# Truncated Cauchy Power Odd Fréchet-G Family of Distributions: Theory and Applications

Authors: M. Shrahili; I. Elbatal
Journal: Complexity
 (2021)

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## Abstract

The truncated Cauchy power odd Fréchet-G family of distributions is presented in this article. This family’s unique models are launched. Statistical properties of the new family are proposed, such as density function expansion, moments, incomplete moments, mean deviation, Bonferroni and Lorenz curves, and entropy. We investigate the maximum likelihood method for predicting model parameters of the new family. Two real-world datasets are used to show the importance and flexibility of the new family by using the truncated Cauchy power odd Fréchet exponential model as example of the family and compare it with some known models, and this model proves the importance and the flexibility for the new family.


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## Body

## 1. Introduction

Many authors have lately made considerable attempts to create new families to expand well-known distributions and give flexible classes to represent data in a wide range of areas, including medical sciences, environmental sciences, engineering, demography, actuarial science, and economics. Many generic families have been developed and are used to explain a wide range of real-world events. Some examples of these families are beta-G [1], gamma-G [2], upper truncated Weibull distribution [3], and truncated Weibull-G more flexible and more reliable than beta-G distribution [4], truncated inverted Kumaraswamy-G by Bantan et al. [5], type II truncated Fréchet-G [6], type II power TL by Bantan et al. [7], odd generalized N-H by Ahmad et al. [8], Topp–Leone (TL) odd Fréchet-G by Al-Marzouki et al. [9], transmuted odd Fréchet-G by Badr et al. [10], and truncated Burr X-G by Bantan et al. [11], among others.Haq and Elgarhy [12] proposed the odd Fréchet-G (OF, −G) with cumulative function (cdf) as follows, for x>0,(1)Hx;α,ξ=e−G¯x,ξ/Gx,ξα,α>0,and probability density function (pdf) is(2)hx;α,ξ=αgx,ξG¯x,ξα−1Gx,ξα+1e−G¯x,ξ/Gx,ξα.The Cauchy (C) distribution plays an important role and has applications in different fields such as econometrics, engineering, spectroscopy, biological analysis, reliability, queueing theory, and stochastic modeling of decreasing hazard rate life devices. There are many authors who have been displayed various generalization and extension forms of Cauchy distribution in the statistical literature, for examples, Rider [13] presented generalized C distribution, A truncated C distribution by Nadarajah et al. [14], the existence of the moments of the C distribution by Ohakwe and Osu [15], Jacob and Jayakumar[16] studied half-C distribution, Kumaraswamy-half-C distribution by Hamedani and Ghosh [17], Alshawarbeh and Famoye [18] presented properties of beta-C distribution, and the power-C negative-binomial distribution by Zubair et al. [19], among others.Recently, Aldahlan et al. [20] proposed the truncated C power-G (TCP-G) family. The cdf of the TCP-G family is given by(3)Fx;λ=4πarctanHxλ,x∈R,where λ>0. The corresponding pdf is(4)fx;λ=4λhxHxλ−1π1+Hx2λ.The aim of this article is to provide a new, broader, and more flexible family of distributions based on the odd Fréchet-G and truncated C power-G (TCP-G) families. The new proposed family has many wide applications in physics, medicine, engineering, and finance. We construct a new family called TCP odd Fréchet-GTCPOF−G family of distributions by inserting equation (1) into equation (3); the cdf and pdf of the TCPOF−G are(5)Fx;λ,α,ξ=4πarctane−λG¯x,ξ/Gx,ξα,x>0,and(6)fx;λ,α,ξ=4λαgx,ξG¯x,ξα−1πGx,ξα+1e−λG¯x,ξ/Gx,ξα,×1+e−2λG¯x,ξ/Gx,ξαα−1.Henceforward, a random variableX having pdf equation (6) will be defined as X∼TCPOFλ,α,ξ. The survival and hazard rate functions for the TCPOF−G family are(7)F¯x;λ,α,ξ=1−4πarctane−λG¯x,ξ/Gx,ξα,τx;λ,α,ξ=4λαgx,ξG¯x,ξα−1e−λG¯x,ξ/Gx,ξαπGx,ξα+11+e−2λG¯x,ξ/Gx,ξα1−4/πarctane−λG¯x,ξ/Gx,ξα.The quantile function (qf) of TCPOF-G family is given by(8)F−1u=QGu=G−111+−1/λlogtanuπ/41/α.The median is given by(9)M=Q2=Q0.5=G−111+−1/λlogtan0.5π/41/α.The structure of this paper is as follows. Section2 has a useful linear explanation of the TCPOF density as well as several specific models. Section 3 looks at structural properties of the TCPOF-G. In Section 4, we discuss the suggested family’s entropy. The maximum likelihood method is used to estimate the model parameters in Section 5. Section 6 provides applications to real-world datasets to illustrate the proposed family’s flexibility.


