# T-Dagum: A Way of Generalizing Dagum Distribution Using Lomax Quantile Function

Authors: Matthew I. Ekum; Muminu O. Adamu; Eno E. Akarawak
Journal: Journal of Probability and Statistics
 (2020)

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## Abstract

Recently, different distributions have been generalized using theT-R {Y} framework but the possibility of using Dagum distribution has not been assessed. The T-R {Y} combines three distributions, with one as a baseline distribution, with the strength of each distribution combined to produce greater effect on the new generated distribution. The new generated distributions would have more parameters but would have high flexibility in handling bimodality in datasets and it is a weighted hazard function of the baseline distribution. This paper therefore generalized the Dagum distribution using the quantile function of Lomax distribution. A member of T-Dagum class of distribution called exponentiated-exponential-Dagum {Lomax} (EEDL) distribution was proposed. The distribution will be useful in survival analysis and reliability studies. Different characterizations of the distribution are derived, such as the asymptotes, stochastic ordering, stress-strength analysis, moment, Shannon entropy, and quantile function. Simulated and real data are used and compared favourably with existing distributions in the literature.


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## Body

## 1. Introduction

The quality of the result of a statistical model depends so much on the fitness of the assumed probability distribution to the data. Thus, significant effort has been made in developing different families of probability distributions along with their relevant statistical methods [1].However, there are still many important real problems where any of the existing standard and newly developed distributions do not fit the data appropriately, especially in the areas of finance, engineering, medicine, and environmental hazards. The Dagum distribution is one of the most important distributions in modeling income and wealth distribution, especially personal income, and is mostly associated with the study of income distribution [2]. It is related to the Gini index (see [3]); it is not the only three-parameter distribution used to model income distribution but it is often most appropriate [4].The following Dagum distributions have been proposed: beta-Dagum distribution [5], Mc-Dagum Distribution [6], weighted Dagum distribution [7], gamma-Dagum distribution [8], exponentiated Kumaraswamy–Dagum distribution [9], extended Dagum distribution [10], transmuted Dagum distribution [11], Dagum–Poisson distribution [12], exponentiated generalized exponential Dagum distribution [13], and power log-Dagum distribution [14].Johnson et al. [15] asserted that the use of four-parameter distributions should be sufficient for most practical purposes and that at least three parameters are needed to model any real data, but they doubted any noticeable improvement arising from, including a fifth or sixth parameter. However, we were motivated by the work of [13], with a six-parameter distribution. Their six-parameter distribution performed better than its submodels with fewer parameters. Other authors have also demonstrated and showed that distributions with more parameters have greater flexibility of modelling reliability and survival data than their submodels with fewer parameters, thereby proving Johnson et al. [15] wrong in their statement. Aljarrah et al. [16] mentioned that adding the fifth parameter to the Normal-Weibull{Cauchy} distribution improved the fit of the model to the data with an increase of more than 22 points in the log-likelihood value. Several works by the following authors, Paranaíba et al. [17]; Cordeiro and Lemonte [18]; Domma and Condino [5]; Oluyede et al. [8]; Silva et al. [10]; and Bakouch et al. [14], further supported the work of Nasiru et al. [13].The motivation of this work is that the Dagum distribution despite being one of the most important and appropriate distributions in modelling income and wealth, and its relationship with Gini Index, is not being generalized via theT-R {Y} framework. The T-R {Y} framework is a combination of 3 distributions, T, R, and Y, where the quantile function of Y is used as a frame to hold the cdf of R, which is being transformed by T, with some parameters of each distribution having effect on the newly formed distribution. One major importance of providing new distribution through the quantile function of an existing distribution is that the newly formed distribution has the tendency of having higher flexibility in handling bimodality in datasets and it is a weighted hazard function of the baseline distribution (Dagum distribution in this case). For more detailed informations on the importance of using this method, T-R {Y}, see Aljarrah et al. [16]; Alzaatreh et al. [19, 20]; Zubair et al. [21]; and Famoye et al. [22]. Also, for detailed knowledge of Dagum distribution, see Bandourian et al. [4]; Kleiber and Kotz [2]; Kleiber [3]; Domma and Condino [5]; Oluyede and Rajasooriya [6]; Oluyede and Ye [7]; Oluyede et al. [8]; Huang and Oluyede [9]; Silva et al. [10]; Shahzad and Asghar [11]; Oluyede et al. [12]; Nasiru et al. [13]; and Bakouch et al. [14].Thus, in this study, a new generalization of the Dagum distribution named that theT-DagumY family is generated and a member of this family, the exponentiated-exponential-Dagum{Lomax} (EEDL) distribution with six parameters, is proposed and its properties are studied. This proposed distribution will not only take into consideration high flexibility in the shape and scale parameters but also takes care of skewness (right and left), kurtosis and tail variation, and sometimes can be stable for some parameter values.The rest of the article is organized as follows. In Section2, the proposed distribution was derived along with some of its characterizations, and the parameters of the proposed distribution were estimated using maximum likelihood estimation (MLE). A simulation study to assess the stability and performance of the parameter estimates was carried out. This is followed by the application of the new model demonstrated using two real datasets, and finally conclusion is given based on the simulation study and the real applications.


## 2. ProposedT-Dagum{Y} Class

The beta-generated family defined asT-X family by [23] was extended by [24] to T-XW family, and further extension was made by [16] to the T-XW by making WFx to be the quantile function of a random variable Y and defined the T-X {Y} family as(1)Gx=∫aQYFxrtdt=RQYFx.TheT-XW in (1) was redefined by Alzaatreh et al. [19] as T-R {Y}. They gave the unified definition of T-R {Y} family. The cdf of the T-R {Y} family is defined by(2)FXx=∫aQYFRxfTtdt=PT≤QYFRx=FTQYFRx,where fTt is the pdf of a random variable T, QY⋅ is the quantile function of a random variable Y and FRx is the cdf of a random variable R. QYFRx is differentiable and monotonically nondecreasing. It is necessary that and fTt have the same support.The pdf corresponding to the cdf in (2) is given by(3)fXx=fRxfTQYFRxfYQYFRx.In the literature, many authors have used thisT-R{Y} framework to develop probability distributions, such as Aljarrah et al. [16]; Alzatraah et al. [19, 20]; Nasir et al. [25, 26]; Jamal et al. [27, 28]; Zubair et al. [21]; Famoye et al. [22]; and Jamal and Nasir [29]. None of these authors has generalized Dagum distribution using this framework.In this research, we letR be a random variable that follows Dagum distribution with cdf, and FRx is given by(4)FRx=1+xv−u−q.The cdf of theT-DagumY family is thus defined by putting equation (4) in (2) to have(5)FXx=FTQY1+xv−u−q.Equation (5) is the cdf, and FXx of the proposed T-DagumY (or simply T-DY) class of distribution. Let the pdf of Dagum distribution be given as(6)fRx=uqxx/vuq1+x/vuq+1,x>0;u,q,v>0.From equation (3), the corresponding pdf to equation (5) is given by(7)fXx=uqxx/vuq1+x/vuq+1×fTQY1+x/v−u−qfYQY1+x/v−u−q.Remark 1.
IfX follows T-DY class of distributions, it is easy to see that(i)
X=dvFYT−1/q−1−1/u(ii)
QXp=vFYQTp−1/q−1−1/u(iii)
IfT=dY, then X=d Dagumu,q,v(iv)
IfY=dDagumu,q,v, then X=dTwhere QX⋅ is the quantile function of X, FY⋅ is the cdf of Y and QT⋅ is the quantile function of T; where p is generated from a standard uniform distribution. Remark 1(i) is a random variable, while Remark 1(ii) is a quantile function (see [21]).
The cdf in (5) can be used to generate many distributions, who are members of the T-D{Y} class of distributions.


### 2.1. TheT-{Lomax} Family

LetY be a Lomax random variable with pdf given by(8)fYx=γθ1+xθ−γ+1,x>0,γ,θ>0.The cdf and quantile function of Lomax distribution are given byFYx=1−1+x/θ−γ and QYx=θ1−p−1/γ−1, respectively. From equations (5) and (7), the cdf and pdf of the T-D {Lomax} distribution are, respectively, given as(9)fXx=FTθ1−1+xv−u−q−1/γ−θ,(10)fXxuqxx/vuq1+x/vuq+1×fTθ1−1+x/v−u−q−1/γ−θfYθ1−1+x/v−u−q−1/γ−θ.The cdf of theT-Dagum{Lomax} class of distributions, using Lomax quantile function, is defined in (9). This equation (9) is a new way of generalizing Dagum distribution. So, T can be any univariate probability distribution with support 0,∞.


### 2.2. Some Properties ofT-D{Lomax} Class of Distribution

Some general properties of theT-D{Lomax} class are discussed in this section.Lemma 1.
Given any random variableT with pdf fTx, then the random variable X=v1−1+T/θ−γ−1/q−1−1/u follows T-Dagum{Lomax} distribution in (9).Proof.
It is easy to see the result from Remark1(i). Lemma 1 shows the relationships between X and T random variables. Random variable X can be generated from random variable T using these relationships. If, for instance, random variable T is a known standard random variable, in which quantile function is known, then random variates X can be simulated by first simulating T values.Lemma 2.
The quantile functions for theT-D{Lomax} distribution is given by(11)QXp=v1−1+QTp/θ−γ−1/q−1−1/u.Proof.
It is easy to see the result from Remark1(ii).Theorem 1.
Shannon’s entropy of theT-D{Lomax} class of distribution can be expressed as(12)ηX=ηT+ElnfYT+uqlnv−lnuq+Elnx−uqElnx+q+1Eln1+xvu.Proof.
SinceXd=vFYT−1/q−1−1/u, it follows that T=QYFRx. Hence, based on the pdf in equation (3), we can write(13)fXx=fRxfTtfYt.
This implies that(14)ηX=ηT+ElnfYT+ElnfRX.
For theT-DY class of distribution, we have(15)lnfRx=lnuq−lnx+uqlnx−q+1ln1+xvu.
So, the result in Theorem1 follows from (14) and (15).


## 2.1. TheT-{Lomax} Family

LetY be a Lomax random variable with pdf given by(8)fYx=γθ1+xθ−γ+1,x>0,γ,θ>0.The cdf and quantile function of Lomax distribution are given byFYx=1−1+x/θ−γ and QYx=θ1−p−1/γ−1, respectively. From equations (5) and (7), the cdf and pdf of the T-D {Lomax} distribution are, respectively, given as(9)fXx=FTθ1−1+xv−u−q−1/γ−θ,(10)fXxuqxx/vuq1+x/vuq+1×fTθ1−1+x/v−u−q−1/γ−θfYθ1−1+x/v−u−q−1/γ−θ.The cdf of theT-Dagum{Lomax} class of distributions, using Lomax quantile function, is defined in (9). This equation (9) is a new way of generalizing Dagum distribution. So, T can be any univariate probability distribution with support 0,∞.


