Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications

This paper introduces a novel alpha power exponentiated Weibull-Pareto distribution based on the alpha power transformation. We derive several properties of the new distribution, including moments, quantile function, mean residual life, mean waiting time, and order statistics. Estimating model parameters is performed using the method of maximum likelihood. Then, for the purpose of evaluating the eﬀectiveness of the estimates, we conduct some simulation studies. Finally, we demonstrate the superiority of this new model by analyzing three real-life data sets. The results indicate the APENWPD provides the best ﬁt among some other competitive models. Thus, the proposed distribution possesses great potential for wider applications in statistics. In the future, it might be possible to develop regression model based on the APENWPD and compare it with some existing models.


Introduction
Probability distributions are required for many theoretical and practical statistical methods, including inference, modeling, survival analysis, and reliability analysis. Statistical distributions are useful in many fields. In engineering, for instance, they can be used to model the life cycle of a machine. In medical sciences, statistical distributions have been used to study the duration to recurrence of some types of cancer after surgical removal and survival times of patients after surgery. The probability distribution therefore provides essential information about statistical inference and data analysis. Such information may be useful for making some critical decisions. Accordingly, selecting the appropriate distribution to employ when modeling the data is extremely important. Identifying a suitable distribution for a set of data significantly enhances the accuracy of the statistical analysis. Frequently, data may exhibit certain characteristics that cannot be adequately explained by a classical distribution. It is therefore crucial to develop new methods for modifying existing distributions in order to improve the goodness of fit. In recent years, there has been an increased interest in extending existing classical distributions to obtain greater flexibility in modeling data from different fields of study. Most of these extensions are obtained through developing methods for generating new classes of distributions that extend the existing standard models. Various techniques have been proposed in the literature to generate new distributions by adding one or more additional shape parameter(s). Examples of such well-known generators include the beta-G family proposed by Eugene et al. (2002), Gamma-G (type 1) by Zografos and Balakrishnan (2009), the Kw-G by Jones (2009), McDonald-G by Alexander et al. (2012), Gamma-G (type 2) by Ristić and Balakrishnan (2012), Gamma-G (type 3) by Torabi and Hedesh (2012), among others.
Additionally, Alzaatreh et al. (2013) developed a general technique that allows the use of any baseline distribution as a generator. This innovative technique is described as the transformed transformer (T − X) family of distributions.
Based on the T − X technique, Nasiru and Luguterah (2015) suggested the new Weibull-Pareto (NWP) distribution. Following that, Al-Omari et al. (2019) combined the two methods of the exponentiated class and the T − X family in order to introduce the exponentiated new Weibull-Pareto (ENWP) distribution. Thus, the cumulative distribution function (CDF) of the ENWP distribution is expressed as The corresponds probability distribution function (PDF) is expressed as where β, θ, ω, δ > 0 and x > 0.
Recently, Mahdavi and Kundu (2017) have proposed a new technique called the alpha power transformation (APT) for adding an extra parameter to a family of distributions. This parameter provides more flexibility to the CDF and PDF of an APT family which can be expressed as where F (x) and f (x) represent the CDF and PDF of any continuous distribution.
This family of distributions has been applied by many authors, include Mahdavi and Kundu (2017) who applied the APT technique to the exponential distribution to obtain the alpha power exponential (APE) distribution, Nassar et al. (2017) proposed a three-parameter distribution, known as the alpha power Weibull distribution, Dey et al. (2018) introduced the alpha power transformed Lindley distribution, Ihtisham et al. (2019) introduced the alpha power Pareto distribution, Dey et al. (2019) proposed the alpha power transformed inverse Lindley distribution and Eghwerido et al. (2020) presented the alpha power Gompertz distribution.
The main aim of this article is to introduce a new probability distribution, called the alpha power exponentiated new Weibull-Pareto distribution (APENWPD), based on a new technique. Particularly, this technique combines the three approaches of T − X, exponentiated, and APT in order to increase the flexibility of modelling real data.
This article is planned as follows: in Section 2, we define the APENWPD and provide some plots for its PDF and hazard rate function. We derive in Section 3 some fundamental statistical properties of the APENWPD. In Section 4, we discuss the estimation of the unknown model parameters using the maximum likelihood method and Section 5 provides some simulation studies that evaluate these estimates. In Section 6, we consider three applications that show the efficiency of the introduced distribution. Finally, in Section 7, we offer some concluding remarks.
The survival function, S(x) = 1 − F (x), of the APENWPD is expressed as The hazard rate function, , of the APENWPD is expressed as