## 2. Linear Representation of TCPOF

Ifz<1 and b>0 is a real noninteger, then the following power series hold:(10)1+z−b=∑k=0∞−bkzk.Applying equation (10) in equation (6), we get(11)fTCPOFrx;λ,α,ξ=4λαgx,ξG¯x,ξα−1πGx,ξα+1∑i=0∞−1ie−λ2i+1G¯x,ξ/Gx,ξα.By using the power series for the exponential function, the last term in equation (11) gives(12)e−λ2i+1G¯x,ξ/Gx,ξα=∑j=0∞−1jλj2i+1jj!G¯x,ξαjGx,ξαj.TheTCPOF−G density reduces to(13)fTCPOFrx;λ,α,ξ=4αgx,ξπ∑i,j=0∞−1i+jλj+12i+1jj!G¯x,ξαj+1−1Gx,ξαj+1+1.Using the general binomial expansion, we can write(14)Gx,ξ−αj+1+1=∑k=0∞Γαj+1+k+1k!Γαj+1+11−Gx,ξk,and(15)1−Gx,ξαj+1+k−1=∑m=0∞−1mΓαj+1+km!Γαj+1+k−mGx,ξm.Inserting equations (14) and (15) in equation (13), the TCPOF−G density becomes(16)fTCPOFrx;λ,α,ξ=∑m=0∞ψmπm+1x,where hυx=υgx;ξGx;ξυ−1 denotes the pdf of the exponentiated generalized (Exp-G) distribution with power parameter ν, and(17)ψm=4απ∑i,j,k=0∞−1i+j+mλj+12i+1jj!k!m!m+1Γαj+1+k+1Γαj+1+1×Γαj+1+km!Γαj+1+k−m.


### 2.1. Four Special Models of the TCPOF Family

In this part, we presented three distinct models of the TCPOF family of distributions. When the cdfGx and pdf gx have simple analytic expressions, the pdf equation (8) will be most tractable. Based on the baseline distributions, we propose four submodels of this family: Weibull, exponential, Rayleigh, and Lomax. The cdf and pdf files for these baseline models are provided in Table 1.Table 1
Some new models of the new family.
ModelGx;ξgx;ξG¯x;ξ/Gx;ξWeibull1−e−βxμμβμxμ−1e−βxμe−βxμ/1−e−βxμz<11−e−βxβe−βxe−βx/1−e−βxRayleigh1−e−βx2β2xe−βx2e−βx2/1−e−βx2Lomax1−1+x/b−aa/b1+x/b−a−11+x/b−a/1−1+x/b−a


#### 2.1.1. TCPOF Weibull (TCPOFW) Distribution

The cdf and pdf of TCPOFW distribution are(18)Fx;λ,α,μ,β=4πarctane−λe−βxμ/1−e−βxμα,x>0,fx;λ,α,μ,β=4λαμβμxμ−1e−βxμe−βxμα−1π1−e−βxμα+1e−λe−βxμ/1−e−βxμα,1+e−2λe−βxμ/1−e−βxμα−1.


