Int. J. Anal. Appl. (2023), 21:45 The New Dagum-X Family of Distributions: Properties and Applications Amani S. Alghamdi∗, Huda Alghamdi, Aisha Fayomi Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia ∗Corresponding author: amaalghamdi@kau.edu.sa Abstract. Various statistical distributions are still being used extensively over the previous decades for modeling data in numerous areas such as engineering, sciences, and finance. Nonetheless, in a lot of applied areas, there is a continuous need for expanded forms of these distributions. However, many common distributions do not fit the data well. Thus, new distributions have been constructed in literature. The purpose of this article is to present a new family of distributions using the Dagum distribution as a generator and to study its properties such as hazard rate functions, moments, quan- tile function, ordered statistics and Renyi entropy. Moreover, one sub model called Dagum-Frechet distribution is discussed with some of its properties. The maximum likelihood estimation is employed to estimate the parameters of the proposed distribution, and the confidence intervals are obtained. Finally, two real data sets are analyzed to illustrate the performance of the purposed distribution. 1. Introduction Statistical literature is abounding with many statistical distributions that are used for data modeling in various areas of applied life, such as engineering, actuarial sciences, education, demography, eco- nomics, finance, insurance, environmental, medical, and biological studies. The quality of statistical distribution is based on fitting the assumed probability distribution to the data. However, there are various issues where any of these distributions do not fit the data appropriately, especially in the areas of engineering, finance, medicine and environmental hazards. Therefore, a significant effort has been made in developing different families of distributions. Recently, there has been a growing interest of generating wide families of distributions from existing families of distributions by adding one or more additional parameter(s) to the baseline distribution. There are a lot of well-known family of distribu- tions, such as Beta-G by [1], KumaraSwamy-G by [2], Exponentiated generalized-G by [3], Gamma-X Received: Feb. 19, 2023. 2020 Mathematics Subject Classification. 62E10. Key words and phrases. estimation; Fréchet distribution; T-X family; hazard function; moment; maximum likelihood. https://doi.org/10.28924/2291-8639-21-2023-45 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-45 2 Int. J. Anal. Appl. (2023), 21:45 family by [4] and Logistic-X by [5]. Moreover, [6] and [7] introduced the odd Lomax-G family of dis- tributions and the Lomax Gumbel distribution respectively. The Zubair-G family of distributions was studied by [8] and the Zubair-Weibull distribution is obtained. [9] proposed the exponentiated Gumbel family of distributions and evolved three separate models. The transformed transformer (T-X) method is considered as one of the most important ways to generalize distributions, which many have relied on in their researches. It is introduced by [10] for generating families of continuous distributions. Let r(t) be the probability density function (PDF) of a random variable T, where T ∈ [a,b], for −∞ ≤ a ≤ b ≤∞. Assume W(G(x)) be a function of the cumulative density function (CDF) G(x) of any random variable X, where W(G(x)) satisfies the following: i. W(G(x)) ∈ [a,b]. ii. W(G(x)) is differentiable and monotonically nondecreasing. iii. W(G(x)) → a as x→ −∞ and W(G(x)) → b as x→ ∞. The CDF and PDF of the T-X family of distributions are given respectively as: F (x) = ∫ W(G(x)) a r(t)dt, (1.1) f (x) = [ d dx W (G(x)) ] r[W (G(x))]. (1.2) The definition of W(G(x)) depends on the support of the random variable T as follows: (1) When the support of T is bounded: W(G(x)) can be defined as G(x) or G(x)α. (2) When the support of T is [a,∞), for a ≥ 0: W(G(x)) can be defined as −log(1 −G(x)) or G(x)/(1 −G(x)) or −log(1 −G(x)α). (3) When the support of T is (−∞,∞): W(G(x)) can be defined as log[−log(1 − G(x))] or log[G(x)/(1 −G(x))] ( [10]). The main purpose of this article is to introduce a new family of distributions, called the Dagum- X family of distributions that are more adaptable to data in a wide range of applications. This article is organized as follows: In Section 2, Dagum-X family of distribution is defined. Also, its propability and cumulative distribution functions are introduced. Some special models of this family are presented in Section 3. Section 4 shows some mathematical properties of the Dagum-X family of distributions including survival, hazard function, rth moments, quantile function, Renyi entropy and order statistics. The characterizations of one sub-model of this family are studied in