# Extended Odd Fréchet-G Family of Distributions

Authors: Suleman Nasiru
Journal: Journal of Probability and Statistics
 (2018)

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## Abstract

The need to develop generalizations of existing statistical distributions to make them more flexible in modeling real data sets is vital in parametric statistical modeling and inference. Thus, this study develops a new class of distributions called the extended odd Fréchet family of distributions for modifying existing standard distributions. Two special models named the extended odd Fréchet Nadarajah-Haghighi and extended odd Fréchet Weibull distributions are proposed using the developed family. The densities and the hazard rate functions of the two special distributions exhibit different kinds of monotonic and nonmonotonic shapes. The maximum likelihood method is used to develop estimators for the parameters of the new class of distributions. The application of the special distributions is illustrated by means of a real data set. The results revealed that the special distributions developed from the new family can provide reasonable parametric fit to the given data set compared to other existing distributions.


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## Body

## 1. Introduction

The fundamental reason for parametric statistical modeling is to identify the most appropriate model that adequately describes a data set obtained from experiment, observational studies, surveys, and so on. Most of these modeling techniques are based on finding the most suitable probability distribution that explains the underlying structure of the given data set. However, there is no single probability distribution that is suitable for different data sets. Thus, this has triggered the need to extend the existing classical distributions or develop new ones. Barrage of methods for defining new families of distributions have been proposed in literature for extending or generalizing the existing classical distributions in recent time. Some of these methods include Weibull-G [1], odd generalized exponential family [2], odd Lindley-G family [3], Topp-Leone odd log-logistic-G family [4], odd Burr-G family [5], odd Fréchet-G family [6], odd gamma-G family [7], transformed-transformer method [8], exponentiated transformed-transformer method [9], exponentiated generalized transformed-transformer method [10], alpha power transformed family [11], alpha logarithmic transformed family [12], Kumaraswamy-G family [13], beta-G family [14], Kumaraswamy transmuted-G family [15], transmuted geometric-G family [16], and beta extended Weibull family [17]. These methods are developed with the motivation of defining new models with different kinds of failure rates (monotonic and nonmonotonic), constructing heavy-tailed distributions for modeling different kinds of data sets, developing distributions with symmetric, right skewed, left skewed, reversed J shape, and consistently providing a reasonable parametric fit to given data sets.Recently, [6] developed the odd Fréchet family of distributions and defined its cumulative distribution function (CDF) as(1)Hx=e-1-Gx;ψ/Gx;ψθ,x∈R,where G(x;ψ) is the baseline CDF and ψ is a p×1 vector of associated parameters. Using the transformed-transformer method proposed by [8], an extension of the odd Fréchet family of distributions called the extended odd Fréchet-G (EOF-G) family of distributions is developed by integrating the Fréchet probability density function (PDF). Hence, the CDF of the EOF-G family is defined as(2)Fx=∫0Gx;ψα/1-Gx;ψαθx-θ-1e-x-θdx=e-1-Gx;ψα/Gx;ψαθ,α>0,θ>0,x∈R,where α and θ are extra shape parameters. The corresponding PDF of the new family is obtained by differentiating equation (2) and is given by(3)fx=αθgx;ψ1-Gx;ψαθ-1Gx;ψαθ+1e-1-Gx;ψα/Gx;ψαθ,α>0,θ>0,x∈R.The associated hazard rate function of the EOF-G family is defined as(4)hx=αθgx;ψ1-Gx;ψαθ-1Gx;ψαθ+11-e-1-Gx;ψα/Gx;ψαθe-1-Gx;ψα/Gx;ψαθ,α>0,θ>0,x∈R.Hereafter, a random variable X following the EOF-G distribution is denoted by X~EOF-G(x;α,θ,ψ) and for the purpose of simplicity, G(x;ψ) can be written as G(x). The CDF of the EOF-G family of distributions is tractable which makes it easy to generate random numbers provided that the CDF of the baseline distribution is also tractable. The uth quantile of the EOF-G family is given by(5)xu=G-111+-log⁡u1/θ1/α,u∈0,1,where G-1(u) is the baseline quantile function. When α=1, the EOF-G family of distributions reduces to the odd Fréchet family of distributions. Adopting the interpretation of the CDF of the odd Weibull family as given in [18], the physical interpretation of the CDF of the EOF-G family is given as follows: Suppose Y is a lifetime random variable with continuous CDF, G(x;ψ)α. The odds ratio that an individual (component) having the lifetime Y will die (fail) at time x is G(x;ψ)α/1-G(x;ψ)α. Given that the variability of these odds of death is denoted by the random variable X and that it follows the Fréchet distribution, then(6)PY≤x=PX≤Gx;ψα1-Gx;ψα=Fx,which is given in (2). The rest of the paper is organized as follows: In Section 2, special distributions of the EOF-G family are discussed. In Section 3, the mixture representation of the PDF and CDF of the EOF-G family is given. The statistical properties of the new family are derived in Section 4. In Section 5, the estimators for the parameters of the family are developed using the technique of maximum likelihood estimation. Monte Carlo simulations are performed in Section 6 to assess the performance of the estimators. In Section 7, the application of the special distributions is demonstrated using real data set. Finally, the concluding remarks of the study are given in Section 8.