## 2.2. Some Properties ofT-D{Lomax} Class of Distribution

Some general properties of theT-D{Lomax} class are discussed in this section.Lemma 1.
Given any random variableT with pdf fTx, then the random variable X=v1−1+T/θ−γ−1/q−1−1/u follows T-Dagum{Lomax} distribution in (9).Proof.
It is easy to see the result from Remark1(i). Lemma 1 shows the relationships between X and T random variables. Random variable X can be generated from random variable T using these relationships. If, for instance, random variable T is a known standard random variable, in which quantile function is known, then random variates X can be simulated by first simulating T values.Lemma 2.
The quantile functions for theT-D{Lomax} distribution is given by(11)QXp=v1−1+QTp/θ−γ−1/q−1−1/u.Proof.
It is easy to see the result from Remark1(ii).Theorem 1.
Shannon’s entropy of theT-D{Lomax} class of distribution can be expressed as(12)ηX=ηT+ElnfYT+uqlnv−lnuq+Elnx−uqElnx+q+1Eln1+xvu.Proof.
SinceXd=vFYT−1/q−1−1/u, it follows that T=QYFRx. Hence, based on the pdf in equation (3), we can write(13)fXx=fRxfTtfYt.
This implies that(14)ηX=ηT+ElnfYT+ElnfRX.
For theT-DY class of distribution, we have(15)lnfRx=lnuq−lnx+uqlnx−q+1ln1+xvu.
So, the result in Theorem1 follows from (14) and (15).


## 3. Proposed Six-Parameter Exponentiated-Exponential-Dagum{Lomax} Distribution

The six-parameter exponentiated-exponential-Dagum{Lomax} (EEDL) distribution is proposed and we derived some of its characterizations.


### 3.1. Cumulative Distribution Function (cdf) of EEDL Distribution

Gupta and Kundu [30] defined the pdf of exponentiated exponential distribution as(16)fTx=αλe−λx1−e−λxα−1,where λ is the scale parameter and α is the shape parameter, and the cdf is given by(17)FTx=1−e−λxα;x≥0,α,λ>0.Substituting equations (16) and (17) into equation (9), we have(18)FXx=1−exp−λθ1−1+vux−u−q−1/γ−1α.From (18), it is notable that λθ is a constant and can be replaced with β and vu is also a constant and can be replaced with ϕ without any loss of generality so that (18) is reduced to(19)FXx=1−exp−β1−1+ϕx−u−q−1/γ−1α,where α, γ, u, and q are shape parameters, which define the shape (skewness, kurtosis, and mode) of the distribution, while β and ϕ are scale parameters, which defines the spread of the distribution. Thus, (19) is now the cdf of the new probability distribution called the exponentiated-exponential-dagum{lomax} (EEDL) distribution.


### 3.2. Probability Density Function (pdf) of EEDL Distribution

The pdf of the new probability distribution, that is, EEDL distribution can be obtained by differentiating equation (19) with respect to x or by substituting equations (8) and (16) directly into equation (10) to have(20)fXx=αβquϕγexp−β1−1+ϕx−u−q−1/γ−11−exp−β1−1+ϕx−u−q−1/γ−1α−1xu+11+ϕx−uq+11−1+ϕx−u−q1/γ+1,where x≥0,α,β,q,u,ϕ,γ>0. Equation (20) is the pdf of the new probability distribution, EEDL. The six-parameter distribution will be a good distribution to model any environmental hazard, survival, and time to failure data parameters.If(21)w=1−1+ϕx−u−q−1/γ−1,w+1=1−1+ϕx−u−q−1/γ,w+1−γ=1−1+ϕx−u−q,1+ϕx−u−q=1−w+1−γ,1+ϕx−u=1−w+1−γ−1/q,then(22)fXx=αβquϕγw+1γ+1e−βw1−e−βwα−1xu+11+ϕx−uq+1,x,w≥0,α,β,q,u,ϕ,γ>0,and the cdf becomes(23)FXx=1−e−βwα,where w is a function of x.


### 3.3. Quantile Function of EEDL Distribution

In probability theory, we can characterize a random variable by its quantile function. It is much important in deriving measures of partition such as the median, quartiles, octiles, deciles, and percentiles.Lemma 3.
The quantile function of the EEDL distribution forp random variable, uniformly distributed on the interval [0, 1], is given by(24)QXp=ϕ1/u1−1−1βlog1−p1/α−γ−1/q−1−1/u.Proof.
Letp=Fx=PX≤x, by making x the subject of the formula in equation (19), and the inverse function of p is given by(25)x=QXp=ϕ1/u1−1−1βlog1−p1/α−γ−1/q−1−1/u.
This quantile function in (25) will be used to generate random variates in the simulation study. The median and 1st and 3rd quartiles can be obtained by setting p = 0.5, 0.25, and 0.75, respectively. Other measures of partitions can also be obtained by setting p appropriately.


### 3.4. Survival Function of EEDL Distribution

IfX follows an EEDL distribution and FXx be the probability that any given device of interest will survive to a given point in time, x, such that x∈X, that is, PX≤x, then the survival function, SXx, is a function that gives the probability that such a device will survive beyond x.Suppose thatFXx is the cdf of EEDL distribution supported on the interval 0,∞ as proposed in equation (19); then, the survival function of EEDL is given by(26)SXx=PX>x=1−1−exp−β1−1+ϕx−u−q−1/γ−1α,and in terms of w, equation (26) becomes(27)SXx=PX>x=1−1−e−βwα.


### 3.5. Hazard Function of EEDL Distribution

LetX be a random variable that follows an EEDL distribution, with pdf and survival function given in (22) and (27), respectively; then, its hazard function is given by(28)hXx=αβquϕγw+1γ+1e−βw1−e−βwα−11−1−e−βwα,where w is a function of x as defined in equation (21).


### 3.6. Cumulative Hazard Function of EEDL Distribution

LetX be a random variable that follows an EEDL distribution, with survival function given in (27), then its cumulative hazard function is given by(29)HXx=−log1−1−e−βwα,where w is a function of x as defined in equation (17).Figure1 depicts the pdfs of EEDL distribution for various values of the parameters and depicts that the distribution can be stable (normal), positively skewed, or negatively skewed. Figure 2 depicts that the EEDL distribution can be stable (symmetric), positively or negatively skewed, unimodal, or bimodal. The behaviour of EEDL can help to model any environmental hazard data or any data with high degree of variability. The four shape parameters can capture any features or variation in a dataset.Figure 1
EEDL probability density function: (a) alpha = 0.8,q = phi = gamma = 2, (b) alpha = 3, q = phi = gamma = 2, (c) q = phi = gamma = 2, and (d) alpha = 8, u = phi = gamma = 2.
(a)
(b)
(c)
(d)Figure 2
Histogram of EEDL distribution with various parameter values. (a)α = 1.5, β = 2.5, γ = 1.5, ϕ = 3.5, u = 0.5, and q = 1, (b) β = 4.5, γ = 1.5, ϕ = 2.5, u = 0.9, and q = 2.6, (c) β = 4.5, γ = 1.5, ϕ = 2.5, and u = 0.9, (d) β = 0.5, γ = 0.5, ϕ = 0.8, and u = 1.6, (e) ϕ = 2.5, u = 3.5, and q = 1.6, (f) γ = 1.5, ϕ = 2.5, and u = 4.2, (g) α = 5.5 and γ = 1.5, and (h) u = 10.2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)A distribution can also be characterized by its asymptotes, stochastic ordering, stress, and strength properties.


### 3.7. Asymptotes of EEDL Distribution

#### 3.7.1. Vertical Asymptotes of EEDL Distribution

Theorem 2.
IffXx and hXx are the pdf and hazard rate of EEDL distribution, then line x=−ϕ1/u, if it exist, is a vertical asymptote of the graph of the functions fXx and hXx if the following statements hold:(30)limx⟶−ϕ1/ufXx=∞.
The horizontal asymptote forhXx is given by(31)limx⟶−ϕ1/uhXx=∞.Proof.
The denominator of the pdf of EEDL distribution in (22) is equated to zero to have(32)xu+11+ϕx−uq+1=0,and to solve for x, we have(33)1+ϕx−uq+1=0,ϕx−u=−1,x=−ϕ1/u.
Also, the denominator of the hazard rate of EEDL distribution in (28) is equated to zero to have(34)γ1−1−e−βwα=0,1−e−βwα=0,e−βw=1,−βw=log1,w=0,but(35)w=1−1+ϕx−u−q−1/γ−1,1−1+ϕx−u−q−1/γ−1=0,1+ϕx−u=0,ϕx−u=−1,x=−ϕ1/u.
The proof is complete. Note that ifu is an even real number, then x is a complex number. Also, if u is an odd real number, then x will be negative because ϕ>0. Take note that the limit is one-sided, since x≥0. This implies that the vertical asymptote of EEDL distribution does not exist for all real values of x≥0.


#### 3.7.2. Horizontal Asymptotes of EEDL Distribution

LetfXx and hXx be the pdf and hazard functions of EEDL distribution.The horizontal asymptotes are horizontal lines that the function approaches asx⟶∞.The horizontal asymptote forfXx is given by(36)limx⟶∞fXx=0.The horizontal asymptote forhXx is given by(37)limx⟶∞hXx=0.Take note that the limit is one-sided, sincex≥0.