Section 5. The maximum likelihood method for parameter estimation is discussed in section 6. In section 7 simulation studies will be conducted to show the performance of the maximum likelihood estimation (MLE) method. Section 8 provides a real data application to show the flexibility of the Dagum-X family. Finally, concluding remarks are presented in Section 9. Int. J. Anal. Appl. (2023), 21:45 3 2. The Dagum - X Family The importance of a statistical model lies in the fitness of the probability distribution to the data. Thus, different families of probability distributions have been developed for fitting different types of data. However, there are still several constraints that affect on fitting the formed distributions to the data appropriately especially in particular applications. As mentioned earlier, Dagum distribution has received interest from researchers because of its competition with other models. Different extensions that include Dagum distribution have been proposed and developed using different approaches in attempt to provide more flexibility in fitting data. Using the kurtosis diagram provided by [11] and [12], [13] presented the log-Dagum distribution and examined the changes in the kurtosis. More structural properties and parameter estimates for the log-Dagum distribution were addressed by [14]. [15] proposed a new class of distributions called Mc-Dagum distribution. Several distributions, including the beta-Dagum, beta-Burr III, beta-Fisk, Dagum, Burr III, and Fisk distributions, are included in this class of distributions as special cases. They obtained the properties of the model and the maximum likelihood estimates of the model parameters. [16] proposed a new class of weighted Dagum and related distributions and discussed this class in detail. [17] studied a new five-parameter model called the extended Dagum distribution and discussed the features of the model. The proposed model contains as special cases the log-logistic and Burr III distributions among others. [18] proposed a new four parameter distribution called the Dagum-Poisson (DP) distribution by compounding Dagum and Poisson distributions. The structural properties and the maximum likelihood estimates (MLEs) of the parameters are obtained. [19] pro- posed the exponentiated generalized exponential Dagum distribution. There are several sub-models in this family of distributions, including the Dagum distribution, Burr III distribution, exponentiated generalized Dagum distribution, Fisk distribution, and exponentiated generalized exponential Burr III distribution. [20] introduced a new model called a power log-Dagum distribution. The model consists of many new sub-models such as: linear log-Dagum, power logistic, log-Dagum distributions and linear logistic among them. Three distinct estimating procedures are given along with the model’s properties. The odd Dagum-G family, which [21] introduced, is a new family of continuous distributions with three additional shape parameters. The properties of the suggested family and the model parameters estimates are attained. Using the T-X approach, several new distributions have been introduced in the literature. We generalized the Dagum distribution using T-X method by [10]. The new family of Dagum distribution, called Dagum-X, can be defined as follows: Let F (t) and f (t) be the CDF and the PDF for a Dagum random variable T ∈ [0,∞), given by 4 Int. J. Anal. Appl. (2023), 21:45 F (t; λ,δ,β) = (1 + λt−δ) −β ,t > 0, λ,δ,β > 0, (2.1) and f (t; λ,δ,β) = βλδt−δ−1(1 + λt−δ) −β−1 ,t > 0, λ,δ,β > 0, (2.2) where ≥ is a scale parameter and ◦ and � are shape parameters. By replacing t in Equation (2.1) by the W (G(X)) = G(x;θ) G(x;θ) , we obtained the CDF of a new family namely, Dagum-X family, where G(x; θ) and G(x; θ)=1-G(x; θ) are the baseline CDF and survival function (SF) depending on a parameter vector θ. F (x; λ,δ,β,θ) = ( 1 + λ [ G(x; θ) G(x; θ) ]−δ)−β , (2.3) The PDF is obtained by differentiating Equation (2.3) with respect to (w. r. t.) x as follows: f (x; λ,δ,β,θ) = βλδg(x) [G(x)] −δ−1 [1 −G(x)]−δ+1 [ 1 + λ ( G(x) 1 −G(x) )−δ]−β−1 , t > 0, λ,δ,β > 0. (2.4) 3. Special Models One of the main reasons for the desire to generate different families of distributions is to provide different extensions of appropriate distributions that are more flexible to use with data in various applications. In this section, some Dagum-X special distributions are introduced, such as Dagum- Weibull(D-W), Dagum-exponential(D-exp), Dagum-Rayleigh(D-R) and Dagum-Fréchet(D-Fr). 