#### 2.1.2. TCPOF Exponential (TCPOFE) Distribution

The cdf and pdf of the TCPOFE model (forx>0 ) are(19)Fx;λ,α,β=4πarctane−λe−βx/1−e−βxα,x>0,fx;λ,α,β=4λαβe−βxe−βxα−1π1−e−βxα+1e−λe−βx/1−e−βxα,1+e−2λe−βx/1−e−βxα−1.


#### 2.1.3. TCPOF Rayleigh (TCPOFR) Distribution

The cdf and pdf of the TCPOFR model are(20)Fx;λ,α,β=4πarctane−λe−βx2/1−e−βx2α,x>0,fx;λ,α,β=4λαμβ2xe−βx2e−βx2α−1π1−e−βx2α+1e−λe−βx2/1−e−βx2α,1+e2−λe−βx2/1−e−βx2α−1.


#### 2.1.4. TCPOF Lomax (TCPOFL) Distribution

The cdf and pdf of the TCPOFL model are(21)Fx;λ,α,a,b=4πarctane−λ1+x/b−a/1−1+x/b−aα,x>0,fx;λ,α,a,b=4λαa/b1+x/b−a−11+x/b−aα−1π1−1+x/b−aα+1e−λ1+x/b−a/1−1+x/b−aα×1+e−2λ1+x/b−a/1−1+x/b−aα−1.Plots of the TCPOFW, TCPOFE, TCPOFR, and TCPOFL densities are represented in Figure1.Figure 1
TCPOFW, TCPOFE, TCPOFR, and TCPOFL densities for different values of parameters.


## 2.1. Four Special Models of the TCPOF Family

In this part, we presented three distinct models of the TCPOF family of distributions. When the cdfGx and pdf gx have simple analytic expressions, the pdf equation (8) will be most tractable. Based on the baseline distributions, we propose four submodels of this family: Weibull, exponential, Rayleigh, and Lomax. The cdf and pdf files for these baseline models are provided in Table 1.Table 1
Some new models of the new family.
ModelGx;ξgx;ξG¯x;ξ/Gx;ξWeibull1−e−βxμμβμxμ−1e−βxμe−βxμ/1−e−βxμz<11−e−βxβe−βxe−βx/1−e−βxRayleigh1−e−βx2β2xe−βx2e−βx2/1−e−βx2Lomax1−1+x/b−aa/b1+x/b−a−11+x/b−a/1−1+x/b−a


### 2.1.1. TCPOF Weibull (TCPOFW) Distribution

The cdf and pdf of TCPOFW distribution are(18)Fx;λ,α,μ,β=4πarctane−λe−βxμ/1−e−βxμα,x>0,fx;λ,α,μ,β=4λαμβμxμ−1e−βxμe−βxμα−1π1−e−βxμα+1e−λe−βxμ/1−e−βxμα,1+e−2λe−βxμ/1−e−βxμα−1.


### 2.1.2. TCPOF Exponential (TCPOFE) Distribution

The cdf and pdf of the TCPOFE model (forx>0 ) are(19)Fx;λ,α,β=4πarctane−λe−βx/1−e−βxα,x>0,fx;λ,α,β=4λαβe−βxe−βxα−1π1−e−βxα+1e−λe−βx/1−e−βxα,1+e−2λe−βx/1−e−βxα−1.


### 2.1.3. TCPOF Rayleigh (TCPOFR) Distribution

The cdf and pdf of the TCPOFR model are(20)Fx;λ,α,β=4πarctane−λe−βx2/1−e−βx2α,x>0,fx;λ,α,β=4λαμβ2xe−βx2e−βx2α−1π1−e−βx2α+1e−λe−βx2/1−e−βx2α,1+e2−λe−βx2/1−e−βx2α−1.


### 2.1.4. TCPOF Lomax (TCPOFL) Distribution

The cdf and pdf of the TCPOFL model are(21)Fx;λ,α,a,b=4πarctane−λ1+x/b−a/1−1+x/b−aα,x>0,fx;λ,α,a,b=4λαa/b1+x/b−a−11+x/b−aα−1π1−1+x/b−aα+1e−λ1+x/b−a/1−1+x/b−aα×1+e−2λ1+x/b−a/1−1+x/b−aα−1.Plots of the TCPOFW, TCPOFE, TCPOFR, and TCPOFL densities are represented in Figure1.Figure 1
TCPOFW, TCPOFE, TCPOFR, and TCPOFL densities for different values of parameters.