### 3.8. Stochastic Ordering of EEDL Distribution

#### 3.8.1. General Order Statistics of EEDL Distribution

Theorem 3.
The pdf of the general order statistics of EEDL distributionfXjx exists and it is given by(38)fXjx=n!αβquϕγj−1!n−j!w+1γ+1e−βw1−e−βwαj−11−1−e−βwαn−jxu+11+ϕx−uq+1.Proof.
LetX1,X2,…,Xn denote the order statistics of a random sample that follows EEDL distribution, from a continuous population with cdf FXx and pdf fXx. Then, the pdf Xj is(39)fXjx=n!j−1!n−j!fXxFXxj−11−FXxn−j.
Substitute (22) and (23) into (39) to have (40):(40)fXjx=n!j−1!n−j!αβquϕγw+1γ+1e−βw1−e−βwα−1xu+11+ϕx−uq+11−e−βwαj−11−1−e−βwαn−j,(41)fXjx=n!αβquϕγj−1!n−j!w+1γ+1e−βw1−e−βwαj−11−1−e−βwαn−jxu+11+ϕx−uq+1.
Thus, equation (41) completes the proof.
LetX and Y be two random variables that follow the EEDL distribution. X is less than Y if PX>x≤PY>y,∀x∈−∞,∞, where P (.) denotes the probability of an event.Theorem 4.
LetX1 and X2 be two random variables that follow the EEDL distribution. If X1≤stX2 and EX1=EX2, then X1=dX2 (X1 is equal to X2 in distribution).Proof.
From the pdf of the general order statistic of the EEDL distribution given in (41), set j  = 1 to arrive at the 1st order statistic given by(42)fX1x=nαβquϕγw+1γ+1e−βw1−e−βwα−11−1−e−βwαn−1xu+11+ϕx−uq+1.
Also, from (41), set j  = 2 to arrive at the 2nd order statistic given by(43)fX2x=nn−1αβquϕγw+1γ+1e−βw1−e−βw2α−11−1−e−βwαn−2xu+11+ϕx−uq+1.
ForX1≤X2, we can show that EX1=EX2:(44)e−βw1−e−βwα−11−1−e−βwαn−1≤n−1e−βw1−e−βw2α−11−1−e−βwαn−2.
Using series expansion, we have the inequality as(45)∑i,k=0−1i+kn−1iαi+α−1ke−βwk+1≤∑i,k=0−1i+kn−2iαi+2α−1ke−βwk+1.
Inequality (45) can be reduced to(46)∑i,k=0n−1iαi+α−1k≤∑i,k=0n−2iαi+2α−1k.
Take the expectation of both sides to have(47)E∑i,k=0n−1iαi+α−1k=E∑i,k=0n−2iαi+2α−1k.
Note that the expectation of a constant is a constant.
To make equality prevail, we test wheni=k=0 to arrive at(48)n−1!α−1!n−1!α−1!=n−2!2α−1!n−2!2α−1!,1=1.
Thus, the proof is complete. Hence,X1 and X2 are random samples from EEDL distribution.


### 3.9. Stress-Strength Reliability Analysis of EEDL Distribution

Theorem 5.
LetX and Z be two independent random variables that follow the EEDL distribution with pdfs fXx and fZz, respectively. If Z represents the stress and X represents the strength, then the stress-strength reliability of EEDL distribution with X∼EEDLα1,β1,ϕ,u,q,r and Z∼EEDLα2,β2,ϕ,u,q,r does not depend on β1 or β2.Proof.
The probability thatX>Z is the reliability of the stress-strength of the EEDL distribution, and it is given by(49)R=PZ<X=∫0∞∫0xfXxfZzdzdx,(50)R=PZ<X=∫0∞fXxFZxdx,where(51)FZz=1−e−βwzα2,(52)fXx=α1β1e−β1wx1−e−β1wxα1−11+wxγ+1.
Put equations (51) and (52) in (50) to have(53)R=∫0∞α1β1e−β1wx1−e−β1wxα1−11+wxγ+11−e−βwxα2dx.
By using linear expansion, we have(54)R=α1β1∑i,j,k∞−1i+jα2iα1−1jγ+1k∫0∞wxke−β1+β2i+β1jdx.
The integral in (54) using gamma function becomes(55)R=α1β1∑i,j,k∞−1i+jα2iα1−1jγ+1kΓk+1−β1+β2i+β1jk+1,where i≤α2, j≤α1−1, and k≤γ+1. Equation (55) is the stress-strength reliability function of the EEDL distribution with X∼EEDLα1,β1,ϕ,u,q,r and X∼EEDLα2,β2,ϕ,u,q,r.
However, ifi=j=k=0, equation (55) is reduced to(56)R=α1.
Thus, equation (56) completes the proof. Hence, the stress-strength reliability, R, does not depend on β.
The probability density, cumulative distribution, quantile, survival, hazard, cumulative hazard functions, asymptotes, stochastic ordering, and stress-strength analysis are some of the different ways of characterizing a random variable.


### 3.10. Related Distributions

Most generalized distributions have relationship with their base distributions by varying one or more of its parameters’ value. EEDL is also related with its parent distribution.


#### 3.10.1. Transformations

(1) Exponentiated Exponential DistributionLemma 4.
IfX∼EEDLα,β,γ,u,q,ϕ, then the random variable Y=1−1+ϕX−u−q1/γ−1 has an exponentiated exponential distribution with parameters α and β.Proof.
By using the transformation method, the result is shown as follows:(57)fy=αβe−βy1−e−βyα−1.
(2) Ratio of Exponential and Lomax DistributionsLemma 5.
IfX∼EEDL1,β,γ,u,q,1, then the pdf of random variable Z=1−1+ϕX−u−q1/γ−1 is the ratio of two pdfs of random variables that have an exponential distribution with parameter β and Lomax distribution with parameter γ, respectively.Proof.
By using the transformation method, the result is shown as follows:(58)fz=βe−βzγ1+z−γ.


### 3.11. Moments of EEDL Distribution

The moments of a probability distribution are a very important property in describing the distribution. The mean, variance, standard deviation, measure of skewness and kurtosis, and other parametric measures that describe the distribution can be derived from it.Theorem 6.
LetX follow an EEDL distribution, and the rth moment of X can be expressed in terms of gamma function of W with parameters k+1 and βl+1, and it is given by(59)EXr=αβΓαAϕr/uΓr/u∫0∞wke−βw1+ldw,where(60)A=∑j=0∞∑i=0∞∑k=0γ+1+jr/q∑l=0α−1−1r/u−jΓγ+jr/q+1Γj/q+iΓr/u+jΓj+1Γi+1Γj/qΓα−lΓl+1Γγ+2−k+jr/q.Proof.
Recall thatw=1−1+ϕx−u−q−1/γ−1 and x=1/ϕ1−w+1−γ−1/q−1−1/u so that the pdf of EEDL distribution is reduced to(61)fXx=αβe−βw1−e−βwα−11+w−γ+1,(62)fXx=αβe−βw1−e−βwα−11+wγ+1.
See the complete proof of equation (62) in Appendix I.(63)EXr=∫0∞xrαβe−βw1−e−βwα−11+wγ+1dx,(64)δwδx=quϕx−u+11+ϕx−u−q+1γw+1−γ+1,dx=dwγw+1−γ+1dxquϕx−u+11+ϕx−u−q+1.
Substitute (64) into (63) to have(65)EXr=αβ∫0∞ϕr/u1−1+w−γ−1/q−1−r/ue−βw1−eβwα−11+w−γ+1dw,(66)EXr=αβ∫0∞∑j=0∞∑i=0∞−1r/u−jΓj/q+iΓr/u+jΓj+1Γr/uΓi+1Γj/qϕr/u1+wγ+11+wjγ/qe−βw1−eβwα−1dw,EXr=αβ∫0∞∑j=0∞∑i=0∞∑k=0∞−1r/u−jΓγ+jr/q+1Γj/q+iΓr/u+jΓj+1Γr/uΓi+1Γj/qΓk+1Γγ+2−k+jr/qϕr/uwke−βw1−eβwα−1dw,EXr=αβΓαAϕr/uΓr/u∫0∞wke−βw1+ldw.
Equation (66) completes the proof, where(67)A=∑j=0∞∑i=0∞∑k=0∞∑l=0∞−1r/u−jΓγ+jr/q+1Γj/q+iΓr/u+jΓj+1Γi+1Γj/qΓα−lΓl+1Γγ+2−k+jr/q.
Recall gamma function:(68)Γaba=∫0∞xa−1e−bxdx.
Thus, therth moment of the EEDL distribution is given by(69)EXr=αβΓαAϕr/uΓk+1Γr/uβl+1k+1.
If the subscriptsi=j=k=l=0, then equation (69) reduces to(70)EXr=αϕr/uΓγ+1.
Ifr=1, we have the mean of the EEDL distribution, which does not depend on variables β and q, and it is given by(71)EX=αϕ1/uΓγ+1,and the variance is(72)VX=αϕ2/uΓγ+11−αΓγ+1.


### 3.12. Shannon Entropy of EEDL Distribution

The Shannon entropy of a random variable is a measure of variation of uncertainty. It is defined by [31] as E−logfx for a random variable X with pdf fx. Let X be a random variable that follows EEDL distribution with pdf fXx as given in (62):(73)fXx=αβe−βw1−e−βwα−11+wγ+1.The Shannon entropy of EEDL distribution is(74)E−logαβe−βw1−e−βwα−11+wγ+1,ηXx=−logα−logβ+βEw−α−1Elog1−e−βw−γ+1Elogw+1,where w is a function of x. It can be written as wx


### 3.13. Maximum Likelihood Estimation of EEDL Distribution Parameters

Recall from (20) the pdf of EEDL distribution, and we derived the log likelihood function as(75)l=logL=nlogΩ+nlogα+nlogβ−β∑i=1nwi−ul∑i=1nlogxi+α−1∑i=1nlog1−e−βw.Remember thatw=1−1+ϕx−u−q−1/γ−1 and(76)Ω=∑i,j,k,l=0∞ϕl−1i+l−jΓi+1Γn+lΓj/γ+kΓr−i+2Γi−j+1Γk+1Γj+1Γj/γΓl+1Γn.We can differentiate (75) easily with respect to α and β.Differentiate (75) partially with respect to α, and equate the result to zero and solve for α:(77)δlδα=nα+∑i=1nlog1−e−βwi,0=nα+∑i=1nlog1−e−βwi,−∑i=1nlog1−e−βwi=nα,α^=−n∑i=1nlog1−e−βwi.Differentiate (75) partially with respect to β, equate the result to zero and solve for β:(78)δlδβ=nβ−∑i=1nwi+βα−1∑i=1ne−βxi1−e−βxi,0=nβ−∑i=1nwi+βα−1∑i=1ne−βxi1−e−βxi,∑i=1nwi=nβ+βα−1∑i=1ne−βxi1−e−βxi.Divide through byn(79)w¯=1β+α−1βn∑i=1ne−βxi1−e−βxi.The estimate ofβ is not in a closed form. From (58), we also differentiate partially with respect to other parameters, equate their results to zero, and solve for each of them. Their solutions are not in closed form as well, so we resolved by using R package “MaxLik.”