3.1. The Dagum- Weibull distribution. The Weibull distribution is one of the lifetime distributions that is most frequently used in different areas, such as economics, biology, hydrology and engineering sciences due to its simplicity and versatility. It generalizes the exponential model to include non constant failure rate functions. In particular, it encompasses both increasing and decreasing failure rate functions. As it is well known that Weibull distribution (with scale and shape parameters a, b > 0) has CDF and PDF given by: G(x; a,b) = 1 −e−( x a ) b , x > 0, a,b > 0 (3.1) and g(x; a,b) = b ab xb−1e−( x a ) b , x > 0, a,b > 0 (3.2) The CDF and PDF of Dagum-Weibull distribution can be obtained by substituting Equations (3.1) and Equations (3.2) in Equations (2.3) and Equations (2.4) as follows: F (x; λ,δ,β,a,b) =  1 + λ [ 1 −e−( x a ) b e−( x a ) b ]−δ−β, x > 0, λ,δ,β,a,b > 0, (3.3) Int. J. Anal. Appl. (2023), 21:45 5 and f (x; λ,δ,β,a,b) = ( βλδb ab ) xb−1 (1 −e−( x a ) b ) −δ−1 (e−( x a ) b ) −δ  1 + λ [ 1 −e−( x a ) b (e−( x a ) b ) ]−δ−β−1, (3.4) x > 0, λ,δ,β,a,b > 0. 3.2. The Dagum-exponential distribution. The exponential distribution is one of the common dis- tributions in reliability analysis. It is a particular case of the gamma distribution and often used to model the time elapsed between events. The exponential distribution (with parameter a > 0) has CDF and PDF given by: G(x; a) = 1 −e−( x a ), x > 0, a > 0 (3.5) and g(x; a) = 1 a e−( x a ), x > 0, a > 0 (3.6) The CDF and PDF of Dagum-exponential distribution can be obtained by substituting Equations (3.5) and Equations (3.6) in Equations (2.3) and Equations (2.4) as follows: F (x; λ,δ,β,a) =  1 + λ [ 1 −e−( x a ) e−( x a ) ]−δ−β, x > 0, λ,δ,β,a > 0, (3.7) and f (x; λ,δ,β,a) = ( βλδ a ) (1 −e−( x a )) −δ−1 (e−( x a )) −δ  1 + λ [ 1 −e−( x a ) (e−( x a )) ]−δ−β−1, x > 0, λ,δ,β,a > 0. (3.8) 3.3. The Dagum-Rayleigh distribution. The Rayleigh distribution is a continuous probability distri- bution introduced by [22]. It is a special case of the Weibull distribution with a scale parameter of 2. It plays an essential role in modeling and analyzing lifetime data such as survival and reliability analysis, theory of communication, physical sciences, technology, diagnostic imaging and applied statistics. The Rayleigh distribution (with scale parameter a > 0) has CDF and PDF given by: G(x; a) = 1 −e−( x a ) 2 , x > 0, a > 0 (3.9) and g(x; a) = 2 a2 xe−( x a ) 2 , x > 0, a > 0 (3.10) The CDF and PDF of Dagum-Rayleigh distribution can be obtained by substituting Equations (3.9) and Equations (3.10) in Equations (2.3) and Equations (2.4) as follows: F (x; λ,δ,β,a) =  1 + λ [ 1 −e−( x a ) 2 e−( x a ) 2 ]−δ−β, x > 0, λ,δ,β,a > 0 (3.11) 6 Int. J. Anal. Appl. (2023), 21:45 and f (x; λ,δ,β,a) = ( 2βλδ a2 ) (x) (1 −e−( x a ) 2 ) −δ−1 (e−( x a ) 2 ) −δ  1 + λ [ 1 −e−( x a ) 2 (e−( x a ) 2 ) ]−δ−β−1, (3.12) x > 0, λ,δ,β,a > 0. 3.4. The Dagum-Fréchet distribution. The Fréchet (Fr) Distribution was developed in the 1920s by French mathematician Maurice René Fréchet to model maximum values in a data set that came from different phenomena such as flood analysis, horse racing, human lifespans, maximum rainfalls and river discharges in hydrology. It is considered as one of the extreme value distributions (EV Ds), known as the EV D Type II. The Fr distribution (with scale and shape parameters a,b > 0 ) has CDF and PDF given by: G(x; a,b) = e−( a x ) b , x > 0, a,b > 0 (3.13) and g(x; a,b) = babx−b−1e−( a x ) b , x > 0, a,b > 0 (3.14) The CDF and PDF of Dagum-Fréchet distribution can be obtained by substituting Equations (3.13) and Equations (3.14) in Equations (2.3) and Equations (2.4) as follows: F (x; λ,δ,β,a,b) =  1 + λ [ e−( a x ) b (1 −e−( a x ) b ) ]−δ−β, x > 0, λ,δ,β,a,b > 0, (3.15) and f (x; λ,δ,β,a,b) = βλδbabx−b−1 (e−( a x ) b ) −δ (1 −e−( a x ) b ) −δ+1  1 + λ [ e−( a x ) b (1 −e−( a x ) b ) ]−δ−β−1,x > 0, λ,δ,β,a,b > 0. 4. Mathematical Properties of the Dagum-X Family This section describes some of mathematical properties of the Dagum-X family of distributions. 4.1. Survival and Hazard rate functions. Let the random variable T be the time to failure of the Dagum-X family of distributions. The survival and hazard rate functions of Dagum-X family of distributions are, respectively, given by: S(t; θ) = 1 − ( 1 + λ [ G(t; θ) G(t; θ) ]−δ)−β , (4.1) and h(t; θ) = βλδg(t) [G(t)] −δ−1 [1−[G(t)]]−δ+1 [ 1 + λ( G(t) 1−G(t)) −δ]−β−1 1 − ( 1 + λ [ G(t;θ) G(t;θ) ]−δ)−β , (4.2) Int. J. Anal. Appl. (2023), 21:45 7 where θ = (β,λ,δ)T is a vector of parameters of baseline distribution. 4.2. Moments. Let X be a random variable follows the Dagum-X family with the density function given in Equation (2.4). The rth moment of X is given by: E(Xr ) = βλδ ∫ ∞ 0 xrg(x) [G(x)] −δ−1 [1 − [G(x)]]−δ+1 [ 1 + λ ( G(x) 1 −G(x) )−δ]−β−1 dx. Using the expansion (see [23]): (1 + x) −(n+1) = ∞∑ k=0 ( n + k k ) (−1)kxk, (4.3) we have E(Xr ) =βλδ ∫ ∞ 0 xrg(x) [G(x)] −δ−1 [1 − [G(x)]]−δ+1 ∞∑ k=0 (−1)kλk ( β + k k )( G(x) 1 −G(x) )−δk dx, =βδ ∫ ∞ 0 xrg(x) ∞∑ k=0 (−1)kλk+1 ( β + k k ) [G(x)] −δ(k+1)−1 [1 −G(x)]δ(k+1)−1dx, and using the following expansion (see [24]) (1 −x)n = ∞∑ k=0 ( n k ) (−1)kxk, (4.4) we have E(Xr ) =βδ ∫ ∞ 0 xrg(x) ∞∑ k=0 (−1)kλk+1 ( β + k k ) [G(x)] −δ(k+1)−1 ∞∑ m=0 (−1)m ( δ(k + 1) − 1 m ) [G(x)] m dx, =βδ ∫ ∞ 0 xrg(x) ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) [G(x)] J dx Therefore, E(Xr ) = C1 ∫ ∞ 0 xrg(x)[G(x)] J dx. (4.5) where C1 = βδ ∑∞ k,m=0 (−1) k+m λk+1 ( β+k k )( δ(k+1)−1 m ) , J = m−δ(k + 1) − 1. 