## 2.1.1. TCPOF Weibull (TCPOFW) Distribution

The cdf and pdf of TCPOFW distribution are(18)Fx;λ,α,μ,β=4πarctane−λe−βxμ/1−e−βxμα,x>0,fx;λ,α,μ,β=4λαμβμxμ−1e−βxμe−βxμα−1π1−e−βxμα+1e−λe−βxμ/1−e−βxμα,1+e−2λe−βxμ/1−e−βxμα−1.


## 2.1.2. TCPOF Exponential (TCPOFE) Distribution

The cdf and pdf of the TCPOFE model (forx>0 ) are(19)Fx;λ,α,β=4πarctane−λe−βx/1−e−βxα,x>0,fx;λ,α,β=4λαβe−βxe−βxα−1π1−e−βxα+1e−λe−βx/1−e−βxα,1+e−2λe−βx/1−e−βxα−1.


## 2.1.3. TCPOF Rayleigh (TCPOFR) Distribution

The cdf and pdf of the TCPOFR model are(20)Fx;λ,α,β=4πarctane−λe−βx2/1−e−βx2α,x>0,fx;λ,α,β=4λαμβ2xe−βx2e−βx2α−1π1−e−βx2α+1e−λe−βx2/1−e−βx2α,1+e2−λe−βx2/1−e−βx2α−1.


## 2.1.4. TCPOF Lomax (TCPOFL) Distribution

The cdf and pdf of the TCPOFL model are(21)Fx;λ,α,a,b=4πarctane−λ1+x/b−a/1−1+x/b−aα,x>0,fx;λ,α,a,b=4λαa/b1+x/b−a−11+x/b−aα−1π1−1+x/b−aα+1e−λ1+x/b−a/1−1+x/b−aα×1+e−2λ1+x/b−a/1−1+x/b−aα−1.Plots of the TCPOFW, TCPOFE, TCPOFR, and TCPOFL densities are represented in Figure1.Figure 1
TCPOFW, TCPOFE, TCPOFR, and TCPOFL densities for different values of parameters.


## 3. Structural Properties

Our study focused on ordinary moments, incomplete moments (IMs), moment generating functions (MGFs), mean deviations (MD), Lorenz and Bonferroni curves (LBCs), and residual life (RL) functions.


### 3.1. Ordinary Moments and Incomplete Moments Functions

A first formula for therth ordinary moment of X, say μr′, is(22)μr′=EXr=∑m=0∞ψmEYm+1r.A second formula depending on the quantile function can be written asEXr=m+1∫−∞∞gxGxmdx=m+1∫01QGu;ξruddu. The sth IMs of X defined by νst for any real s>0 can be expressed from equation (16) as(23)νst=∫−∞txsfxdx=∑m=0∞ψm∫−∞txsπm+1xdx.Equation (23) denotes the sth IMs of πm+1. The MDs about the mean μ=EX and the MDs about the median M are defined by(24)δ1x=EX−μ1′=2μ1′Fμ1′−2ν1μ1′,δ2x=EX−M=μ1′−2ν1M,respectively, where μ1′=EX,M= median X=Q1/2, Fμ1′ is evaluated from equation (5), and ν1t is the first IM given by equation (23) with s=1, where(25)ν1t=∫−∞txfxdx=∑m=0∞ψm∫−∞txπm+1xdx.We can determineδ1x and δ2x by two techniques, the first can be obtained from equation (16) as ν1t=∑m=0∞ψmYm+1t where Ym+1t=∫−∞txπm+1xdx is the first IM of the Exp-G distribution. The second technique is given by ν1t=∑m=0∞ψmδm+1t where(26)δm+1t=m+1∫0Gtum+1QGuau.For a positive random variableX, the LBCs, for a given probability p, are given by Lp=1/μ1′ν1q and Bp=1/pμ1′ν1q, respectively, where μ1′=EX, and q=Qp is the quantile function of X at p.


