini [3] defined skewness in the normal distribution by adding a skewness parameter λεR as follows: where r(x) and R(x) are the probability density function (PDF) and cumulative distribution function (CDF) of the standard normal distribution. Its shape can be positively or negatively skewed depending on the values of λ, and for λ � 0, it becomes standard normal. Mudholkar and Srivastava [4] presented the idea of exponentiated family by taking an extra parameter in the power of the CDF of an existing distribution.
In particular, they derived exponentiated Weibull distribution, which was a more flexible distribution using the expression Furthermore, this idea was adopted by other researchers to obtain different versions of exponentiated distributions including exponentiated Pareto, exponentiated gamma, exponentiated Rayleigh, exponentiated Lomax, exponentiated beta, exponentiated Kumaraswamy, exponentiated Gompertz, exponentiated Lindley, and exponentiated half-normal distribution.
Marshall and Olkin [5] proposed a procedure of adding a new parameter in present distributions by using the generator is generator was initially used to modify exponential and Weibull distributions. Later on, it was used for other well-known distributions as well, for example, Pareto, normal, Lomax, gamma, Lindley, Fréchet, extended Weibull, Rayleigh, beta, and extended generalized Rayleigh.
Eugene et al. [6] suggested the beta-G distributions with the following generator: where H(x) is the CDF of the parent distribution and z(y) is the PDF of the beta distribution. Since its introduction, several distributions have been converted into beta-generated family, for instance, normal, Gumbel, exponential, Fréchet, Weibull, Pareto, modified Weibull, Laplace, Burr XII, generalized Pareto, Cauchy, and extended Weibull. Jones [7] utilized a random variable beta for introducing a general family of univariate distributions. e new distribution family holds greater flexibility for fitting skewed and symmetric models. Zografos and Balakrishnan [8] presented the idea of gamma-generated distributions based on the proposal of Jones [7]. ey used the gamma distribution as the generator and CDF of any random variable as the parent distribution.
Furthermore, for the continuous distributions, Alzaatreh et al. [9] presented the idea of the transformed-transformer family of distributions known as T-X family, where the PDF of some continuous random variable and a function of the CDF instead of the original CDF, satisfying some conditions, were used there in the expression. e generator is given as where v (z) is the PDF of any continuous random variable. Particularly, they derived subfamilies which include new Weibull-X family [10], logistic-X family [11], weighted T-X family [12], and modified T-X family [13]. Mahdavi and Kundu [14] proposed a novel idea, known as alpha power (AP) transformation, introducing the additional parameter to the underlying continuous distribution. e AP transformation is represented as Specifically, the generator was used for the transformation of the exponential distribution with one parameter into the AP exponential distribution with two parameters. Later on, the generator has been used by many researchers so as to have AP generalized exponential [15], AP Weibull distribution [16], AP transformed Pareto distribution [17], AP transformed inverse Lindley distribution [18], and alpha power exponentiated inverse Rayleigh [19].
Cordeiro et al. [20] derived the Ku-Weibull distribution and discussed its several properties. Later on, Cordeiro and de Castro [21] proposed the Kumaraswamy-G family of distributions, defined as follows: where a, b > 0 are the shape parameters. Several distributions have been introduced using (8) such as Kumaraswamy transmuted exponentiated additive Weibull [22], Kumaraswamy generalized power Weibull [23], Kumaraswamy flexible Weibull extension [24], and Kumaraswamy exponentiated inverse Rayleigh [25]. Ahmad [26] introduced a new general family of distributions based on the proposal of Shaw and Buckley [27] as Another recent development in the distribution families is the approach of the cubic rank transmuted family, proposed by Granzotto et al. [28] and given by Using the above generator, several distributions have been introduced, for example, cubic transmuted exponential [29], cubic transmuted Pareto [30], and cubic transmuted Weibull [31]. 2 Mathematical Problems in Engineering Hence, for providing a better fit to the data, it is a common practice to amend the classical probability models so as to model the monotonic and nonmonotonic hazard functions. One such amendment is to produce a generator and use it on existing models to develop new probability classes. e aim behind the development of these generators is to eliminate some of the difficulties found in the present probability models. For this purpose, a new method is proposed in this manuscript, termed as Khalil new generalized family (KNGF) of distributions. e advised model in comparison with the existing probability distributions available in the literature will increase the flexibility as well as produce a better fit and will also be able to model the monotonic and nonmonotonic hazard rate function.

The Khalil New Generalized Family (KNGF)
In this section, we propose a new method for deriving new continuous probability distributions, termed as Khalil new generalized family (KNGF).
Let a random variable X follow Weibull distribution with the CDF as (1 − exp(− αx β )) having α and β as the scale and shape parameters, respectively. Now, replacing x � F(x) in the Weibull density, we have (1 − exp(− αF(x) β )). Henceforth, the CDF of the KNGF is defined as where F(x; ζ) is the considered baseline cumulative distribution function of the baseline model. e probability density function, survival function, hazard rate function, and reversed hazard rate function of the KNGF with scale and shape parameters α and β are, respectively, defined as follows: erefore, a random variable x following PDF (12) is defined as the Khalil new generalized family of distributions, denoted as X ∼ KNG(x; α, β, ζ), where α, β are the scale and shape parameters and ζ is the parameter of the baseline model. is paper is classified as follows: Section 3 consists of considering a special submodel, i.e., basic Pareto distribution, from the proposed family of distributions, and various statistical properties of the submodel are derived. Section 4 comprises estimating the parameters using the maximum likelihood method. To demonstrate the practicality of the suggested model, simulations and real-life datasets have been used in Section 5. Section 6 finally concludes the paper.