4.3. Quantail function. Let X be a random variable that has the CDF given in Equation (2.3). The quantile function, Q(u) of X can be derived as follows: Let u = F (x) = ( 1 + λ [ G(x; θ) G(x; θ) ]−δ)−β , After simplification, the quantile function is expressed as Q(u) = G−1   ( [u] −1 β −1 λ )−1 δ 1 + ( [u] −1 β −1 λ )−1 δ   , (4.6) 8 Int. J. Anal. Appl. (2023), 21:45 where, u is a uniform random number on the interval (0, 1) and G−1(.) is the inverse function of G(.). In particular, Q(0.5) is the median of the family and defined by substituting u = 0.5 in Equation (4.6): Q(0.5) = G−1   ( [0.5] −1 β −1 λ )−1 δ 1 + ( [0.5] −1 β −1 λ )−1 δ   . The first and third quartiles can be obtained also by substituting u = 0.25 and u = 0.75, respectively, in Equation (4.6), as follows: Q(0.25) = G−1   ( [0.25] −1 β −1 λ )−1 δ 1 + ( [0.25] −1 β −1 λ )−1 δ   , and Q(0.75) = G−1   ( [0.75] −1 β −1 λ )−1 δ 1 + ( [0.75] −1 β −1 λ )−1 δ   . 4.4. Rényi Entropys. The entropy of a random variable X represents a measure of uncertainty vari- ation. Let X be a random variable that has the PDF given in Equation (2.4), then the Rényi entropy of the random variable X is defined as: Rθ(x) = (1 −θ) −1 log [∫ ∞ 0 f (x) θ dx ] , θ > 0 and θ 6= 1. (4.7) Therefore, by applying Equation (2.4) into Equation (4.7), we have: Rθ(x) = 1 1 −θ log  ∫ ∞ 0 (βλδ) θ [g(x)] θ [ [G(x)] −θ(δ+1) [1 −G(x)]−θ(δ−1) ][ 1 + λ ( G(x) 1 −G(x) )−δ]−θ(β+1) dx   . Using the expansions in Equation (4.3) and Equation (4.4), the Rényi entropy of the Dagum-X, Rθ(x), can be written as: Rθ(x) = 1 1 −θ log ∫ ∞ 0 (βλδ) θ [g(x)] θ ∞∑ k,r=0 ( θ(β + 1) + k − 1 k )( θ(δ − 1) + δk r ) × (−1)r+kλk[G(x)]−θ(δ+1)−δk+rdx Thus, Rθ(x) = 1 1 −θ log [ C1 ∫ ∞ 0 [g(x)] θ [G(x)] J1dx ] (4.8) Int. J. Anal. Appl. (2023), 21:45 9 where J1 and C1 are resbectivaly, defined as follow: J1 = −θ(δ + 1) −δk + r C1 =(βλδ) θ ∞∑ k,r=0 ( θ(β + 1) + k − 1 k )( θ(δ − 1) + δk r ) (−1)r+kλk 4.5. Order Statistics. Let x1:n,x2:n, ...,xn:n be the order statistics obtained from the Dagum-X with CDF F (x) and PDF f (x), respectively, given in Equation (2.3) and Equation (2.4). The PDF of the ith order statistics can be expressed as: fi:n(x) = n!βλδg(x) (i − 1)!(n− i)! [G(x)] −δ−1 [1 − [G(x)]]−δ+1 [ 1 + λ ( G(x) 1 −G(x) )−δ]−β−1 ×  [1 + λ( G(x) 1 −G(x) )−δ]−βi−1  1 − [ 1 + λ ( G(x) 1 −G(x) )−δ]−βn−i Let u = [ 1 + λ ( G(x) 1 −G(x) )−δ]−β , (4.9) then fi:n(x) = n!βλδg(x) (i − 1)!(n− i)! [G(x)] −δ−1 [1 − [G(x)]]−δ+1 u (1+1 β ) [u] i−1 [1 −u]n−i, = n!βλδg(x) (i − 1)!(n− i)! [G(x)] −δ−1 [1 − [G(x)]]−δ+1 u (i+1 β ) [1 −u]n−i, By applying the expansion in Equation (4.4), we have fi:n(x) = n!βδg(x) (i − 1)!(n− i)! ∞∑ k,m=0 ( n− i k )( βi + βk + m m ) (−1)k+m(λ)m+1G(x)−(δm+δ+1) × ∞∑ l=0 ( δm + δ − 1 l ) (−1)lG(x)l, = n!βδg(x) (i − 1)!(n− i)! ∞∑ k,m,l=0 ( n− i k )( βi + βk + m m )( δm + δ − 1 l ) (−1)k+m+l(λ)m+1G(x)l−(δm+δ+1). (4.10) 5. Dagum-Fréchet Distribution and Its Properties The Fréchet distribution is becoming increasingly a preferred distribution in extending new statistical models. [25] introduced a distribution that generalizes the Fréchet distribution, known as the exponen- tiated Fréchet distribution and included a detailed analysis of the mathematical properties of this new distribution. [26] introduced and studied three component mixtures of the Fréchet distributions when 10 Int. J. Anal. Appl. (2023), 21:45 the shape parameter is known under Bayesian view point. [27] developed a new compound continuous distribution named the Gompertz Fréchet distribution which extends the Frèchet distribution. [28] proposed a new four-parameter Fréchet distribution called the odd Lomax Fréchet distribution. The new model can be expressed as a linear mixture of Fréchet densities. The D-Fr distribution is introduced briefly in (3.4) as a special model of the Dagum-X family. The CDF and PDF of the distribution are given in Equations (3.15) and (??), respectively. In this section, mathematical properties of the new distribution are presented and the maximum likelihood estimation is employed to estimate the parameters of the new distribution. Monte Carlo Simulation by using R program to assess the performance of the maximum likelihood estimation is applied and discussed. Finally, real data sets are analyzed to illustrate the performance of the proposed distribution. The plot of the PDF is presented using different values for the five parameters to study its behaviour, as shown in Figure (1). 