## 2. Special Distributions of the EOF-G Family

In this section, two special distributions of the EOF-G family are discussed.


### 2.1. EOF-Nadarajah-Haghighi (EOFNH) Distribution

Suppose the baseline CDF is that of the Nadarajah-Haghighi distribution; that is,G(x;β,λ)=1-e(1-(1+λx)β) with corresponding PDF g(x;β,λ)=βλ(1+λx)β-1e(1-(1+λx)β) and positive parameters β,λ>0. The PDF of the EOFNH distribution is given by(7)fx=αβλθ1+λxβ-1e1-1+λxβ1-1-e1-1+λxβαθ-11-e1-1+λxβαθ+1e-1-e1-1+λxβ-α-1θ,where α,β,θ>0 are shape parameters, λ>0 is a scale parameter, and x>0. Figure 1 shows the plots of the PDF of the EOFNH distribution for some selected parameter values. The density function exhibits different kinds of shapes.Figure 1
Plots of the EOFNH distribution density function.The corresponding hazard rate function is given by(8)hx=αβλθ1+λxβ-1e1-1+λxβ1-1-e1-1+λxβαθ-11-e1-1+λxβαθ+11-e-1-e1-1+λxβ-α-1θe-1-e1-1+λxβ-α-1θ,x>0.The plots of the hazard rate function of the EOFNH distribution for some selected parameter values are shown in Figure 2. The hazard rate function can assume decreasing, bathtub, upside down bathtub, and other nonmonotonic failure rate forms.Figure 2
Plots of the EOFNH distribution hazard rate function.The quantile function of the EOFNH distribution is given by(9)xu=1-log1-1+-log⁡u1/θ-1/α1/β-1λ,u∈0,1.Equation (9) can be used to generate random numbers from the EOFNH distribution. The first quartile, median, and upper quartile of the distribution are obtained by substituting u=0.25,0.5, and 0.75, respectively, into (9).


### 2.2. EOF-Weibull (EOFW) Distribution

Consider the Weibull distribution with shape parameterβ>0 and scale parameter λ>0, where the CDF and PDF for x>0 are given by G(x;β,λ)=1-e-λxβ and g(x;β,λ)=βλxβ-1e-λxβ. Substituting the PDF and CDF of the Weibull distribution in (3), the PDF of the EOFW distribution is defined as(10)fx=αβλθxβ-1e-λxβ1-1-e-λxβαθ-11-e-λxβαθ+1e-1-e-λxβ-α-1θ,where α,β,θ>0 are shape parameters, λ>0 is scale parameter, and x>0. Figure 3 displays some of the possible shapes of the density function of the EOFW distribution. The density exhibits unimodal and reversed J-shape among others.Figure 3
Plots of the EOFW distribution density function.The hazard rate function of the EOFW distribution is given by(11)hx=αβλθxβ-1e-λxβ1-1-e-λxβαθ-11-e-λxβαθ+11-e-1-e-λxβ-α-1θe-1-e-λxβ-α-1θ,x>0.The hazard rate function can assume decreasing, bathtub, and upside down bathtub forms for some selected parameter values as shown in Figure 4.Figure 4
Plots of the EOFW distribution hazard rate function.The quantile function of the EOFW distribution is defined as(12)xu=-log⁡1-1+-log⁡u1/θ-1/αλ1/β,u∈0,1.The generation of random numbers from the EOFW distribution can easily be done using (12).