### 3.14. Simulation

A simulation study is carried out to investigate the performance of the estimators. The standard error of estimate (SE), average absolute bias (AAB), and root mean square error (RMSE) of the maximum likelihood estimators of the parameters of the EEDL were examined. The simulation study was repeated forN = 1000 times each with sample sizes n  = 20, 50, 100, 250, and 500 and parameter values: α = 1.5, β = 0.1, γ = 2.0, ϕ = 0.3, q = 0.01, and u = 0.9.Table1 presents the MLE estimates, standard error of estimate, and AAB and RMSE values of the parameters α, β, γ, ϕ, q, and u for different sample sizes. The result shows that as the sample size approaches infinity, the AAB and RMSE decrease asymptotically to zero, proving their consistency. It is consistent in the sense that it converges to the true parameter value as the number of observations becomes larger and the error reduces to zero. See Figure 3 for pictorial view.Table 1
Simulation results of MLE.
Parameter
n
Estimate
SE
AAB
RMSE
α = 1.5
20
1.5020
0.0000549
0.001610
0.00720
50
1.5013
0.0000204
0.000660
0.00466
100
1.5017
0.0000132
0.000360
0.00357
250
1.5020
0.0000032
0.000120
0.00184
500
1.5001
0.0000001
0.000010
0.00010
β = 0.1
20
0.0367
1.0443120
0.222721
0.99604
50
0.1447
0.3889741
0.087315
0.61741
100
0.1437
0.1935787
0.043777
0.43777
250
0.1596
0.0765768
0.017467
0.27617
500
0.4500
0.0002455
0.000700
0.01565
γ = 2.0
20
2.0110
0.0142591
0.026025
0.11639
50
2.0111
0.0053693
0.010259
0.07254
100
2.0013
0.0026316
0.005104
0.05104
250
2.0010
0.0010761
0.002071
0.03274
500
2.0001
0.0005010
0.001000
0.02236
ϕ = 0.3
20
0.3015
0.3286991
0.124953
0.55881
50
0.3012
0.1237652
0.049252
0.34827
100
0.3011
0.0626044
0.024895
0.24895
250
0.2996
0.0246000
0.009900
0.15653
500
0.2999
0.0001804
0.000600
0.01342
q = 0.01
20
0.0002
0.0624754
0.054476
0.24362
50
0.0154
0.0241751
0.021768
0.15392
100
0.0130
0.0119727
0.010887
0.10887
250
0.0116
0.0047506
0.004350
0.06879
500
0.0101
0.0000000
0.000000
0.00000
u = 0.9
20
0.9014
0.0000152
0.000849
0.00379
50
0.9008
0.0000052
0.000319
0.00226
100
0.9005
0.0000021
0.000144
0.00144
250
0.9002
0.0000012
0.000070
0.00111
500
0.9001
0.0000010
0.000001
0.00000Figure 3
(a)α = 1.5. (b) β = 0.1. (c) γ = 2.0. (d) ϕ = 0.3. (e) q = 0.01. (f) u = 0.9.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.15. Applications

The EEDL distribution is fitted to two real datasets retrieved from [13]. The first application is a data on failure times for 36 appliances subjected to an automatic life test, while the second data is on failure time data on 100 cm yarn subjected to 2 : 3% strain level. EEDL distribution is compared with that of exponentiated generalized exponential Dagum distribution (EGEDD), the exponentiated Kumaraswamy Dagum (EKD) distribution and the Mc-Dagum (McD) distribution using Log-likelihood, Akaike information criterion (AIC), and Kolmogorov–Smirnov (K-S) statistic criteria.


#### 3.15.1. Application 1: Appliances Data

The appliances data in Table2 was obtained from [13, 32]. The dataset consists of failure times for 36 appliances subjected to an automatic life test. The dataset is depicted in Figure 4 and shows that there is a gap in the histogram with positive skewness (2.279) and highly leptokurtic (9.669). Table 3 displays the maximum likelihood estimates of the parameters with their corresponding standard errors in brackets. Table 3 shows all the parameters of the EEDL distribution and other distributions.Table 2
Failure times for 36 appliances subjected to an automatic life test.
11
35
49
170
1990
2831
2223
3034
2327
3059
2400
3112
329
2451
3214
381
2471
3478
708
2551
3504
958
2565
4329
1062
2568
6367
1167
2694
6976
1594
2702
7846
1925
2761
13403Figure 4
Failure time on appliances.Table 3
MLE of parameters and standard errors for appliances data.
Model
Parameter estimates
EEDL
α
⌢
β
⌢
γ
⌢
ϕ
⌢
q
⌢
u
⌢
0.491(0.0822)
0.011(0.0001)
0.260(0.0189)
1.055(0.0729)
1.041(0.0666)
0.163(0.0163)
EGEDD
α
⌢
λ
⌢
β
⌢
θ
⌢
c
⌢
d
⌢
0.001(0.0001)
27.198(0.001)
4.560(0.847)
2.838(0.123)
20.866(0.010)
0.070(0.003)
EKD
α
⌢
λ
⌢
δ
⌢
ϕ
⌢
θ
⌢
5.562(1.517)
12.683(2.158)
3.716(0.755)
0.128(0.029)
11.609(3.922)
McD
λ
⌢
δ
⌢
β
⌢
a
⌢
b
⌢
c
⌢
1.427(0.092)
3.455(0.212)
1.275(6.875)
10.505(56.906)
0.064(0.012)
500.556(6.976)Table4 clearly shows that the EEDL distribution provides a better fit to the appliances data than the other models. Its log-likelihood, AIC and approach zero faster than other distributions. It also has the smallest K-S statistic compared with other distribution in this research. Thus, the EEDL is a better fit to the appliances data.Table 4
Log-likelihood, information, and goodness-of-fit criteria for appliances data.
Model
LogLik
AIC
K-S stat
KSp value
EEDL
−320.756
653.512
0.222
0.340
EGEDD
−328.870
669.740
0.253
0.124
EKD
−341.650
693.295
0.269
0.178
McD
−356.480
724.955
0.347
0.128


#### 3.15.2. Application 2: Yarn Data

Table5 represents the data on time to failure of a 100 cm polyster/viscose yarn subjected to 2 : 3% strain level in textile experiment in order to assess the tensile fatigue characteristics of the yarn. The dataset can be found in [13, 33, 34]. The skewness and kurtosis of the data are 1.336119 and 5.802452, respectively. The data is positively skewed and very peaked, as depicted in Figure 5.Table 5
Failure time data on 100 cm yarn subjected to 2 : 3% strain level.
86
146
251
653
98
249
400
292
131
169
175
176
76
264
15
364
195
262
88
264
157
220
42
321
180
198
38
20
61
121
282
224
149
180
325
250
196
90
229
166
38
337
65
151
341
40
40
135
597
246
211
180
93
315
353
571
124
279
81
286
497
182
423
185
229
400
338
290
398
71
246
185
188
568
55
55
61
244
20
289
393
396
203
829
239
236
286
194
277
143
198
264
105
203
124
137
135
350
193
188Figure 5
Failure time on yarn.The maximum likelihood estimates of the parameters of the fitted models with their corresponding standard errors in brackets are given in Table6. All the parameters of the EEDL are significant at the 5% significance level. The EEDL provides a better fit to the yarn data than the EGEDD, EKD, and McD distributions, as shown in Table 7.Table 6
Maximum likelihood estimates of parameters and standard errors for yarn data.
Model
Parameter estimates
EEDL
α
⌢
β
⌢
γ
⌢
ϕ
⌢
q
⌢
u
⌢
1.0921(0.2133)
0.0004(0.0004)
1.5196(0.1457)
3.4906(0.0005)
0.7205(0.0924)
2.3297(0.0228)
EGEDD
α
⌢
λ
⌢
β
⌢
θ
⌢
c
⌢
d
⌢
0.026(0.007)
75.310(0.007)
0.017(0.005)
3.513(0.631)
45.692(0.036)
0.090(0.011)
EKD
α
⌢
λ
⌢
δ
⌢
ϕ
⌢
θ
⌢
46.109(1.295)
39.413(5.006)
5.188(0.961)
0.203(0.040)
31.169(11.023)
McD
λ
⌢
δ
⌢
β
⌢
a
⌢
b
⌢
c
⌢
0.027(0.0848)
0.600(0.09647)
98.780(0.00002)
0.333(0.1504)
25.042(0.0004507)
46.276(0.00004654)Table 7
Log-likelihood, information, and goodness-of-fit criteria for yarn data.
Model
LogLik
AIC
K-S stat
KSp value
EEDL
−625.459
199.719
0.110
0.581
EGEDD
−628.170
1268.336
0.249
0.124
EKD
−653.960
1317.913
0.985
0.178
McD
−628.200
1268.399
0.285
0.128Table7 shows that EEDL log-likelihood and its AIC approaches zero than that of others and has the smallest K-S statistic compared to the other models. Thus, the EEDL is a better fit.The kernel density graphs of the data and the EEDL distribution are super-imposed on the histograms on Figures4 and 5. This shows that the newly proposed EEDL distribution fits the two data.Figures6 to 9 show that EEDL distribution fit the two data well. The EEDL distribution fits the yarn data more than the appliances data. The QQ plots in Figures 10 and 11 shows that most of the data points, especially the ones at the middle, fall on the theoretical line, but the few ones outside the unit circle (−1 and 1) of the theoretical quantiles fall outside the line. This means that if the edges of the data are trimmed, the EEDL distribution would be a perfect fit for the data. However, Tables 4 and 7 shows a confirmatory test that the EEDL distribution is a good fit to both the appliances and yarn data via the K-S statistic and p value.Figure 6
TTT plot of appliances.Figure 7
TTT plot of yarn.Figure 8
PP plot of appliances.Figure 9
PP plot of yarn.Figure 10
QQ plot of appliances.Figure 11
QQ plot for EEDL distribution on yarn data.


## 3.1. Cumulative Distribution Function (cdf) of EEDL Distribution

Gupta and Kundu [30] defined the pdf of exponentiated exponential distribution as(16)fTx=αλe−λx1−e−λxα−1,where λ is the scale parameter and α is the shape parameter, and the cdf is given by(17)FTx=1−e−λxα;x≥0,α,λ>0.Substituting equations (16) and (17) into equation (9), we have(18)FXx=1−exp−λθ1−1+vux−u−q−1/γ−1α.From (18), it is notable that λθ is a constant and can be replaced with β and vu is also a constant and can be replaced with ϕ without any loss of generality so that (18) is reduced to(19)FXx=1−exp−β1−1+ϕx−u−q−1/γ−1α,where α, γ, u, and q are shape parameters, which define the shape (skewness, kurtosis, and mode) of the distribution, while β and ϕ are scale parameters, which defines the spread of the distribution. Thus, (19) is now the cdf of the new probability distribution called the exponentiated-exponential-dagum{lomax} (EEDL) distribution.


## 3.2. Probability Density Function (pdf) of EEDL Distribution

The pdf of the new probability distribution, that is, EEDL distribution can be obtained by differentiating equation (19) with respect to x or by substituting equations (8) and (16) directly into equation (10) to have(20)fXx=αβquϕγexp−β1−1+ϕx−u−q−1/γ−11−exp−β1−1+ϕx−u−q−1/γ−1α−1xu+11+ϕx−uq+11−1+ϕx−u−q1/γ+1,where x≥0,α,β,q,u,ϕ,γ>0. Equation (20) is the pdf of the new probability distribution, EEDL. The six-parameter distribution will be a good distribution to model any environmental hazard, survival, and time to failure data parameters.If(21)w=1−1+ϕx−u−q−1/γ−1,w+1=1−1+ϕx−u−q−1/γ,w+1−γ=1−1+ϕx−u−q,1+ϕx−u−q=1−w+1−γ,1+ϕx−u=1−w+1−γ−1/q,then(22)fXx=αβquϕγw+1γ+1e−βw1−e−βwα−1xu+11+ϕx−uq+1,x,w≥0,α,β,q,u,ϕ,γ>0,and the cdf becomes(23)FXx=1−e−βwα,where w is a function of x.