0 2 4 6 8 10 0 .0 0 .2 0 .4 0 .6 0 .8 x P D F parameters β=1.5, λ=0.5, δ=0.75, a=1.0, b=1.5 β=2.0, λ=1.0, δ=0.85, a=1.5, b=2.0 β=2.5, λ=1.5, δ=0.95, a=2.0, b=2.5 β=3.0, λ=2.0, δ=1.15, a=2.5, b=3.0 Figure 1. The D-Fr density function when all shape and scale parameters are changing. Figure (1) displays the density function of the D-Fr for different values of the shape and scale parameters. It is right skewed and has different levels of kurtosis which shows the flexibility of the distribution for modelling skew data. 5.1. Survival and Hazrd functions. The survival and hazard rate functions of D-Fr are given by substituting Equation (3.13) and (3.14) in Equation (4.1) and (4.2) respectively as follows: S(x; θ) = 1 −  1 + λ [ e−( a x ) b (1 −e−( a x ) b ) ]−δ−β, (5.1) Int. J. Anal. Appl. (2023), 21:45 11 and h(x; θ) = βλδbabx−b−1 ( (e−( a x ) b ) −δ (1−e−( a x ) b ) −δ+1 )( 1 + λ [ e−( a x ) b (1−e−( a x ) b ) ]−δ)−β−1 1 − ( 1 + λ [ e−( a x ) b (1−e−( a x ) b ) ]−δ)−β . (5.2) 0 2 4 6 8 10 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 x h (x ) parameters β=1.5, λ=0.5, δ=0.75, a=1.0, b=1.5 β=2.0, λ=1.0, δ=0.85, a=1.5, b=2.0 β=2.5, λ=1.5, δ=0.95, a=2.0, b=2.5 β=3.0, λ=2.0, δ=1.15, a=2.5, b=3.0 Figure 2. The D-Fr hazard rate function when all shape and scale parameters are changing. Various curves of the hazard function of Dagum-Fréchet distribution are shown in Figure (2). By assuming different values of the shape and scale parameters, the curves appear to be unimodal and positive skewed with different levels of skewness and kurtosis. 5.2. Moments. The rth moment of the D-Fr distribution is obtained by substituting Fréchet distri- bution’s CDF and PDF in Equations (3.13) and (3.14) into the rth moment of Dagum-X in Equation (4.5). As a result, the rth moment of D-Fr distribution is given as E(Xr ) = C1 ∫ ∞ 0 xrg(x)[G(x)] J dx, = C1 ∫ ∞ 0 xr ( babx−b−1e−( a x ) b )[ e−( a x ) b ]J dx, = C2 ∫ ∞ 0 xr−b−1 [ e−( a x ) b ]J2 dx, where, C1 = βδ ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) , C2 = C1ba b, 12 Int. J. Anal. Appl. (2023), 21:45 and J2 = J + 1 = m−δ(k + 1) Using integration by substitution, let u = J2 [ ( a x ) b ] then x = [ a( u J2 ) −(1 b ) ] and dx = ( −a bJ2 )( u J2 ) (−1 b −1) du. Hence, E(Xr ) = C2 ∫ ∞ 0 xr−b−1 [ e−( a x ) b ]J2 dx = C2 ∫ 0 ∞ [ a( u J2 ) −(1 b ) ]r−b−1 e−u( −a bJ2 )( u J2 ) (−1 b −1) du = C2 ar−b b ( 1 J2 )−r b +1∫ ∞ 0 u −r b e−udu = C2 ar−b b ( 1 J2 )−r b +1 Γ(− r b + 1) = βδbab ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) ar−b b ( 1 J2 )−r b +1 Γ(− r b + 1) = βδar ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m )( 1 J2 )−r b +1 Γ(− r b + 1) Then, the moment of Dagum-Fréche distribution is given as E(Xr ) = C3 ( 1 J2 )−r b +1 ar Γ(− r b + 1), [1 − r b ] > 0 (5.3) where, C3 = βδ ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) , and J2 = m−δ(k + 1). 5.2.1. Mean and Variance. The mean of D-Fr distribution can be obtained by setting (r = 1) in Equation (5.3), which results in the following form: E(X) = C3 ( 1 J2 )−1 b +1 aΓ(− 1 b + 1), (5.4) where, C3 = βδ ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) , Int. J. Anal. Appl. (2023), 21:45 13 and J2 = m−δ(k + 1). The 2nd moment E(X2) can be found by setting (r = 2) in Equation (5.3), then the variance of D-Fr distribution can be obtained as follows: V ar(X) = [C3 ( 1 J2 )−2 b +1 a2Γ(− 2 b + 1)] − [C3 ( 1 J2 )−1 b +1 aΓ(− 1 b + 1)] 2 . (5.5) where, C3 = βδ ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) , and J2 = m−δ(k + 1). 5.3. Quantail function. The quantail function of the D-Fr distribution, x = F−1(u), can be obtained by inverting the CDF in Equation (3.15) as follows: x = Q(u) = a −log [ (u) −1 β −1 λ ]−1 δ  1+ ( (u) −1 β −1 λ )−1 δ     1 b . (5.6) where, u is a uniform random number on the interval (0, 1). Therefore, the median of the D-Fr can be found by substituting u = 0.5 in Equation (5.6) as follows: Q(0.5) = a −log [ (0.5) −1 β −1 λ ]−1 δ  1+ ( (0.5) −1 β −1 λ )−1 δ     1 b . (5.7) The first and third quartiles can also be obtained by substituting u = 0.25 and u = 0.75 in Equation (5.6), respectively as follows: Q(0.25) = a −log [ (0.25) −1 β −1 λ ]−1 δ  1+ ( (0.25) −1 β −1 λ )−1 δ     1 b , 14 Int. J. Anal. Appl. (2023), 21:45 and Q(0.75) = a −log [ (0.75) −1 β −1 λ ]−1 δ  1+ ( (0.75) −1 β −1 λ )−1 δ     1 b . 