## 2.1. EOF-Nadarajah-Haghighi (EOFNH) Distribution

Suppose the baseline CDF is that of the Nadarajah-Haghighi distribution; that is,G(x;β,λ)=1-e(1-(1+λx)β) with corresponding PDF g(x;β,λ)=βλ(1+λx)β-1e(1-(1+λx)β) and positive parameters β,λ>0. The PDF of the EOFNH distribution is given by(7)fx=αβλθ1+λxβ-1e1-1+λxβ1-1-e1-1+λxβαθ-11-e1-1+λxβαθ+1e-1-e1-1+λxβ-α-1θ,where α,β,θ>0 are shape parameters, λ>0 is a scale parameter, and x>0. Figure 1 shows the plots of the PDF of the EOFNH distribution for some selected parameter values. The density function exhibits different kinds of shapes.Figure 1
Plots of the EOFNH distribution density function.The corresponding hazard rate function is given by(8)hx=αβλθ1+λxβ-1e1-1+λxβ1-1-e1-1+λxβαθ-11-e1-1+λxβαθ+11-e-1-e1-1+λxβ-α-1θe-1-e1-1+λxβ-α-1θ,x>0.The plots of the hazard rate function of the EOFNH distribution for some selected parameter values are shown in Figure 2. The hazard rate function can assume decreasing, bathtub, upside down bathtub, and other nonmonotonic failure rate forms.Figure 2
Plots of the EOFNH distribution hazard rate function.The quantile function of the EOFNH distribution is given by(9)xu=1-log1-1+-log⁡u1/θ-1/α1/β-1λ,u∈0,1.Equation (9) can be used to generate random numbers from the EOFNH distribution. The first quartile, median, and upper quartile of the distribution are obtained by substituting u=0.25,0.5, and 0.75, respectively, into (9).


## 2.2. EOF-Weibull (EOFW) Distribution

Consider the Weibull distribution with shape parameterβ>0 and scale parameter λ>0, where the CDF and PDF for x>0 are given by G(x;β,λ)=1-e-λxβ and g(x;β,λ)=βλxβ-1e-λxβ. Substituting the PDF and CDF of the Weibull distribution in (3), the PDF of the EOFW distribution is defined as(10)fx=αβλθxβ-1e-λxβ1-1-e-λxβαθ-11-e-λxβαθ+1e-1-e-λxβ-α-1θ,where α,β,θ>0 are shape parameters, λ>0 is scale parameter, and x>0. Figure 3 displays some of the possible shapes of the density function of the EOFW distribution. The density exhibits unimodal and reversed J-shape among others.Figure 3
Plots of the EOFW distribution density function.The hazard rate function of the EOFW distribution is given by(11)hx=αβλθxβ-1e-λxβ1-1-e-λxβαθ-11-e-λxβαθ+11-e-1-e-λxβ-α-1θe-1-e-λxβ-α-1θ,x>0.The hazard rate function can assume decreasing, bathtub, and upside down bathtub forms for some selected parameter values as shown in Figure 4.Figure 4
Plots of the EOFW distribution hazard rate function.The quantile function of the EOFW distribution is defined as(12)xu=-log⁡1-1+-log⁡u1/θ-1/αλ1/β,u∈0,1.The generation of random numbers from the EOFW distribution can easily be done using (12).


## 3. Mixture Representation

In this section, the mixture representation of the PDF and CDF of the EOF-G family of distributions is discussed. The mixture representation is useful when deriving the statistical properties of this new family of distributions. Using the Taylor series expansion, the PDF can be written as(13)fx=αθ∑i=0∞-1igx;ψ1-Gx;ψαθi+1-1i!Gx;ψαθi+1+1.Equation (13) can be written as(14)fx=αθ∑i=0∞-1igx;ψ1-Gx;ψαθi+1-11-1-Gx;ψ-αθi+1+1i!.Applying the generalized binomial series expansion yields(15)fx=αθ∑i,j=0∞-1ii!αθi+1+jjgx;ψ1-Gx;ψj1-Gx;ψαθi+1-1.Now using the binomial series expansion, (1-z)η-1=∑j=0∞(-1)jη-1jzj,|z|<1, thrice yields(16)fx=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmqgx;ψGx;ψq,where(17)ωijkmq=-1i+k+m+qi!αθi+1+jjθi+1-1kαkmm+jq.Alternatively (16) can be written in terms of the exponentiated-G (exp-G) density function as(18)fx=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∗πq+1x,where ωijkmq∗=ωijkmq/(q+1) and πq+1(x)=(q+1)g(x;ψ)G(x;ψ)q is the exp-G density function with power parameter q+1. By integrating (18), the mixture representation of the CDF is given by(19)Fx=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∗Πq+1x,where Πq+1(x)=G(x;ψ)q+1 is the CDF of the exp-G family with power parameter q+1.