## 3.3. Quantile Function of EEDL Distribution

In probability theory, we can characterize a random variable by its quantile function. It is much important in deriving measures of partition such as the median, quartiles, octiles, deciles, and percentiles.Lemma 3.
The quantile function of the EEDL distribution forp random variable, uniformly distributed on the interval [0, 1], is given by(24)QXp=ϕ1/u1−1−1βlog1−p1/α−γ−1/q−1−1/u.Proof.
Letp=Fx=PX≤x, by making x the subject of the formula in equation (19), and the inverse function of p is given by(25)x=QXp=ϕ1/u1−1−1βlog1−p1/α−γ−1/q−1−1/u.
This quantile function in (25) will be used to generate random variates in the simulation study. The median and 1st and 3rd quartiles can be obtained by setting p = 0.5, 0.25, and 0.75, respectively. Other measures of partitions can also be obtained by setting p appropriately.


## 3.4. Survival Function of EEDL Distribution

IfX follows an EEDL distribution and FXx be the probability that any given device of interest will survive to a given point in time, x, such that x∈X, that is, PX≤x, then the survival function, SXx, is a function that gives the probability that such a device will survive beyond x.Suppose thatFXx is the cdf of EEDL distribution supported on the interval 0,∞ as proposed in equation (19); then, the survival function of EEDL is given by(26)SXx=PX>x=1−1−exp−β1−1+ϕx−u−q−1/γ−1α,and in terms of w, equation (26) becomes(27)SXx=PX>x=1−1−e−βwα.


## 3.5. Hazard Function of EEDL Distribution

LetX be a random variable that follows an EEDL distribution, with pdf and survival function given in (22) and (27), respectively; then, its hazard function is given by(28)hXx=αβquϕγw+1γ+1e−βw1−e−βwα−11−1−e−βwα,where w is a function of x as defined in equation (21).


## 3.6. Cumulative Hazard Function of EEDL Distribution

LetX be a random variable that follows an EEDL distribution, with survival function given in (27), then its cumulative hazard function is given by(29)HXx=−log1−1−e−βwα,where w is a function of x as defined in equation (17).Figure1 depicts the pdfs of EEDL distribution for various values of the parameters and depicts that the distribution can be stable (normal), positively skewed, or negatively skewed. Figure 2 depicts that the EEDL distribution can be stable (symmetric), positively or negatively skewed, unimodal, or bimodal. The behaviour of EEDL can help to model any environmental hazard data or any data with high degree of variability. The four shape parameters can capture any features or variation in a dataset.Figure 1
EEDL probability density function: (a) alpha = 0.8,q = phi = gamma = 2, (b) alpha = 3, q = phi = gamma = 2, (c) q = phi = gamma = 2, and (d) alpha = 8, u = phi = gamma = 2.
(a)
(b)
(c)
(d)Figure 2
Histogram of EEDL distribution with various parameter values. (a)α = 1.5, β = 2.5, γ = 1.5, ϕ = 3.5, u = 0.5, and q = 1, (b) β = 4.5, γ = 1.5, ϕ = 2.5, u = 0.9, and q = 2.6, (c) β = 4.5, γ = 1.5, ϕ = 2.5, and u = 0.9, (d) β = 0.5, γ = 0.5, ϕ = 0.8, and u = 1.6, (e) ϕ = 2.5, u = 3.5, and q = 1.6, (f) γ = 1.5, ϕ = 2.5, and u = 4.2, (g) α = 5.5 and γ = 1.5, and (h) u = 10.2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)A distribution can also be characterized by its asymptotes, stochastic ordering, stress, and strength properties.


## 3.7. Asymptotes of EEDL Distribution

### 3.7.1. Vertical Asymptotes of EEDL Distribution

Theorem 2.
IffXx and hXx are the pdf and hazard rate of EEDL distribution, then line x=−ϕ1/u, if it exist, is a vertical asymptote of the graph of the functions fXx and hXx if the following statements hold:(30)limx⟶−ϕ1/ufXx=∞.
The horizontal asymptote forhXx is given by(31)limx⟶−ϕ1/uhXx=∞.Proof.
The denominator of the pdf of EEDL distribution in (22) is equated to zero to have(32)xu+11+ϕx−uq+1=0,and to solve for x, we have(33)1+ϕx−uq+1=0,ϕx−u=−1,x=−ϕ1/u.
Also, the denominator of the hazard rate of EEDL distribution in (28) is equated to zero to have(34)γ1−1−e−βwα=0,1−e−βwα=0,e−βw=1,−βw=log1,w=0,but(35)w=1−1+ϕx−u−q−1/γ−1,1−1+ϕx−u−q−1/γ−1=0,1+ϕx−u=0,ϕx−u=−1,x=−ϕ1/u.
The proof is complete. Note that ifu is an even real number, then x is a complex number. Also, if u is an odd real number, then x will be negative because ϕ>0. Take note that the limit is one-sided, since x≥0. This implies that the vertical asymptote of EEDL distribution does not exist for all real values of x≥0.


### 3.7.2. Horizontal Asymptotes of EEDL Distribution

LetfXx and hXx be the pdf and hazard functions of EEDL distribution.The horizontal asymptotes are horizontal lines that the function approaches asx⟶∞.The horizontal asymptote forfXx is given by(36)limx⟶∞fXx=0.The horizontal asymptote forhXx is given by(37)limx⟶∞hXx=0.Take note that the limit is one-sided, sincex≥0.


## 3.7.1. Vertical Asymptotes of EEDL Distribution

Theorem 2.
IffXx and hXx are the pdf and hazard rate of EEDL distribution, then line x=−ϕ1/u, if it exist, is a vertical asymptote of the graph of the functions fXx and hXx if the following statements hold:(30)limx⟶−ϕ1/ufXx=∞.
The horizontal asymptote forhXx is given by(31)limx⟶−ϕ1/uhXx=∞.Proof.
The denominator of the pdf of EEDL distribution in (22) is equated to zero to have(32)xu+11+ϕx−uq+1=0,and to solve for x, we have(33)1+ϕx−uq+1=0,ϕx−u=−1,x=−ϕ1/u.
Also, the denominator of the hazard rate of EEDL distribution in (28) is equated to zero to have(34)γ1−1−e−βwα=0,1−e−βwα=0,e−βw=1,−βw=log1,w=0,but(35)w=1−1+ϕx−u−q−1/γ−1,1−1+ϕx−u−q−1/γ−1=0,1+ϕx−u=0,ϕx−u=−1,x=−ϕ1/u.
The proof is complete. Note that ifu is an even real number, then x is a complex number. Also, if u is an odd real number, then x will be negative because ϕ>0. Take note that the limit is one-sided, since x≥0. This implies that the vertical asymptote of EEDL distribution does not exist for all real values of x≥0.


## 3.7.2. Horizontal Asymptotes of EEDL Distribution

LetfXx and hXx be the pdf and hazard functions of EEDL distribution.The horizontal asymptotes are horizontal lines that the function approaches asx⟶∞.The horizontal asymptote forfXx is given by(36)limx⟶∞fXx=0.The horizontal asymptote forhXx is given by(37)limx⟶∞hXx=0.Take note that the limit is one-sided, sincex≥0.


## 3.8. Stochastic Ordering of EEDL Distribution

### 3.8.1. General Order Statistics of EEDL Distribution

Theorem 3.
The pdf of the general order statistics of EEDL distributionfXjx exists and it is given by(38)fXjx=n!αβquϕγj−1!n−j!w+1γ+1e−βw1−e−βwαj−11−1−e−βwαn−jxu+11+ϕx−uq+1.Proof.
LetX1,X2,…,Xn denote the order statistics of a random sample that follows EEDL distribution, from a continuous population with cdf FXx and pdf fXx. Then, the pdf Xj is(39)fXjx=n!j−1!n−j!fXxFXxj−11−FXxn−j.
Substitute (22) and (23) into (39) to have (40):(40)fXjx=n!j−1!n−j!αβquϕγw+1γ+1e−βw1−e−βwα−1xu+11+ϕx−uq+11−e−βwαj−11−1−e−βwαn−j,(41)fXjx=n!αβquϕγj−1!n−j!w+1γ+1e−βw1−e−βwαj−11−1−e−βwαn−jxu+11+ϕx−uq+1.
Thus, equation (41) completes the proof.
LetX and Y be two random variables that follow the EEDL distribution. X is less than Y if PX>x≤PY>y,∀x∈−∞,∞, where P (.) denotes the probability of an event.Theorem 4.
LetX1 and X2 be two random variables that follow the EEDL distribution. If X1≤stX2 and EX1=EX2, then X1=dX2 (X1 is equal to X2 in distribution).Proof.
From the pdf of the general order statistic of the EEDL distribution given in (41), set j  = 1 to arrive at the 1st order statistic given by(42)fX1x=nαβquϕγw+1γ+1e−βw1−e−βwα−11−1−e−βwαn−1xu+11+ϕx−uq+1.
Also, from (41), set j  = 2 to arrive at the 2nd order statistic given by(43)fX2x=nn−1αβquϕγw+1γ+1e−βw1−e−βw2α−11−1−e−βwαn−2xu+11+ϕx−uq+1.
ForX1≤X2, we can show that EX1=EX2:(44)e−βw1−e−βwα−11−1−e−βwαn−1≤n−1e−βw1−e−βw2α−11−1−e−βwαn−2.
Using series expansion, we have the inequality as(45)∑i,k=0−1i+kn−1iαi+α−1ke−βwk+1≤∑i,k=0−1i+kn−2iαi+2α−1ke−βwk+1.
Inequality (45) can be reduced to(46)∑i,k=0n−1iαi+α−1k≤∑i,k=0n−2iαi+2α−1k.
Take the expectation of both sides to have(47)E∑i,k=0n−1iαi+α−1k=E∑i,k=0n−2iαi+2α−1k.
Note that the expectation of a constant is a constant.
To make equality prevail, we test wheni=k=0 to arrive at(48)n−1!α−1!n−1!α−1!=n−2!2α−1!n−2!2α−1!,1=1.
Thus, the proof is complete. Hence,X1 and X2 are random samples from EEDL distribution.