### 3.2. Moment Generating Function

The MGF ofX is(27)MXt=EetX=∑m=0∞ψmMm+1t,where Mm+1tdenotes the MGF of πm+1.A second alternative formula can be derived from equation (16) as follows: MXt=∑m=0∞ψmγt,m+1, where γt,m+1=m+1∫01umetQGudu.


### 3.3. Moments of RL and Reversed RL

Therth order moment of the RL is given by(28)ψrt=EX−tr|X>t=1F¯t∫t∞x−trfxdx,r≥1=1F¯t∑m=0∞ψm∗∫t∞xrπm+1xdx,where ψm∗=∑m=0∞∑m=0rψmrm−tr−m. The mean RL MRL of TCPOF, −G family can be obtained by setting r=1 in equation (28), defined as(29)ψ1t=EXt=EX|X>t.Therth order moment of the reversed RL (or inactivity time) can be obtained by the next equation(30)mrt=Et−Xr|X≤t=1Ft∫0tt−xrfxdx,r≥1=1Ft∑m=0∞ψm∗∫0txrπm+1xdx.


## 3.1. Ordinary Moments and Incomplete Moments Functions

A first formula for therth ordinary moment of X, say μr′, is(22)μr′=EXr=∑m=0∞ψmEYm+1r.A second formula depending on the quantile function can be written asEXr=m+1∫−∞∞gxGxmdx=m+1∫01QGu;ξruddu. The sth IMs of X defined by νst for any real s>0 can be expressed from equation (16) as(23)νst=∫−∞txsfxdx=∑m=0∞ψm∫−∞txsπm+1xdx.Equation (23) denotes the sth IMs of πm+1. The MDs about the mean μ=EX and the MDs about the median M are defined by(24)δ1x=EX−μ1′=2μ1′Fμ1′−2ν1μ1′,δ2x=EX−M=μ1′−2ν1M,respectively, where μ1′=EX,M= median X=Q1/2, Fμ1′ is evaluated from equation (5), and ν1t is the first IM given by equation (23) with s=1, where(25)ν1t=∫−∞txfxdx=∑m=0∞ψm∫−∞txπm+1xdx.We can determineδ1x and δ2x by two techniques, the first can be obtained from equation (16) as ν1t=∑m=0∞ψmYm+1t where Ym+1t=∫−∞txπm+1xdx is the first IM of the Exp-G distribution. The second technique is given by ν1t=∑m=0∞ψmδm+1t where(26)δm+1t=m+1∫0Gtum+1QGuau.For a positive random variableX, the LBCs, for a given probability p, are given by Lp=1/μ1′ν1q and Bp=1/pμ1′ν1q, respectively, where μ1′=EX, and q=Qp is the quantile function of X at p.


## 3.2. Moment Generating Function

The MGF ofX is(27)MXt=EetX=∑m=0∞ψmMm+1t,where Mm+1tdenotes the MGF of πm+1.A second alternative formula can be derived from equation (16) as follows: MXt=∑m=0∞ψmγt,m+1, where γt,m+1=m+1∫01umetQGudu.


## 3.3. Moments of RL and Reversed RL

Therth order moment of the RL is given by(28)ψrt=EX−tr|X>t=1F¯t∫t∞x−trfxdx,r≥1=1F¯t∑m=0∞ψm∗∫t∞xrπm+1xdx,where ψm∗=∑m=0∞∑m=0rψmrm−tr−m. The mean RL MRL of TCPOF, −G family can be obtained by setting r=1 in equation (28), defined as(29)ψ1t=EXt=EX|X>t.Therth order moment of the reversed RL (or inactivity time) can be obtained by the next equation(30)mrt=Et−Xr|X≤t=1Ft∫0tt−xrfxdx,r≥1=1Ft∑m=0∞ψm∗∫0txrπm+1xdx.