## 4. Statistical Properties

In this section, the moments, incomplete moments, generating function, entropies, and order statistics of the EOF-G family are derived.


### 4.1. Moments

Therth noncentral moment of a random variable X is given by E(Xr)=∫-∞∞xrf(x)dx. Hence, using this definition the rth noncentral moment of the EOF-G random variable is given by(20)EXr=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmqτr,q,where τr,q=∫-∞∞xrg(x;ψ)G(x;ψ)qdx is the probability weighted moment of the baseline distribution. The rth noncentral moment can also be expressed in terms of the quantile of the baseline distribution. Letting G(x;(ψ))=u, the rth noncentral moment in terms of the quantile is given by(21)EXr=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫01QGuruqdu,where QG(u) is the quantile function of the baseline distribution.


### 4.2. Incomplete Moments

Therth incomplete moment of a random variable X is defined as mr(y)=∫-∞yxrf(x)dx. Thus, the rth incomplete moment of the EOF-G random variable is given by(22)mry=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫0yxrgx;ψGx;ψqdxIn terms of the quantile function of the baseline distribution, the rth incomplete moment is given by(23)mry=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫0GyQGuruqdu.Utilize the power series expansion of the quantile of the baseline; that is,(24)QGu=∑h=0∞ehuh,where eh(h=0,1,…) are suitably chosen real numbers that depend on the parameters of the G(x;ψ) distribution. Furthermore, for positive integer r(r≥1),(25)QGur=∑h=0∞ehuhr=∑h=0∞er,h′uh,where er,h′=(he0)-1∑z=1h[z(r+1)-h]ezer,h-z′ and er,0′=(e0)h. For more details on quantile power series expansion, see [19]. Hence,(26)mry=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫0Gy∑h=0∞er,h′uh+qdu=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmqer,h′Gyh+q+1h+q+1.The incomplete moments are used in the computation of other useful statistical measures such as the mean deviations about the mean (δ1=E(|X-μ1′|)) and about the median (δ2=E(|X-M|)). The mean deviation about the mean and about the median can further be expressed as(27)δ1=2μ1′Fμ1′-2m1μ1′,δ2=μ1′-2m1M,where μ1′=μ is the mean obtained by putting r=1 into (20), M is the median obtained by substituting u=0.5 into (5), and m1(y)=∫-∞yxf(x)dx is the first incomplete moment which can be obtained from (23) by substituting r=1.


### 4.3. Generating Function

In this subsection, two formulae for the computation of the moment generating functionMX(t)=E(etX) are given. Using the Taylor series expansion, MX(t)=E(etX)=∑r=0∞tr/r!E(Xr). Thus, the moment generating function is given by(28)MXt=∑i,j=0∞∑k,m=0∞∑r=0∞∑q=0m+jωijkmqτr,q.Alternatively, the moment generating function can be expressed in terms of the quantile function of the baseline distribution as(29)MXt=∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫01etQGuuqdu.


### 4.4. Entropy Measures

Entropies are measures of uncertainty or variation of a random variable. In this subsection, the Rényi, Shannon, andδ entropies are studied. The Rényi entropy [20] of a random variable X with PDF f(x) is defined as(30)IRδ=11-δlog⁡∫-∞∞fxδdx,δ>0,δ≠1.Using similar concepts for expanding the PDF,(31)fxδ=αθδ∑i,j=0∞∑k,m=0∞∑q=0m+jϖijkmqgx;ψδGx;ψq,where(32)ϖijkmq=-1i+k+m+qδii!αθi+δ+δ+j-1jθi+δ-δkαkmm+jq.Hence,(33)IRδ=11-δlog⁡αθδ∑i,j=0∞∑k,m=0∞∑q=0m+jϖijkmq∫-∞∞gx;ψδGx;ψqdx,δ>0,δ≠1.The Shannon entropy [21] of a random variable X, say ηX=E(-log⁡f(X)). The Shannon entropy is a special case of the Rényi entropy when δ↑1. The δ-entropy is given by(34)Hδ=1δ-1log⁡1-∫-∞∞fxδdx,δ>0,δ≠1.Thus, the δ-entropy is(35)Hδ=1δ-11-αθδ∑i,j=0∞∑k,m=0∞∑q=0m+jϖijkmq∫-∞∞gx;ψδGx;ψqdx,δ>0,δ≠1.