5.4. Rényi Entropys. Using the definition of the Rényi entropy in Equation (4.7), and applying the CDF and PDF of Frechet distribution in Equation (3.13) and (3.14), we have: Rθ(x) = 1 1 −θ log [ C1 ∫ ∞ 0 (babx−b−1e−( a x ) b ) θ (e−( a x ) b ) J1 dx ] = 1 1 −θ log [ C2 ∫ ∞ 0 (x−θ(b+1))(e−( a x ) b ) θ (e−( a x ) b ) J1 dx ] = 1 1 −θ log [ C2 ∫ ∞ 0 x−θ(b+1)(e−( a x ) b J2)dx ] where, C1 =(βλδ) θ ∞∑ k,r=0 ( θ(β + 1) + k − 1 k )( θ(δ − 1) + δk r ) (−1)r+kλk, C2 =C1(ba b) θ , J1 =−θ(δ + 1) −δk + r, and J2 =θ + J1. Using integration by substitution and after simplification, we get Rθ(x) = 1 1 −θ log [ C3 ∫ ∞ 0 uθ+ θ b −1 b −1e−udu ] , Rθ(x) = 1 (1 −θ) log [ C3Γ(θ + θ b − 1 b ) ] , [θ + θ b − 1 b ] > 0 (5.8) where C3 and J2 are, respectively, as follows: C3 =(βδ) θ bθ−1a1−θ ∞∑ k,r=0 ( θ(β + 1) + k − 1 k )( θ(δ − 1) + δk r ) (−1)r+kλk+θ ( 1 J2 )(θ+θ−1 b ) , J2 =−δ(θ + k) + r. Int. J. Anal. Appl. (2023), 21:45 15 5.5. Order Statistics. The order statistics of D-Fr distribution is obtained by substituting the Fr distribution’s CDF and PDF in Equation (3.13) and (3.14) in the order statistics of Dagum-X family, in Equation (4.10), as follows: fi:n(x) = n! (i − 1)!(n− i)! βδbabx(−b−1) ∞∑ k,m,l=0 ( n− i k )( βi + βk + m m )( δm + δ − 1 l ) (−1)k+m+l × (λ)m+1[e−( a x ) b ] l−δm−δ . (5.9) 6. Maximum Likelihood Estimation In this section, the MLE method will be applied to estimate the unknown parameters of the D-Fr distribution. Assume that x1,x2, ...xn is a random sample of the D-Fr distribution, then the likelihood function for the vector of parameters θ = (β,λ,δ,a,b)T is given by: L(θ) = n∏ i=1 βλδbabx−b−1 (e−( a x ) b ) −δ (1 −e−( a x ) b ) −δ+1  1 + λ [ e−( a x ) b (1 −e−( a x ) b ) ]−δ−β−1, (6.1) then the log likelihood function can be written as: l = logL = nlogβ + nlogλ + nlogδ + nlogb + nbloga− (b + 1) n∑ i=1 log[x] −δ n∑ i=1 log[e−( a x ) b ] +(δ − 1) n∑ i=1 log[1 −e−( a x ) b ] − (β + 1) n∑ i=1 log  1 + λ ( e−( a x ) b 1 −e−( a x ) b )−δ. (6.2) The first partial derivatives of the log likelihood function in Equation (6.2) with respect to �, ≥, ◦, a and b are respectively given as follows: ∂l ∂β = n β − n∑ i=1 log  1 + λ   e−( axi )b 1 −e−( a xi ) b  −δ  , (6.3) ∂l ∂λ = n λ − (β + 1) n∑ i=1 ( e −( axi ) b 1−e −( axi ) b )−δ 1 + λ ( e −( axi ) b 1−e −( axi ) b )−δ , (6.4) ∂l ∂δ = n δ − n∑ i=1 log[e −( a xi ) b ] − (β + 1) n∑ i=1 − ( e −( axi ) b 1−e −( axi ) b )−δ λlog ( e −( axi ) b 1−e −( axi ) b ) 1 + λ ( e −( axi ) b 1−e −( axi ) b )−δ + n∑ i=1 log[1 −e−( a xi ) b ], (6.5) 16 Int. J. Anal. Appl. (2023), 21:45 ∂l ∂a = nb a − (β + 1) n∑ i=1  λ   be −2( axi ) b   e−( axi )b 1−e −( axi ) b  −δ−1δ( a xi ) −1+b xi(1−e −( axi ) b )2 + be −( axi ) b   e−( axi )b 1−e −( axi ) b  −δ−1δ( a xi ) −1+b x(1−e −( axi ) b )     [ 1 + λ ( e −( axi ) b 1−e −( axi ) b )−δ] −δ ∑n i=1 b[ a xi ]−1+b xi + (−1 + δ) ∑n i=1 be −( axi ) b ( a xi ) −1+b xi[1−e −( axi ) b ] , (6.6) ∂l ∂b = n b + nloga− n∑ i=1 log[xi ] −(β + 1) n∑ i=1  λ   e −2( axi ) b   e−( axi )b 1−e −( axi ) b  −δ−1δlog( a xi )( a xi ) b (1−e −( axi ) b )2 + e −( axi ) b   e−( axi )b 1−e −( axi ) b  −δ−1δlog( a xi )( a xi ) b 1−e −( axi ) b     [ 1 + λ ( e −( axi ) b 1−e −( axi ) b )−δ] −δ ∑n i=1−log( a xi )( a xi ) b + (−1 + δ) ∑n i=1 e −( axi ) b log( a xi )( a xi ) b 1−e −( axi ) b . (6.7) The MLEs β̂, λ̂, δ̂, â, b̂ of β,λ,δ,a,b can be obtained by equating the results to zero and solving the system of nonlinear equations numerically. For interval estimation of the model parameters, inverting Fisher information matrix is required, but finding the expectation of the Fisher information matrix is not easy. Therefore, the 5x5 observed information matrix is used to generate confidence intervals for the model parameters. The observed information matrix is given as follows: I(θ̂)=   − ∂ 2l ∂β2 − ∂ 2l ∂β∂λ − ∂ 2l ∂β∂δ − ∂ 2l ∂β∂a − ∂ 2l ∂β∂b − ∂ 2l ∂λ∂β − ∂ 2l ∂λ2 − ∂ 2l ∂λ∂δ − ∂ 2l ∂λ∂a − ∂ 2l ∂λ∂b − ∂ 2l ∂δ∂β − ∂ 2l ∂δ∂λ −∂ 2l ∂δ2 − ∂ 2l ∂δ∂a − ∂ 2l ∂δ∂b − ∂ 2l ∂a∂β − ∂ 2l ∂a∂λ − ∂ 2l ∂a∂δ −∂ 2l ∂a2 − ∂ 2l ∂a∂b − ∂ 2l ∂b∂β − ∂ 2l ∂b∂λ − ∂ 2l ∂b∂δ − ∂ 2l ∂b∂a −∂ 2l ∂b2   The expectation of the observed information matrix can be solved iteratively using R software. Therefore, the multivariate normal distribution N5(0, I−1) can be used to construct 100(1-�)% two sided approximate confidence intervals for the model parameters �, ≥, ◦, a and b where α is the significant level. 7. Simulation Study In this Section, simulation studies have been performed using R program to evaluate the theoretical results of the estimation process. The performance of the MLEs of the parameters has been consid- ered. Furthermore, the approximate confidence intervals with confidence level 90% are obtained. The Int. J. Anal. Appl. (2023), 21:45 17 algorithm for the simulation procedure is described below: Step 1: 5000 random samples of size n=75, 100, 200, 300, 600 and 1000 are generated from the D-Fr distribution. The true parameter values are assumed as ( �=0.75, ≥=0.2, ◦=0.1, a = 0.9 and b = 0.7). Step 2: The parameters of the distribution are estimated using the MLE method for each sample. Step 3: The R function (nlminb) is used to solve the five nonlinear likelihood for �, ≥, ◦, a and b. Step 4: For each simulation, the average biases (ABs) and the mean sqare errors (MSEs) are calcu- lated by: Bias(ŷ) = ∑5000 i=1 1 5000 (ŷ −y), MSE(ŷ) = ∑5000 i=1 1 5000 (ŷ −y)2. Table 1. MLEs, ABs, MSE and 90% confidence limits of the