## 4. Entropy

The Rényi entropy is defined by (θ>0,θ≠1(31)IRθ=11−θlog∫−∞∞fθxdx.Using the general binomial expansion, applying the same method of the linear representation equation (16) and after some simplifications, we get(32)IRθ=11−θlog∫−∞∞fθxdx,where(33)Δm=4λαπθ∑i,j,k=0∞−1i+j+mλj2i+θji!j!k!m!×Γθ+iΓαθ+j+k+θΓαθ+j+k−θ+1Γαθ+j+θΓαθ+j+k−θ−m+1.Thus, Rényi entropy ofTCPOF, −G family is given by(34)IRθ=11−θlog∑m=0∞Δm∫−∞∞gxθGxmdx.Also,θ−entropy can be obtained as(35)ωθX=11−θlog1−∑m=0∞Δm∫−∞∞gxθGxmdx.


## 5. Maximum Likelihood Estimation

Letx1,…,xn be a random sample of size n from the TCPOF−G given by equation (11). Let Ω=λ,α,ξT be q×1 vector of parameters. The log-likelihood function is(36)Ln=nlog4λπ+nlogα+∑i=1nloggxi;ξ+α−1∑i=1nlogG¯xi;ξ−α+1∑i=1nlogGxi;ξ−λ∑i=1ndiα−∑i=1nlog1+e−2λdiα.di=G¯xi;ξ/Gxi;ξ. The score vector components, say, UΩ=∂Ln/∂Ω=∂Ln/∂λ,∂Ln/∂α,∂Ln/∂ξ are given by(37)Uλ=∂Ln∂λ=nλ−∑i=1ndiα+∑i=1n2diαe−2λdiα1+e−2λdiα,Uα=∂Ln∂α=nα+∑i=1nlogG¯xi;ξ−∑i=1nlogGxi;ξ−λ∑i=1ndiαlogdi+∑i=1n2λdiαe−2λdiαlogdi1+e−2λdiα,Uξ=∂Ln∂ξk=∑i=1ng′xi;ξgxi;ξ+α−1∑i=1nG¯′xi;ξG¯xi;ξ−α+1∑i=1nG′xi;ξGxi;ξ−λα∑i=1ndiα−1∂di∂ξk+∑i=1n2λαdiα−1e−2λdiα1+e−2λdiα∂di∂ξk,where g′xi;ξ=∂gxi;δ/∂ξk,G′xi;ξ=∂Gxi;ξ/∂ξk, and G¯′xi;ξ=∂G¯xi;ξ/∂ξk. The maximum likelihood estimation (MLE) of parameters is obtained by setting ∂Ln/∂λ=∂Ln/∂α=∂Ln/∂ξk=0 and solving these equations simultaneously to get theMLEΩ^.