Special cases of the APENWPD
The APENWPD reduces to the ENWP distribution at α = 1.
The APENWPD reduces to the new Weibull Pareto distribution at α = ω = 1.

Expansion for the PDF.
This subsection provides an expansion for the APENWPD PDF in (6). Particularly, using the following series representation Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications we obtain the PDF as follows Furthermore, applying the following binomial series expansion in (10), we can rewrite the PDF of APENWPD as follows where Figure 1 and Figure 2 illustrate some various shapes of the PDF and h(x) of the APENWPD for some particular parameters. Several shapes such as symmetric, near symmetric, inverted J-shaped, right-skewed, and left-skewed shape are observed for the density of the APENWPD in Figure 1. Additionally, Figure 2 indicates that the hazard rate function for the APENWPD features a wide variety of asymmetrical shapes. This indicate to the flexibility of the APENWPD for modeling data set with various shapes.

Properties of the APENWPD
In this section, we will discuss some distributional properties of the APENWPD. These properties are discussed in the following subsections.

Moment generating function
If X follows APENWPD, then the moment generating function (MGF) (M x (t)) can be derived as

Characteristic function
Let X ∼ AP EN W P D(α, β, θ, ω, δ), then the characteristic function, φ x (t), of X can be obtained as Proof. In order to determine the characteristic function of the APENWPD, we apply Then, we have Applying the series representation of e itx given in (22), we obtain where µ r is computed based on (18). Thus, the characteristic function of the APENWPD can be described as follows Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications

Mean residual life and mean waiting time
The mean residual life (MRL) function of the APENWPD, say µ(t), with f (x) given by (6), is obtained from where S(t) is the survival function, E(t) is the mean and I = t 0 xf (x)dx.
where γ (a, b) = a 0 x b−1 e −x dx denotes the lower incomplete gamma function. Thus, the MRL of APENWPD is obtained by substituting (7), (19), and (25) into (24), as follows Similarly, if X has the CDF (5), then its mean waiting time (MWT),μ(t) can be defined as follows where I = t 0 xf (x)dx denotes the first incomplete moment given by (25). Thus, the MWT of the APENWPD can be derived by substituting the equations (5) and (25) in equation (26) as follows

Shannon and Rényi entropies
Entropy is a measure of variation or uncertainty of the RV X. The Shannon entropy, SE X , of an RV X with PDF f (x) is defined as follows Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications Using the PDF specified in (6), SE X can be derived as where k is the Euler constant.
In addition, the Rényi entropy, (RE X (v)) might be obtained as follows Inserting (6) into (30) yields Then, after solving the integral, the Rényi entropy of the APENWPD can be written as Table 1 displays the mean, variance, skewness and Kurtosis of APENWPD for various values of α, β, θ, ω and δ. For fixed β, θ, ω and δ, the values of the mean and the variance of APENWPD are increasing with the increase of α. While the skewness and Kurtosis values are decreasing as the value of α increases. Also, at fixed α, θ, ω and δ, the variance, skewness and Kurtosis decrease with increasing β.