## 3.8.1. General Order Statistics of EEDL Distribution

Theorem 3.
The pdf of the general order statistics of EEDL distributionfXjx exists and it is given by(38)fXjx=n!αβquϕγj−1!n−j!w+1γ+1e−βw1−e−βwαj−11−1−e−βwαn−jxu+11+ϕx−uq+1.Proof.
LetX1,X2,…,Xn denote the order statistics of a random sample that follows EEDL distribution, from a continuous population with cdf FXx and pdf fXx. Then, the pdf Xj is(39)fXjx=n!j−1!n−j!fXxFXxj−11−FXxn−j.
Substitute (22) and (23) into (39) to have (40):(40)fXjx=n!j−1!n−j!αβquϕγw+1γ+1e−βw1−e−βwα−1xu+11+ϕx−uq+11−e−βwαj−11−1−e−βwαn−j,(41)fXjx=n!αβquϕγj−1!n−j!w+1γ+1e−βw1−e−βwαj−11−1−e−βwαn−jxu+11+ϕx−uq+1.
Thus, equation (41) completes the proof.
LetX and Y be two random variables that follow the EEDL distribution. X is less than Y if PX>x≤PY>y,∀x∈−∞,∞, where P (.) denotes the probability of an event.Theorem 4.
LetX1 and X2 be two random variables that follow the EEDL distribution. If X1≤stX2 and EX1=EX2, then X1=dX2 (X1 is equal to X2 in distribution).Proof.
From the pdf of the general order statistic of the EEDL distribution given in (41), set j  = 1 to arrive at the 1st order statistic given by(42)fX1x=nαβquϕγw+1γ+1e−βw1−e−βwα−11−1−e−βwαn−1xu+11+ϕx−uq+1.
Also, from (41), set j  = 2 to arrive at the 2nd order statistic given by(43)fX2x=nn−1αβquϕγw+1γ+1e−βw1−e−βw2α−11−1−e−βwαn−2xu+11+ϕx−uq+1.
ForX1≤X2, we can show that EX1=EX2:(44)e−βw1−e−βwα−11−1−e−βwαn−1≤n−1e−βw1−e−βw2α−11−1−e−βwαn−2.
Using series expansion, we have the inequality as(45)∑i,k=0−1i+kn−1iαi+α−1ke−βwk+1≤∑i,k=0−1i+kn−2iαi+2α−1ke−βwk+1.
Inequality (45) can be reduced to(46)∑i,k=0n−1iαi+α−1k≤∑i,k=0n−2iαi+2α−1k.
Take the expectation of both sides to have(47)E∑i,k=0n−1iαi+α−1k=E∑i,k=0n−2iαi+2α−1k.
Note that the expectation of a constant is a constant.
To make equality prevail, we test wheni=k=0 to arrive at(48)n−1!α−1!n−1!α−1!=n−2!2α−1!n−2!2α−1!,1=1.
Thus, the proof is complete. Hence,X1 and X2 are random samples from EEDL distribution.


## 3.9. Stress-Strength Reliability Analysis of EEDL Distribution

Theorem 5.
LetX and Z be two independent random variables that follow the EEDL distribution with pdfs fXx and fZz, respectively. If Z represents the stress and X represents the strength, then the stress-strength reliability of EEDL distribution with X∼EEDLα1,β1,ϕ,u,q,r and Z∼EEDLα2,β2,ϕ,u,q,r does not depend on β1 or β2.Proof.
The probability thatX>Z is the reliability of the stress-strength of the EEDL distribution, and it is given by(49)R=PZ<X=∫0∞∫0xfXxfZzdzdx,(50)R=PZ<X=∫0∞fXxFZxdx,where(51)FZz=1−e−βwzα2,(52)fXx=α1β1e−β1wx1−e−β1wxα1−11+wxγ+1.
Put equations (51) and (52) in (50) to have(53)R=∫0∞α1β1e−β1wx1−e−β1wxα1−11+wxγ+11−e−βwxα2dx.
By using linear expansion, we have(54)R=α1β1∑i,j,k∞−1i+jα2iα1−1jγ+1k∫0∞wxke−β1+β2i+β1jdx.
The integral in (54) using gamma function becomes(55)R=α1β1∑i,j,k∞−1i+jα2iα1−1jγ+1kΓk+1−β1+β2i+β1jk+1,where i≤α2, j≤α1−1, and k≤γ+1. Equation (55) is the stress-strength reliability function of the EEDL distribution with X∼EEDLα1,β1,ϕ,u,q,r and X∼EEDLα2,β2,ϕ,u,q,r.
However, ifi=j=k=0, equation (55) is reduced to(56)R=α1.
Thus, equation (56) completes the proof. Hence, the stress-strength reliability, R, does not depend on β.
The probability density, cumulative distribution, quantile, survival, hazard, cumulative hazard functions, asymptotes, stochastic ordering, and stress-strength analysis are some of the different ways of characterizing a random variable.


## 3.10. Related Distributions

Most generalized distributions have relationship with their base distributions by varying one or more of its parameters’ value. EEDL is also related with its parent distribution.


### 3.10.1. Transformations

(1) Exponentiated Exponential DistributionLemma 4.
IfX∼EEDLα,β,γ,u,q,ϕ, then the random variable Y=1−1+ϕX−u−q1/γ−1 has an exponentiated exponential distribution with parameters α and β.Proof.
By using the transformation method, the result is shown as follows:(57)fy=αβe−βy1−e−βyα−1.
(2) Ratio of Exponential and Lomax DistributionsLemma 5.
IfX∼EEDL1,β,γ,u,q,1, then the pdf of random variable Z=1−1+ϕX−u−q1/γ−1 is the ratio of two pdfs of random variables that have an exponential distribution with parameter β and Lomax distribution with parameter γ, respectively.Proof.
By using the transformation method, the result is shown as follows:(58)fz=βe−βzγ1+z−γ.


## 3.10.1. Transformations

(1) Exponentiated Exponential DistributionLemma 4.
IfX∼EEDLα,β,γ,u,q,ϕ, then the random variable Y=1−1+ϕX−u−q1/γ−1 has an exponentiated exponential distribution with parameters α and β.Proof.
By using the transformation method, the result is shown as follows:(57)fy=αβe−βy1−e−βyα−1.
(2) Ratio of Exponential and Lomax DistributionsLemma 5.
IfX∼EEDL1,β,γ,u,q,1, then the pdf of random variable Z=1−1+ϕX−u−q1/γ−1 is the ratio of two pdfs of random variables that have an exponential distribution with parameter β and Lomax distribution with parameter γ, respectively.Proof.
By using the transformation method, the result is shown as follows:(58)fz=βe−βzγ1+z−γ.


## 3.11. Moments of EEDL Distribution

The moments of a probability distribution are a very important property in describing the distribution. The mean, variance, standard deviation, measure of skewness and kurtosis, and other parametric measures that describe the distribution can be derived from it.Theorem 6.
LetX follow an EEDL distribution, and the rth moment of X can be expressed in terms of gamma function of W with parameters k+1 and βl+1, and it is given by(59)EXr=αβΓαAϕr/uΓr/u∫0∞wke−βw1+ldw,where(60)A=∑j=0∞∑i=0∞∑k=0γ+1+jr/q∑l=0α−1−1r/u−jΓγ+jr/q+1Γj/q+iΓr/u+jΓj+1Γi+1Γj/qΓα−lΓl+1Γγ+2−k+jr/q.Proof.
Recall thatw=1−1+ϕx−u−q−1/γ−1 and x=1/ϕ1−w+1−γ−1/q−1−1/u so that the pdf of EEDL distribution is reduced to(61)fXx=αβe−βw1−e−βwα−11+w−γ+1,(62)fXx=αβe−βw1−e−βwα−11+wγ+1.
See the complete proof of equation (62) in Appendix I.(63)EXr=∫0∞xrαβe−βw1−e−βwα−11+wγ+1dx,(64)δwδx=quϕx−u+11+ϕx−u−q+1γw+1−γ+1,dx=dwγw+1−γ+1dxquϕx−u+11+ϕx−u−q+1.
Substitute (64) into (63) to have(65)EXr=αβ∫0∞ϕr/u1−1+w−γ−1/q−1−r/ue−βw1−eβwα−11+w−γ+1dw,(66)EXr=αβ∫0∞∑j=0∞∑i=0∞−1r/u−jΓj/q+iΓr/u+jΓj+1Γr/uΓi+1Γj/qϕr/u1+wγ+11+wjγ/qe−βw1−eβwα−1dw,EXr=αβ∫0∞∑j=0∞∑i=0∞∑k=0∞−1r/u−jΓγ+jr/q+1Γj/q+iΓr/u+jΓj+1Γr/uΓi+1Γj/qΓk+1Γγ+2−k+jr/qϕr/uwke−βw1−eβwα−1dw,EXr=αβΓαAϕr/uΓr/u∫0∞wke−βw1+ldw.
Equation (66) completes the proof, where(67)A=∑j=0∞∑i=0∞∑k=0∞∑l=0∞−1r/u−jΓγ+jr/q+1Γj/q+iΓr/u+jΓj+1Γi+1Γj/qΓα−lΓl+1Γγ+2−k+jr/q.
Recall gamma function:(68)Γaba=∫0∞xa−1e−bxdx.
Thus, therth moment of the EEDL distribution is given by(69)EXr=αβΓαAϕr/uΓk+1Γr/uβl+1k+1.
If the subscriptsi=j=k=l=0, then equation (69) reduces to(70)EXr=αϕr/uΓγ+1.
Ifr=1, we have the mean of the EEDL distribution, which does not depend on variables β and q, and it is given by(71)EX=αϕ1/uΓγ+1,and the variance is(72)VX=αϕ2/uΓγ+11−αΓγ+1.