### 4.5. Order Statistics

LetX1,X2,…,Xn represent a random sample from EOF-G family and X1:n≤X2:n≤⋯≤Xn:n be the order statistics. Then the PDF, fp:n(x), of the pth order statistic Xp:n is(36)fp:nx=n!p-1!n-p!∑i=0n-p-1iFxp+i-1fx.Substituting the PDF and the CDF of the EOF-G random variable into the last equation yields(37)fp:nx=n!αθp-1!n-p!∑j,k=0∞∑q,s=0∞∑w=0k+s∑i=0n-pφijkqswgx;ψGx;ψw,after some algebraic manipulation, where(38)φijkqsw=-1i+j+q+s+wp+ijj!n-piαθj+1+kkθj+1-1qαqsk+sw.The PDF of the pth order statistic can be expressed in terms of the exp-G density function as(39)fn:px=n!αθp-1!n-p!∑j,k=0∞∑q,s=0∞∑w=0k+s∑i=0n-pφijkqsw∗Δw+1x,where φijkqsw∗=φijkqsw/(w+1) and Δw+1(x)=(w+1)g(x;ψ)G(x;ψ)w is the exp-G density function with power parameter w+1.


## 4.1. Moments

Therth noncentral moment of a random variable X is given by E(Xr)=∫-∞∞xrf(x)dx. Hence, using this definition the rth noncentral moment of the EOF-G random variable is given by(20)EXr=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmqτr,q,where τr,q=∫-∞∞xrg(x;ψ)G(x;ψ)qdx is the probability weighted moment of the baseline distribution. The rth noncentral moment can also be expressed in terms of the quantile of the baseline distribution. Letting G(x;(ψ))=u, the rth noncentral moment in terms of the quantile is given by(21)EXr=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫01QGuruqdu,where QG(u) is the quantile function of the baseline distribution.


## 4.2. Incomplete Moments

Therth incomplete moment of a random variable X is defined as mr(y)=∫-∞yxrf(x)dx. Thus, the rth incomplete moment of the EOF-G random variable is given by(22)mry=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫0yxrgx;ψGx;ψqdxIn terms of the quantile function of the baseline distribution, the rth incomplete moment is given by(23)mry=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫0GyQGuruqdu.Utilize the power series expansion of the quantile of the baseline; that is,(24)QGu=∑h=0∞ehuh,where eh(h=0,1,…) are suitably chosen real numbers that depend on the parameters of the G(x;ψ) distribution. Furthermore, for positive integer r(r≥1),(25)QGur=∑h=0∞ehuhr=∑h=0∞er,h′uh,where er,h′=(he0)-1∑z=1h[z(r+1)-h]ezer,h-z′ and er,0′=(e0)h. For more details on quantile power series expansion, see [19]. Hence,(26)mry=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫0Gy∑h=0∞er,h′uh+qdu=αθ∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmqer,h′Gyh+q+1h+q+1.The incomplete moments are used in the computation of other useful statistical measures such as the mean deviations about the mean (δ1=E(|X-μ1′|)) and about the median (δ2=E(|X-M|)). The mean deviation about the mean and about the median can further be expressed as(27)δ1=2μ1′Fμ1′-2m1μ1′,δ2=μ1′-2m1M,where μ1′=μ is the mean obtained by putting r=1 into (20), M is the median obtained by substituting u=0.5 into (5), and m1(y)=∫-∞yxf(x)dx is the first incomplete moment which can be obtained from (23) by substituting r=1.


## 4.3. Generating Function

In this subsection, two formulae for the computation of the moment generating functionMX(t)=E(etX) are given. Using the Taylor series expansion, MX(t)=E(etX)=∑r=0∞tr/r!E(Xr). Thus, the moment generating function is given by(28)MXt=∑i,j=0∞∑k,m=0∞∑r=0∞∑q=0m+jωijkmqτr,q.Alternatively, the moment generating function can be expressed in terms of the quantile function of the baseline distribution as(29)MXt=∑i,j=0∞∑k,m=0∞∑q=0m+jωijkmq∫01etQGuuqdu.