Order statistics
Suppose X 1 , X 2 , ....., X n are the observed values of a sample from the APENWPD and X i:n denotes the i th order statistic. The density of X i:n can be defined as Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications Substituting by equations (5) and (6) in (32), we have where B(a, b) refers to the beta function. An expansion of the binomial series is given as follows (x − z) n = n y=0 (−1) y n y x n−y z y .
Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications By applying the binomial series expansion given in equation (34), f i:n (x) can be expressed as follows

Maximum likelihood estimates
Assume x 1 , x 2 , x 3 , ...., x n represent a random sample from the APENWPD, then the log-likelihood function ( ) is given as On taking partial derivatives of the log-likelihood in (36) with respect to the parameters and equating the results to zero, we get where Then we can obtain the maximum likelihood estimates (MLEs) for each parameter by the solving system of equations ((37)-(41)). Also, we can find the solution to the equations analytically by using the routine "optim" in R.

Simulation study
In this section, we discuss some simulation studies to investigate the behavior of the MLEs for the unknown parameters of the APENWPD for various sample sizes and different values of the parameters. Particularly, Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications 50, 100, 150, 200, 250 and 500 sample sizes were considered from the APENWPD with 1000 replications. Two different sets of parameters are assumed; Set 1 (α = 0.5, β = 0.4, θ = 0.5, ω = 0.5, δ = 0.5) and Set 2 (α = 1.5, β = 1.5, θ = 1.5, ω = 2, δ = 2). The average estimates and the mean squared errors (MSEs) of the MLEs are calculated for the different sample sizes. The simulation results of the average estimates  Table 2. We can notice that the MSE decreases as n increases and the parameters' MLEs becomes closer to the actual parameter.

Applications
In this section, we consider three real-life data sets. The fit of the APENWPD is compared with some other distributions, namely the exponentiated Weibull Weibull (EWW) distribution by Elgarhy and Hassan (2019) First Data Set.
Second Data Set.
Third Data Set.
The negative log-likelihood value (− ), Cramer-von Mises (W) statistic, Anderson-Darling (A) statistic and the Kolmogorov-Smirnov (K-S) test value with its p-value are considered to compare the fit of the APEN-WPD with the other distributions. The APENWPD performance compared with other distributions for the three real data sets that shown in Tables 3, 5 and 7. In addition, the MLEs and standard errors (SEs) of the parameters for the APENWPD with the other competing distributions for the three data sets are shown in Tables 4, 6 and 8.
Tables 3, 5 and 7 demonstrate that the APENWPD has the lowest values of − , W, A and K-S test values as well as the best p-values. This means that the APENWPD provides the best fit as compared to the other competing distributions for the three real data sets.      it is evident from Figures 3, 4 and 5 that the APENWPD fits the histogram more closely than the other competing models. That is, it is apparent that the APENWPD best fits these data sets when compared to other distributions considered here. Therefore, APENWPD has the potential to compete with other distributions used commonly in literature to fit lifetime data.

Conclusions
When describing and predicting real-world phenomena, statistical distributions are extremely useful. Many distributions have been developed, but there is always the opportunity to develop distributions that are more flexible or that fit specific data scenarios. This has motivated researchers to explore and develop new and more flexible distributions. In this study, a five-parameter APENWPD distribution is introduced based on a new technique for generating distributions. In this method, three techniques are combined: T-X family, exponentiated method, and APT, which provides a greater degree of adaptability to the suitability of practical data sets. Among the characteristics that are relevant to the proposed distribution is the diversity of shapes that the density and the hazard rate functions of the distribution can take. Some mathematical properties of this new distribution are provided. Estimation of the unknown parameters is discussed by employing the maximum likelihood technique. Then, to demonstrate the consistency of the estimates, various simulation studies are conducted. The results indicate that the proposed estimators demonstrate good performance. Furthermore, three real data sets are used to show the new model's flexibility against the competitive models.
The results indicate the APENWPD provides the best fit among some other competitive models. Thus, the proposed distribution possesses great potential for wider applications in statistics. In the future, it might be possible to develop regression model based on the APENWPD and compare it with some existing models.