The Khalil New Generalized Pareto (KNGP) Distribution
Specifically, in this segment, we consider a submodel of the KNG family of distributions, termed as Khalil new generalized Pareto (KNGP) distribution, using the basic Pareto distribution [32] as a baseline model. e cumulative distribution function of the basic Pareto distribution is and its PDF is en, a random variable X following the KNGP distribution is denoted as X ∼ KNGP(x; α, β, λ) with probability density function, cumulative distribution function, and reliability and hazard rate functions given in the following: Probability density function is

Mathematical Problems in Engineering
Figures 1 and 2 illustrate the plots of the density function and cumulative distribution function with scale and shape parameters α, β and scale parameter λ of the KNGP distribution, respectively. e density function varies significantly and changes with respect to scale and shape parameters (α and β). Keeping the scale parameter fixed λ, the PDF becomes more and more symmetric with increasing scale and shape parameter values ( Figure 1). Similarly, Figure 3 demonstrates the hazard rate patterns of the proposed model. e hazard function is nonincreasing for any value of λ and α, β < 2. As the shape parameter values increase, the hazard function first increases and then decreases gradually (nonmonotonic).

Statistical Properties.
e statistical properties of the KNGP distribution are given in the following sections.
3.1.1. Moments. If a random variable X has the KNGP distribution, then the r th moment of the KNGP distribution, say μ r , takes the following form: Proof. By definition, Using the exponent series e x � ∞ m�0 (x m /m!), Use the binomial expansion For r � 1, we can have the mean of the KNGP distribution given by 4 Mathematical Problems in Engineering      Mathematical Problems in Engineering e Taylor series yields the following simplified expression: Using equation (19) in (25), we get

Entropy Measures.
A very important measure in the reliability analysis is the measure of entropy. It is used to measure the amount of variation and uncertainty in the dataset. If the value of entropy is small, it indicates less uncertainty in the data. Hence, for measuring the amount of uncertainty of a random variable x following the KNGP distribution, Renyi [33] and Havarda and Charvat [34] entropies are considered. e entropies are as follows: Proof. By definition, Renyi and q-entropy are e Renyi entropy of the KNGP distribution, for ρ > 1, is as follows: Use the binomial expansion (1 − z) b � ∞ l�0 (− 1) l b l z l in the above expression to have 6 Mathematical Problems in Engineering On simplification of the above integrals, the Renyi entropy yields the following result: e q-entropy or β-entropy, introduced by Havarda and Charvat [30], is defined as Consider the integral Substituting the result of the above integral in H x (q), it reduces to and solving the integral in (35) for the KNGP model, Substituting the above result of the integral in (35), we get e mean residual life function is thus obtained as follows: 8 Mathematical Problems in Engineering 3.1.5. Quantile Function. Let X follow the KNGP distribution with the PDF given in (15). en, the quantile function of X, denoted by Q (u), is as follows: where u follows the uniform distribution over the interval [0, 1]. When u is replaced by q, the median, 1 st quantile, and 3 rd quartile can be obtained from (39) by simply substituting q � 0.5, 0.25, and 0.75, respectively.

Order Statistics.
Let a random sample X of size k from the KNGP distribution have the corresponding order statistics denoted by X 1: k < X 2: k < X 3: k < · · · < X k: k . en, the PDF of the r th order statistics is given by Using equations (15) and (16) in (40), we have

Parameter Estimation of the KNGP Distribution
e subsequent section presents the parametric estimation of the KNGP distribution using the maximum likelihood method.

Maximum Likelihood Estimation.
Consider a random sample X � (X 1 , X 2 , . . . , X n ) from the KNGP (α, β, λ) distribution. en, its likelihood function is e log-likelihood function of the KNGP distribution is obtained by taking the logarithm on both sides of (42) as follows: log e − α − 1 .

Applications
Simulation study and real datasets are used in this section to show the practicality of the suggested KNGP model.

Simulations.
In order to carry out a simulation study for studying the behavior of the MLEs, 1000 samples are generated from the KNGP distribution with various sets of parameter values, i.e., , and (α � 5, β � 7, λ � 2), using the Monte Carlo simulation method. Various sample sizes (n � 50, 100, 500, and 1000) have been considered. e average estimates of the parameters, mean square errors, and biases are reported in Table 1. It can be observed from the results presented in Table 1 that, as the sample size increases (i.e., n > 100), the estimated values of the parameters get quite closer to the assumed parameter values hence proving the property of consistency.

Conclusion
A new family of distributions was suggested in this study which was termed as Khalil new generalized family of distributions. For the purpose of applications, Pareto distribution was used as an input model in the KNGF which resulted in a new family of distributions, named as the KNGP distribution. A number of statistical properties of the suggested distribution have been derived for the submodel. A simulation study was performed by generating data from KNGP, and the maximum likelihood estimates of the unknown parameter were obtained. e simulation results showed consistency of the parameter estimates of the KNGP model. e suggested model was also fitted to three real (monotonic and nonmonotonic) datasets to show its usefulness. KNGP provided a satisfactory fit to the datasets in comparison with basic Pareto, Pareto, generalized Pareto, alpha power Pareto, and exponentiated generalized Pareto. Hence, on the basis of findings, it is concluded that the proposed distribution is a more efficient model as compared to other distributions considered here for modeling real-life datasets.
Data Availability e data used in this Manuscript is obtained from the authors upon request and citing this paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 15