parameters when n= 75, 100, 200, 300, 600 and 1000. Sample Parameter Estimate Bias MSE Lower Limit Upper Limit Length β 0.8054288 0.053428772 0.121979438 0.23019589 1.3706617 1.1404658 λ 0.2755135 0.075513528 0.254675089 0.54778984 1.0988169 0.5510270 n=75 δ 0.1209819 0.020981874 0.002012837 0.05554949 0.1864143 0.1308648 a 0.8423633 -0.057636675 0.153099671 0.20379481 1.4809318 1.2771370 b 0.7091435 0.009143525 0.009290203 0.55082434 0.8674627 0.3166384 β 0.8005119 0.050511859 0.084791214 0.33123212 1.2757916 0.9445595 λ 0.2197650 0.019764989 0.057678640 0.17516100 0.6146910 0.4398520 n=100 δ 0.1173703 0.017370314 0.001444903 0.06158238 0.1731582 0.1115759 a 0.8595773 -0.040422730 0.100288315 0.34132407 1.3778305 1.0365064 b 0.6978595 -0.008140522 0.004583588 0.58620669 0.8095123 0.2233056 β 0.7746440 0.024643973 0.0211146825 0.53835754 1.0109304 0.4725729 λ 0.1953267 -0.004673258 0.0038030341 0.09386590 0.2967876 0.2029217 n=200 δ 0.1090029 0.009002877 0.0007865224 0.06517773 0.1528280 0.0876503 a 0.8856043 -0.014395702 0.0190316874 0.65922062 1.1119880 0.4527674 b 0.6917052 -0.007294776 0.0010332734 0.64046296 0.7429475 0.1024845 β 0.7622825 0.012282484 0.0073377378 0.62240298 0.9021620 0.27975901 λ 0.1955416 -0.004458362 0.0008766267 0.14724566 0.2438376 0.09659195 n=300 δ 0.1051773 0.005177325 0.0001848952 0.08443121 0.1259234 0.04149223 a 0.8932748 -0.006725190 0.0063372406 0.76239317 1.0241565 0.26176328 b 0.6936197 -0.006380267 0.0004032663 0.66220213 0.7250373 0.06283520 β 0.7520984 0.002098386 1.419876e-03 0.69002082 0.8141759 0.12415513 λ 0.1987551 -0.001244911 1.054027e-04 0.18194023 0.2155699 0.03362971 n=600 δ 0.1014451 0.001445127 5.141023e-05 0.08985726 0.1130330 0.02317574 a 0.8981831 -0.001816934 8.178124e-04 0.85109266 0.9452735 0.09418082 b 0.6982087 -0.001791341 6.182002e-05 0.68557661 0.7108407 0.02526411 β 0.7501594 0.0001594468 6.794310e-05 0.73656144 0.7637575 0.027196020 λ 0.1998049 -0.0001951371 7.570256e-06 0.19527647 0.2043333 0.009056788 n=1000 δ 0.1001842 0.0001841564 2.657683e-06 0.09751148 0.1028568 0.005345358 a 0.8996841 -0.0003158548 7.301607e-05 0.88559462 0.9137737 0.028179044 b 0.6997525 -0.0002474826 4.987021e-06 0.69609049 0.7034145 0.007324047 18 Int. J. Anal. Appl. (2023), 21:45 From Table (1), It can be observed that, as the sample size increases, the MLEs approach to the initial values of the parameters. For each sample size n, the MLEs are evaluated using two accuracy measures which are ABs and MSE. As the sample size increases, the ABs and MSEs of the estimated parameters decrease. This indicates that the maximum likelihood estimation method provides consistent estimators for the parameters and approaches the population parameters’ values as the sample size increases. It is also noted that the lengths of the confidence intervals for the estimated parameters decrease as the sample size increases. 8. Application For more illustration, this section compares the efficiency of the goodness-of-fit for the D-Fr dis- tribution with some selected distributions in literature. In particular, two real data sets are used to compare the proposed model with four other distributions, namely, beta-Fréchet(BF) by [29], Gamma- Extended-Fréchet(GEF) by [30], Exponentiated-Exponential-Fréchet(EEF) by [31] and Fréchet(F) dis- tributions by [32] which is also sudied by [33]. The first data set in Table (2) that is used in comparison is provided by Cordeiro and Silva [34]. The data represent the strengths of 1.5 cm glass fibers, measured at the National Physical Laboratory, England. The second data in Table (3) represents breaking stress of carbon fibers of 50 mm length (GPa) and have been previously used by [35]. Table 2. Strength of 1.5 cm glass fibres data (data set 1). 0.55 0.74 0.77 0.81 0.84 1.24 0.93 1.04 1.11 1.13 1.30 1.25 1.27 1.28 1.29 1.48 1.36 1.39 1.42 1.48 1.51 1.49 1.49 1.50 1.50 1.55 1.52 1.53 1.54 1.55 1.61 1.58 1.59 1.60 1.61 1.63 1.61 1.61 1.62 1.62 1.67 1.64 1.66 1.66 1.66 1.70 1.68 1.68 1.69 1.70 1.78 1.73 1.76 1.76 1.77 1.89 1.81 1.82 1.84 1.84 2.00 2.01 2.24 Table 3. Breaking stress of carbon fibers of 50 mm length data (data set 2). 