## 3.12. Shannon Entropy of EEDL Distribution

The Shannon entropy of a random variable is a measure of variation of uncertainty. It is defined by [31] as E−logfx for a random variable X with pdf fx. Let X be a random variable that follows EEDL distribution with pdf fXx as given in (62):(73)fXx=αβe−βw1−e−βwα−11+wγ+1.The Shannon entropy of EEDL distribution is(74)E−logαβe−βw1−e−βwα−11+wγ+1,ηXx=−logα−logβ+βEw−α−1Elog1−e−βw−γ+1Elogw+1,where w is a function of x. It can be written as wx


## 3.13. Maximum Likelihood Estimation of EEDL Distribution Parameters

Recall from (20) the pdf of EEDL distribution, and we derived the log likelihood function as(75)l=logL=nlogΩ+nlogα+nlogβ−β∑i=1nwi−ul∑i=1nlogxi+α−1∑i=1nlog1−e−βw.Remember thatw=1−1+ϕx−u−q−1/γ−1 and(76)Ω=∑i,j,k,l=0∞ϕl−1i+l−jΓi+1Γn+lΓj/γ+kΓr−i+2Γi−j+1Γk+1Γj+1Γj/γΓl+1Γn.We can differentiate (75) easily with respect to α and β.Differentiate (75) partially with respect to α, and equate the result to zero and solve for α:(77)δlδα=nα+∑i=1nlog1−e−βwi,0=nα+∑i=1nlog1−e−βwi,−∑i=1nlog1−e−βwi=nα,α^=−n∑i=1nlog1−e−βwi.Differentiate (75) partially with respect to β, equate the result to zero and solve for β:(78)δlδβ=nβ−∑i=1nwi+βα−1∑i=1ne−βxi1−e−βxi,0=nβ−∑i=1nwi+βα−1∑i=1ne−βxi1−e−βxi,∑i=1nwi=nβ+βα−1∑i=1ne−βxi1−e−βxi.Divide through byn(79)w¯=1β+α−1βn∑i=1ne−βxi1−e−βxi.The estimate ofβ is not in a closed form. From (58), we also differentiate partially with respect to other parameters, equate their results to zero, and solve for each of them. Their solutions are not in closed form as well, so we resolved by using R package “MaxLik.”


## 3.14. Simulation

A simulation study is carried out to investigate the performance of the estimators. The standard error of estimate (SE), average absolute bias (AAB), and root mean square error (RMSE) of the maximum likelihood estimators of the parameters of the EEDL were examined. The simulation study was repeated forN = 1000 times each with sample sizes n  = 20, 50, 100, 250, and 500 and parameter values: α = 1.5, β = 0.1, γ = 2.0, ϕ = 0.3, q = 0.01, and u = 0.9.Table1 presents the MLE estimates, standard error of estimate, and AAB and RMSE values of the parameters α, β, γ, ϕ, q, and u for different sample sizes. The result shows that as the sample size approaches infinity, the AAB and RMSE decrease asymptotically to zero, proving their consistency. It is consistent in the sense that it converges to the true parameter value as the number of observations becomes larger and the error reduces to zero. See Figure 3 for pictorial view.Table 1
Simulation results of MLE.
Parameter
n
Estimate
SE
AAB
RMSE
α = 1.5
20
1.5020
0.0000549
0.001610
0.00720
50
1.5013
0.0000204
0.000660
0.00466
100
1.5017
0.0000132
0.000360
0.00357
250
1.5020
0.0000032
0.000120
0.00184
500
1.5001
0.0000001
0.000010
0.00010
β = 0.1
20
0.0367
1.0443120
0.222721
0.99604
50
0.1447
0.3889741
0.087315
0.61741
100
0.1437
0.1935787
0.043777
0.43777
250
0.1596
0.0765768
0.017467
0.27617
500
0.4500
0.0002455
0.000700
0.01565
γ = 2.0
20
2.0110
0.0142591
0.026025
0.11639
50
2.0111
0.0053693
0.010259
0.07254
100
2.0013
0.0026316
0.005104
0.05104
250
2.0010
0.0010761
0.002071
0.03274
500
2.0001
0.0005010
0.001000
0.02236
ϕ = 0.3
20
0.3015
0.3286991
0.124953
0.55881
50
0.3012
0.1237652
0.049252
0.34827
100
0.3011
0.0626044
0.024895
0.24895
250
0.2996
0.0246000
0.009900
0.15653
500
0.2999
0.0001804
0.000600
0.01342
q = 0.01
20
0.0002
0.0624754
0.054476
0.24362
50
0.0154
0.0241751
0.021768
0.15392
100
0.0130
0.0119727
0.010887
0.10887
250
0.0116
0.0047506
0.004350
0.06879
500
0.0101
0.0000000
0.000000
0.00000
u = 0.9
20
0.9014
0.0000152
0.000849
0.00379
50
0.9008
0.0000052
0.000319
0.00226
100
0.9005
0.0000021
0.000144
0.00144
250
0.9002
0.0000012
0.000070
0.00111
500
0.9001
0.0000010
0.000001
0.00000Figure 3
(a)α = 1.5. (b) β = 0.1. (c) γ = 2.0. (d) ϕ = 0.3. (e) q = 0.01. (f) u = 0.9.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.15. Applications

The EEDL distribution is fitted to two real datasets retrieved from [13]. The first application is a data on failure times for 36 appliances subjected to an automatic life test, while the second data is on failure time data on 100 cm yarn subjected to 2 : 3% strain level. EEDL distribution is compared with that of exponentiated generalized exponential Dagum distribution (EGEDD), the exponentiated Kumaraswamy Dagum (EKD) distribution and the Mc-Dagum (McD) distribution using Log-likelihood, Akaike information criterion (AIC), and Kolmogorov–Smirnov (K-S) statistic criteria.


### 3.15.1. Application 1: Appliances Data

The appliances data in Table2 was obtained from [13, 32]. The dataset consists of failure times for 36 appliances subjected to an automatic life test. The dataset is depicted in Figure 4 and shows that there is a gap in the histogram with positive skewness (2.279) and highly leptokurtic (9.669). Table 3 displays the maximum likelihood estimates of the parameters with their corresponding standard errors in brackets. Table 3 shows all the parameters of the EEDL distribution and other distributions.Table 2
Failure times for 36 appliances subjected to an automatic life test.
11
35
49
170
1990
2831
2223
3034
2327
3059
2400
3112
329
2451
3214
381
2471
3478
708
2551
3504
958
2565
4329
1062
2568
6367
1167
2694
6976
1594
2702
7846
1925
2761
13403Figure 4
Failure time on appliances.Table 3
MLE of parameters and standard errors for appliances data.
Model
Parameter estimates
EEDL
α
⌢
β
⌢
γ
⌢
ϕ
⌢
q
⌢
u
⌢
0.491(0.0822)
0.011(0.0001)
0.260(0.0189)
1.055(0.0729)
1.041(0.0666)
0.163(0.0163)
EGEDD
α
⌢
λ
⌢
β
⌢
θ
⌢
c
⌢
d
⌢
0.001(0.0001)
27.198(0.001)
4.560(0.847)
2.838(0.123)
20.866(0.010)
0.070(0.003)
EKD
α
⌢
λ
⌢
δ
⌢
ϕ
⌢
θ
⌢
5.562(1.517)
12.683(2.158)
3.716(0.755)
0.128(0.029)
11.609(3.922)
McD
λ
⌢
δ
⌢
β
⌢
a
⌢
b
⌢
c
⌢
1.427(0.092)
3.455(0.212)
1.275(6.875)
10.505(56.906)
0.064(0.012)
500.556(6.976)Table4 clearly shows that the EEDL distribution provides a better fit to the appliances data than the other models. Its log-likelihood, AIC and approach zero faster than other distributions. It also has the smallest K-S statistic compared with other distribution in this research. Thus, the EEDL is a better fit to the appliances data.Table 4
Log-likelihood, information, and goodness-of-fit criteria for appliances data.
Model
LogLik
AIC
K-S stat
KSp value
EEDL
−320.756
653.512
0.222
0.340
EGEDD
−328.870
669.740
0.253
0.124
EKD
−341.650
693.295
0.269
0.178
McD
−356.480
724.955
0.347
0.128


### 3.15.2. Application 2: Yarn Data

Table5 represents the data on time to failure of a 100 cm polyster/viscose yarn subjected to 2 : 3% strain level in textile experiment in order to assess the tensile fatigue characteristics of the yarn. The dataset can be found in [13, 33, 34]. The skewness and kurtosis of the data are 1.336119 and 5.802452, respectively. The data is positively skewed and very peaked, as depicted in Figure 5.Table 5
Failure time data on 100 cm yarn subjected to 2 : 3% strain level.
86
146
251
653
98
249
400
292
131
169
175
176
76
264
15
364
195
262
88
264
157
220
42
321
180
198
38
20
61
121
282
224
149
180
325
250
196
90
229
166
38
337
65
151
341
40
40
135
597
246
211
180
93
315
353
571
124
279
81
286
497
182
423
185
229
400
338
290
398
71
246
185
188
568
55
55
61
244
20
289
393
396
203
829
239
236
286
194
277
143
198
264
105
203
124
137
135
350
193
188Figure 5
Failure time on yarn.The maximum likelihood estimates of the parameters of the fitted models with their corresponding standard errors in brackets are given in Table6. All the parameters of the EEDL are significant at the 5% significance level. The EEDL provides a better fit to the yarn data than the EGEDD, EKD, and McD distributions, as shown in Table 7.Table 6
Maximum likelihood estimates of parameters and standard errors for yarn data.
Model
Parameter estimates
EEDL
α
⌢
β
⌢
γ
⌢
ϕ
⌢
q
⌢
u
⌢
1.0921(0.2133)
0.0004(0.0004)
1.5196(0.1457)
3.4906(0.0005)
0.7205(0.0924)
2.3297(0.0228)
EGEDD
α
⌢
λ
⌢
β
⌢
θ
⌢
c
⌢
d
⌢
0.026(0.007)
75.310(0.007)
0.017(0.005)
3.513(0.631)
45.692(0.036)
0.090(0.011)
EKD
α
⌢
λ
⌢
δ
⌢
ϕ
⌢
θ
⌢
46.109(1.295)
39.413(5.006)
5.188(0.961)
0.203(0.040)
31.169(11.023)
McD
λ
⌢
δ
⌢
β
⌢
a
⌢
b
⌢
c
⌢
0.027(0.0848)
0.600(0.09647)
98.780(0.00002)
0.333(0.1504)
25.042(0.0004507)
46.276(0.00004654)Table 7
Log-likelihood, information, and goodness-of-fit criteria for yarn data.
Model
LogLik
AIC
K-S stat
KSp value
EEDL
−625.459
199.719
0.110
0.581
EGEDD
−628.170
1268.336
0.249
0.124
EKD
−653.960
1317.913
0.985
0.178
McD
−628.200
1268.399
0.285
0.128Table7 shows that EEDL log-likelihood and its AIC approaches zero than that of others and has the smallest K-S statistic compared to the other models. Thus, the EEDL is a better fit.The kernel density graphs of the data and the EEDL distribution are super-imposed on the histograms on Figures4 and 5. This shows that the newly proposed EEDL distribution fits the two data.Figures6 to 9 show that EEDL distribution fit the two data well. The EEDL distribution fits the yarn data more than the appliances data. The QQ plots in Figures 10 and 11 shows that most of the data points, especially the ones at the middle, fall on the theoretical line, but the few ones outside the unit circle (−1 and 1) of the theoretical quantiles fall outside the line. This means that if the edges of the data are trimmed, the EEDL distribution would be a perfect fit for the data. However, Tables 4 and 7 shows a confirmatory test that the EEDL distribution is a good fit to both the appliances and yarn data via the K-S statistic and p value.Figure 6
TTT plot of appliances.Figure 7
TTT plot of yarn.Figure 8
PP plot of appliances.Figure 9
PP plot of yarn.Figure 10
QQ plot of appliances.Figure 11
QQ plot for EEDL distribution on yarn data.