## 6. Applications

In this section, we used two datasets taken from Bjerkedal [21] and Gross and Clark [22]. The TCPOFE model as example of the new family is compared with some other known competitive models to demonstrate its importance in data modeling. The MLE method is used to estimate the parameters of the competitive models. Some information criterions (IC) are as follows: the Akaike IC (Z1), Bayesian IC (Z2), consistent Akaike IC (Z3), Hannan–Quinn IC (Z4), Cramér–Von Mises (Z5), Anderson–Darling (Z6), Kolmogorov–Smirnov (Z7) statistics with its p value (Z8) model selection criteria, and goodness of fit tests are used to identify the best model.The goodness of fits of the TCPOFE model is compared with those of other competitive models, namely, Marshall–Olkin (MO) E (MOE), Burr X-E (BrXE), Kumaraswamy E (KwE), generalized MO exponential (GMOE), beta exponential (BE), Kumaraswamy MO E (KMOE), MO Kumaraswamy E (MOKE), moment exponential (ME), and exponential (E) models.The MLEs and standard errors (SEs) are calculated for both datasets. The numerical results are listed in Tables2 and 3. From the numerical values provided in Tables 4 and 5 and the information shown in Figures 2 and 3, the TCPOFE model performs considerably better than the other considered extensions of the E model. Hence, the TCPOFE model is a good alternative over these models for both datasets.Table 2
Estimates for dataset I.
DistributionsMLEs and SEsTCPOFE (λ, α, β)0.151, 6.011, 8.084(0.168), (1.024), (0.6141)MOE (a, b)8.780, 1.380(3.560), (0.194)BrXE (a, β)0.475, 0.2055(0.060), (0.012)KE (a, b, β)3.3041, 1.1002, 1.0371(1.1061), (0.7642), (0.6141)GMOE (a, b, β)0.1789, 47.6350, 4.4652(0.0702), (44.9011), (1.3270)BE (a, b, β)0.8073, 3.4612, 1.3311(0.6961), (1.0032), (0.8551)KMOE (α, a, b, β)0.3731, 3.4782, 3.3063, 0.2990(0.1358), (0.862), (0.781), (1.113)MOKE (α, a, b, β)0.0081, 2.7162, 1.9861, 0.0992(0.0021), (1.3158), (0.7839), (0.0481)ME (β)0.9252(0.0768)E (β)0.540(0.063)Table 3
Estimates for dataset II.
DistributionsMLEs and SEsTCPOFE (λ, α, β)1.475, 21.125, 1.295(5.068), (0.990), (3.816)MOE (a, b)54.470, 2.320(35.580), (0.370)BrXE (a, β)1.1635, 0.3207(0.330), (0.030)KE (a, b, β)83.7558, 0.5679, 3.3329(42.3612), (0.3261), (1.1880)GMOE (a, b, β)0.5192, 89.4623, 3.1691(0.2561), (66.2782), (0.7721)BE (a, b, β)81.6333, 0.5421, 3.5142(120.4104), (0.3272), (1.4101)KMOE (α, a, b, β)8.8679, 34.8258, 0.2989, 4.8988(9.1459), (22.3119), (0.2387), (3.1757)MOKE (α, a, b, β)0.1333, 33.2322, 0.5711, 1.6691(0.3320), (57.8371), (0.7211), (1.8141)ME (β)0.9502(0.1501)E (β)0.526(0.117)Table 4
Some IC for dataset I.
DistributionsZ1Z2Z3Z4Z7Z8TCPOFE191.815191.387192.168194.5340.085(0.674)MOE210.36214.92210.53212.160.10(0.430)BrXE235.30239.90235.50237.100.22(0.002)KE209.42216.24209.77212.120.09(0.500)GMOE210.54217.38210.89213.240.09(0.510)BE207.38214.22207.73210.080.11(0.340)KMOE207.82216.94208.42211.420.09(0.530)MOKE209.44218.56210.04213.040.10(0.440)ME210.40212.68210.45211.300.14(0.130)E234.63236.91234.68235.540.27(0.060)Table 5
Some IC for dataset II.
DistributionsZ1Z2Z3Z4Z7Z8TCPOFE37.48535.38838.98538.0680.108(0.975)MOE43.5145.5144.2243.900.18(0.55)BrXE48.1050.1048.8048.500.25(0.17)KE41.7844.7543.28,42.320.14(0.86)GMOE42.7545.7444.2543.340.15(0.78)BE43.4846.4544.9844.020.16(0.80)KMOE42.8046.8445.5543.600.15(0.86)MOKE41.5845.5444.2542.300.14(0.87)ME54.3255.3154.5454.500.32(0.07)E67.6768.6767.8967.870.44(0.004)Figure 2
Fitted cdf, pdf, and pp plots for dataset I.Figure 3
Fitted cdf, pdf, and pp plots for dataset II.


## 7. Conclusion

In this article, we have shown how the use of the “truncated Cauchy power” scheme applied to the odd Fréchet family can lead to new flexible statistical models for the analysis of right-skewed data. Precisely, we have proceeded as follows. First, we have defined the TCPOF-G family of distributions, discussed the motivations behind it, and studied its main properties of interest. Four particular examples of the new family are proposed. Rényi entropy of the TCPOF is calculated. We used the maximum likelihood technique to estimate the parameters. Examples from real-world data demonstrate empirically the significance and promise of the proposed family. In the future, we are planning to use the new family to generate a new model and study its statistical properties and prove the flexibility of it by using more real datasets.



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*Source: 4256945-2021-10-27.xml*