0.39 0.85 1.08 1.25 1.47 1.57 1.61 1.61 1.69 1.80 1.84 1.87 1.89 2.03 2.03 2.05 2.12 2.35 2.41 2.43 2.48 2.50 2.53 2.55 2.55 2.56 2.59 2.67 2.73 2.74 2.79 2.81 2.82 2.85 2.87 2.88 2.93 2.95 2.96 2.97 3.09 3.11 3.11 3.15 3.15 3.19 3.22 3.22 3.27 3.28 3.31 3.31 3.33 3.39 3.39 3.56 3.60 3.65 3.68 3.70 3.75 4.20 4.38 4.42 4.70 4.90 Certain criteria are used in order to compare between the distributions. The distribution with best fit is the one that has the lowest value of the information criteria (AIC, AICc, BIC and HQIC) that are defined as Int. J. Anal. Appl. (2023), 21:45 19 AIC = −2l(θ̂) + 2p, BIC = −2l(θ̂) + plog(n), AICc = AIC + 2p(p + 1) n−p− 1 , HQIC = −2l(θ̂) + 2plog(log(n)). where l(θ̂) is denoted by the log likelihood function evaluated at the maximum likelihood estimates, p is the number of parameters in the model and n is the sample size. Table 4. The log likelihood, AIC, AICc, BIC and HQIC for the data set 1 Distribution l̂ AIC AICc BIC HQIC D-Fr 12.7337 35.46739 36.52003 46.18307 39.68192 BF 30.22177 68.44353 69.13319 77.01607 71.81516 GEF 30.66209 69.32417 70.01383 77.89671 72.69579 F 46.85336 97.70672 97.90672 101.993 99.39253 EEF 47.63954 103.2791 103.9687 111.8516 106.6507 Table 5. The log likelihood, AIC, AICc, BIC and HQIC for the data set 2 Distribution l̂ AIC AICc BIC HQIC D-Fr 85.27445 180.5489 181.5489 191.4972 184.8751 BF 100.276 208.552 209.2077 217.3106 212.0129 EEF 100.4079 208.8158 209.4716 217.5744 212.2768 GEF 101.1724 210.3448 211.0006 219.1035 213.8058 F 121.195 246.39 246.5805 250.7693 248.1205 Table 6. The MLEs for the data sets 1 and 2 Data Distribution β λ δ a b D-Fr 0.2231043 15.0981157 15.6678754 1.0476744 0.9305450 BF 0.6071596 1.9876964 ... 17.9057801 41.5883328 Data 1 GEF 0.7120245 1.7786321 ... 35.0796598 13.7818068 F ... ... ... 1.263794 2.887747 EEF 33.4764197 4.7270858 ... 27.9266876 0.3634021 D-Fr 0.2502454 0.7341762 22.7850195 1.1852932 0.3223076 BF 0.3798751 3.1899014 ... 20.5420435 40.0668395 Data 2 GEF 0.5395719 3.5837019 ... 23.1357231 8.5334421 F ... ... ... 2.034154 1.649719 EEF 30.5722159 2.2994984 ... 34.5713778 0.4487714 20 Int. J. Anal. Appl. (2023), 21:45 Tables (4) and (5), demonstrate that the D-Fr model has the lowest value of the information criteria which implies that the proposed model provides a better fit than the other comparative models. x1 D e n s it y 0 1 2 3 4 0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 Distributions Dagum−Frechet Frechet Beta−Frechet Gamma Extended Frechet Exponentiated−Exponential Frechet Distributions Dagum−Frechet Frechet Beta−Frechet Gamma Extended Frechet Exponentiated−Exponential Frechet Figure 3. Fitted density curves to the first real data. x1 D e n s it y 0 1 2 3 4 5 6 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 Distributions Dagum−Frechet Frechet Beta−Frechet Gamma Extended Frechet Exponentiated−Exponential Frechet Figure 4. Fitted density curves to the second real data. Plots of the fitted densities of the five distributions are shown in Figures (3) and (4). The plots illustrate that the D-Fr distribution provides a better fit to the data than other distributions. 9. Conclusion The development of generalizing families of distributions have attracted the attention of both theoretical and applied statisticians. In this paper, a new family of distributions, called the Dagum-X family of distribution is introduced. The mathematical properties of Dagum-X family of distributions are discussed. A sub model called Dagum-Frechet distribution is presented with some of its properties. The maximum likelihood estimation method was employed for estimating the model parameters and Int. J. Anal. Appl. (2023), 21:45 21 investigated through a simulation study. The simulation study indicates that the maximum likelihood estimation method provides consistent estimators for the parameters. The performance of the Dagum- Frechet distribution was compared to that of beta Frechet, Gamma-Extended-Frechet, Exponentiated- Exponential-Frechet, and Frechet distributions using two real-life data sets for demonstration purposes. 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J. 8 (2019), 139-143. https://doi.org/10.15406/bbij.2019.08.00282. https://doi.org/10.6339/jds.201501_13(1).0004 https://doi.org/10.1285/I20705948V9N1P169 https://doi.org/10.1285/I20705948V9N1P169 https://doi.org/10.1016/j.jksus.2017.09.009 https://doi.org/10.1080/02664763.2018.1523376 https://doi.org/10.1080/14786448008626893 https://doi.org/10.5539/ijsp.v2n3p31 https://doi.org/10.18187/pjsor.v13i3.1703 https://doi.org/10.1080/25742558.2019.1568662 https://doi.org/10.18187/pjsor.v16i2.3097 https://doi.org/10.2991/jsta.2013.12.1.4 https://doi.org/10.2991/jsta.2013.12.1.4 https://doi.org/10.6339/jds.201601_14(1).0010 https://doi.org/10.6339/jds.201601_14(1).0010 https://doi.org/10.1080/09720510.2019.1645400 https://doi.org/10.5902/2179460x13194 https://doi.org/10.15406/bbij.2019.08.00282 1. Introduction 2. The Dagum - X Family 3. Special Models 3.1. The Dagum- Weibull distribution 3.2. The Dagum-exponential distribution 3.3. The Dagum-Rayleigh distribution 3.4. The Dagum-Fréchet distribution 4. Mathematical Properties of the Dagum-X Family 4.1. Survival and Hazard rate functions 4.2. Moments 4.3. Quantail function 4.4. Rényi Entropys 4.5. Order Statistics 5. Dagum-Fréchet Distribution and Its Properties 5.1. Survival and Hazrd functions 5.2. Moments 5.3. Quantail function 5.4. Rényi Entropys 5.5. Order Statistics 6. Maximum Likelihood Estimation 7. Simulation Study 8. Application 9. Conclusion References