## 3.15.1. Application 1: Appliances Data

The appliances data in Table2 was obtained from [13, 32]. The dataset consists of failure times for 36 appliances subjected to an automatic life test. The dataset is depicted in Figure 4 and shows that there is a gap in the histogram with positive skewness (2.279) and highly leptokurtic (9.669). Table 3 displays the maximum likelihood estimates of the parameters with their corresponding standard errors in brackets. Table 3 shows all the parameters of the EEDL distribution and other distributions.Table 2
Failure times for 36 appliances subjected to an automatic life test.
11
35
49
170
1990
2831
2223
3034
2327
3059
2400
3112
329
2451
3214
381
2471
3478
708
2551
3504
958
2565
4329
1062
2568
6367
1167
2694
6976
1594
2702
7846
1925
2761
13403Figure 4
Failure time on appliances.Table 3
MLE of parameters and standard errors for appliances data.
Model
Parameter estimates
EEDL
α
⌢
β
⌢
γ
⌢
ϕ
⌢
q
⌢
u
⌢
0.491(0.0822)
0.011(0.0001)
0.260(0.0189)
1.055(0.0729)
1.041(0.0666)
0.163(0.0163)
EGEDD
α
⌢
λ
⌢
β
⌢
θ
⌢
c
⌢
d
⌢
0.001(0.0001)
27.198(0.001)
4.560(0.847)
2.838(0.123)
20.866(0.010)
0.070(0.003)
EKD
α
⌢
λ
⌢
δ
⌢
ϕ
⌢
θ
⌢
5.562(1.517)
12.683(2.158)
3.716(0.755)
0.128(0.029)
11.609(3.922)
McD
λ
⌢
δ
⌢
β
⌢
a
⌢
b
⌢
c
⌢
1.427(0.092)
3.455(0.212)
1.275(6.875)
10.505(56.906)
0.064(0.012)
500.556(6.976)Table4 clearly shows that the EEDL distribution provides a better fit to the appliances data than the other models. Its log-likelihood, AIC and approach zero faster than other distributions. It also has the smallest K-S statistic compared with other distribution in this research. Thus, the EEDL is a better fit to the appliances data.Table 4
Log-likelihood, information, and goodness-of-fit criteria for appliances data.
Model
LogLik
AIC
K-S stat
KSp value
EEDL
−320.756
653.512
0.222
0.340
EGEDD
−328.870
669.740
0.253
0.124
EKD
−341.650
693.295
0.269
0.178
McD
−356.480
724.955
0.347
0.128


## 3.15.2. Application 2: Yarn Data

Table5 represents the data on time to failure of a 100 cm polyster/viscose yarn subjected to 2 : 3% strain level in textile experiment in order to assess the tensile fatigue characteristics of the yarn. The dataset can be found in [13, 33, 34]. The skewness and kurtosis of the data are 1.336119 and 5.802452, respectively. The data is positively skewed and very peaked, as depicted in Figure 5.Table 5
Failure time data on 100 cm yarn subjected to 2 : 3% strain level.
86
146
251
653
98
249
400
292
131
169
175
176
76
264
15
364
195
262
88
264
157
220
42
321
180
198
38
20
61
121
282
224
149
180
325
250
196
90
229
166
38
337
65
151
341
40
40
135
597
246
211
180
93
315
353
571
124
279
81
286
497
182
423
185
229
400
338
290
398
71
246
185
188
568
55
55
61
244
20
289
393
396
203
829
239
236
286
194
277
143
198
264
105
203
124
137
135
350
193
188Figure 5
Failure time on yarn.The maximum likelihood estimates of the parameters of the fitted models with their corresponding standard errors in brackets are given in Table6. All the parameters of the EEDL are significant at the 5% significance level. The EEDL provides a better fit to the yarn data than the EGEDD, EKD, and McD distributions, as shown in Table 7.Table 6
Maximum likelihood estimates of parameters and standard errors for yarn data.
Model
Parameter estimates
EEDL
α
⌢
β
⌢
γ
⌢
ϕ
⌢
q
⌢
u
⌢
1.0921(0.2133)
0.0004(0.0004)
1.5196(0.1457)
3.4906(0.0005)
0.7205(0.0924)
2.3297(0.0228)
EGEDD
α
⌢
λ
⌢
β
⌢
θ
⌢
c
⌢
d
⌢
0.026(0.007)
75.310(0.007)
0.017(0.005)
3.513(0.631)
45.692(0.036)
0.090(0.011)
EKD
α
⌢
λ
⌢
δ
⌢
ϕ
⌢
θ
⌢
46.109(1.295)
39.413(5.006)
5.188(0.961)
0.203(0.040)
31.169(11.023)
McD
λ
⌢
δ
⌢
β
⌢
a
⌢
b
⌢
c
⌢
0.027(0.0848)
0.600(0.09647)
98.780(0.00002)
0.333(0.1504)
25.042(0.0004507)
46.276(0.00004654)Table 7
Log-likelihood, information, and goodness-of-fit criteria for yarn data.
Model
LogLik
AIC
K-S stat
KSp value
EEDL
−625.459
199.719
0.110
0.581
EGEDD
−628.170
1268.336
0.249
0.124
EKD
−653.960
1317.913
0.985
0.178
McD
−628.200
1268.399
0.285
0.128Table7 shows that EEDL log-likelihood and its AIC approaches zero than that of others and has the smallest K-S statistic compared to the other models. Thus, the EEDL is a better fit.The kernel density graphs of the data and the EEDL distribution are super-imposed on the histograms on Figures4 and 5. This shows that the newly proposed EEDL distribution fits the two data.Figures6 to 9 show that EEDL distribution fit the two data well. The EEDL distribution fits the yarn data more than the appliances data. The QQ plots in Figures 10 and 11 shows that most of the data points, especially the ones at the middle, fall on the theoretical line, but the few ones outside the unit circle (−1 and 1) of the theoretical quantiles fall outside the line. This means that if the edges of the data are trimmed, the EEDL distribution would be a perfect fit for the data. However, Tables 4 and 7 shows a confirmatory test that the EEDL distribution is a good fit to both the appliances and yarn data via the K-S statistic and p value.Figure 6
TTT plot of appliances.Figure 7
TTT plot of yarn.Figure 8
PP plot of appliances.Figure 9
PP plot of yarn.Figure 10
QQ plot of appliances.Figure 11
QQ plot for EEDL distribution on yarn data.


## 4. Concluding Remarks

This research proposed a new univariate continuous probability distribution called exponentiated-exponential-Dagum {Lomax} distribution, EEDL, which is a member of theT-Dagum{lomax} family and presented results on its statistical properties, such as the cumulative distribution function, density function, the quantile function, survival function, hazard function, cumulative hazard function, asymptotes, stochastic ordering, stress-strength analysis, moments, and Shannon entropy. The maximum likelihood estimation of the parameters of the model was derived. The newly proposed EEDL distribution was applied to two datasets that are positively skewed and very peaked and the results of its performance were compared favourably with EGEDD, EKD, and McD. Further research can be carried out on submodels of EEDL.See Table8 for seven of its submodels with fewer parameters. These submodels will be investigated in subsequent work. Exponential-DagumLomax distribution is one of its submodels. The more significant a parameter of a distribution is, the more likely its fit is better to some datasets (see [8, 10, 13] and [14]).Table 8
Submodels of EEDL distribution.
SM
Density function
Np
1
f
X
x
=
β
q
u
ϕ
/
γ
exp
−
β
1
−
1
+
ϕ
x
−
u
−
q
−
1
/
γ
−
1
/
x
u
+
1
1
+
ϕ
x
−
u
q
+
1
1
−
1
+
ϕ
x
−
u
−
q
1
/
γ
+
1
,
α
=
1
5
2
f
X
x
=
β
u
ϕ
/
γ
exp
−
β
1
−
1
+
ϕ
x
−
u
−
q
−
1
/
γ
−
1
/
x
u
+
1
1
+
ϕ
x
−
u
2
1
−
1
+
ϕ
x
−
u
−
1
1
/
γ
+
1
,
α
=
q
=
1
4
3
f
X
x
=
β
q
u
ϕ
exp
−
β
1
−
1
+
ϕ
x
−
u
−
q
−
1
−
1
/
x
u
+
1
1
+
ϕ
x
−
u
q
+
1
1
−
1
+
ϕ
x
−
u
−
q
2
,
α
=
γ
=
1
4
4
f
X
x
=
q
u
ϕ
exp
−
1
−
1
+
ϕ
x
−
u
−
q
−
1
−
1
/
x
u
+
1
1
+
ϕ
x
−
u
q
+
1
1
−
1
+
ϕ
x
−
u
−
q
2
,
α
=
γ
=
β
=
1
3
5
f
X
x
=
u
ϕ
/
γ
exp
−
1
−
1
+
ϕ
x
−
u
−
1
−
1
/
γ
−
1
/
x
u
+
1
1
+
ϕ
x
−
u
2
1
−
1
+
ϕ
x
−
u
−
1
1
/
γ
+
1
,
α
=
q
=
β
=
1
3
6
f
X
x
=
α
/
γ
exp
−
1
−
1
+
ϕ
x
−
1
−
1
−
1
/
γ
−
1
1
−
exp
−
1
−
1
+
ϕ
x
−
1
−
1
−
1
/
γ
−
1
α
−
1
/
x
2
1
+
x
−
u
2
1
−
1
+
ϕ
x
−
1
−
1
1
/
γ
+
1
,
ϕ
=
β
=
u
=
q
=
1
2
7
f
X
x
=
u
exp
−
1
−
1
+
ϕ
x
−
u
−
1
−
1
/
γ
−
1
/
x
u
+
1
1
+
x
−
u
2
1
−
1
+
x
−
u
−
1
2
,
α
=
γ
=
ϕ
=
β
=
q
=
1
1
SM denotes submodel and Np denotes number of parameters. Model 1 is exponential-Dagum{Lomax} distribution (EDD) with parametersβ,γ,ϕ,q,andu; model 2 is a four-parameter EDD with parameters β,γ,ϕ,andu; model 3 is a four-parameter EDD with parameters β,ϕ,u,andq; model 4 is a three-parameter EDD with parameters ϕ,q,andu; model 5 is a three-parameter EDD with parameters γ,ϕ,andu; model 6 is a two-parameter EEDL distribution with parameters αandγ; model 7 is a one-parameter EEDL distribution with parameter u.



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*Source: 1641207-2020-04-24.xml*