## 4.4. Entropy Measures

Entropies are measures of uncertainty or variation of a random variable. In this subsection, the Rényi, Shannon, andδ entropies are studied. The Rényi entropy [20] of a random variable X with PDF f(x) is defined as(30)IRδ=11-δlog⁡∫-∞∞fxδdx,δ>0,δ≠1.Using similar concepts for expanding the PDF,(31)fxδ=αθδ∑i,j=0∞∑k,m=0∞∑q=0m+jϖijkmqgx;ψδGx;ψq,where(32)ϖijkmq=-1i+k+m+qδii!αθi+δ+δ+j-1jθi+δ-δkαkmm+jq.Hence,(33)IRδ=11-δlog⁡αθδ∑i,j=0∞∑k,m=0∞∑q=0m+jϖijkmq∫-∞∞gx;ψδGx;ψqdx,δ>0,δ≠1.The Shannon entropy [21] of a random variable X, say ηX=E(-log⁡f(X)). The Shannon entropy is a special case of the Rényi entropy when δ↑1. The δ-entropy is given by(34)Hδ=1δ-1log⁡1-∫-∞∞fxδdx,δ>0,δ≠1.Thus, the δ-entropy is(35)Hδ=1δ-11-αθδ∑i,j=0∞∑k,m=0∞∑q=0m+jϖijkmq∫-∞∞gx;ψδGx;ψqdx,δ>0,δ≠1.


## 4.5. Order Statistics

LetX1,X2,…,Xn represent a random sample from EOF-G family and X1:n≤X2:n≤⋯≤Xn:n be the order statistics. Then the PDF, fp:n(x), of the pth order statistic Xp:n is(36)fp:nx=n!p-1!n-p!∑i=0n-p-1iFxp+i-1fx.Substituting the PDF and the CDF of the EOF-G random variable into the last equation yields(37)fp:nx=n!αθp-1!n-p!∑j,k=0∞∑q,s=0∞∑w=0k+s∑i=0n-pφijkqswgx;ψGx;ψw,after some algebraic manipulation, where(38)φijkqsw=-1i+j+q+s+wp+ijj!n-piαθj+1+kkθj+1-1qαqsk+sw.The PDF of the pth order statistic can be expressed in terms of the exp-G density function as(39)fn:px=n!αθp-1!n-p!∑j,k=0∞∑q,s=0∞∑w=0k+s∑i=0n-pφijkqsw∗Δw+1x,where φijkqsw∗=φijkqsw/(w+1) and Δw+1(x)=(w+1)g(x;ψ)G(x;ψ)w is the exp-G density function with power parameter w+1.


## 5. Parameter Estimation

In this section, the maximum likelihood technique is employed to develop estimators for estimating the parameters of the EOF-G family of distributions. Supposex1,x2,…,xn are possible outcomes of a random sample obtained from X~EOF-G(x;α,θ,ψ) and ϑ=(α,θ,ψ)T is a parameter vector; then the total log-likelihood function is given by(40)l=nlog⁡αθ+∑i=1nlog⁡gxi;ψ+θ-1∑i=1nlog⁡1-Gxi;ψα-αθ+1∑i=1nlog⁡Gxi;ψ-∑i=1n1-Gxi;ψαGxi;ψαθ.By finding the partial derivatives of (40), the components of the score vector U(ϑ)=∂l/∂α,∂l/∂θ,∂l/∂ψT are(41)∂l∂α=nα+θ-1∑i=1nGxi;ψαlog⁡Gxi;ψ1-Gxi;ψ-θ∑i=1nlog⁡Gxi;ψ+θ∑i=1n1-Gxi;ψαθ-1log⁡Gxi;ψGxi;ψαθ,(42)∂l∂θ=nθ+∑i=1nlog⁡1-Gxi;ψα-α∑i=1nlog⁡Gxi;ψ-∑i=1n1-Gxi;ψαGxi;ψαθlog⁡1-Gxi;ψαGxi;ψα,(43)∂l∂ψ=∑i=1ng′xi;ψgxi;ψ+αθ-1∑i=1nG′xi;ψGxi;ψα-11-Gxi;ψ-αθ+1∑i=1nG′xi;ψGxi;ψ+αθ∑i=1nG′xi;ψ1-Gxi;ψαθ-1Gxi;ψαθ+1,where g′(xi;ψ)=∂g(xi;ψ)/∂ψ and G′(xi;ψ)=∂G(xi;ψ)/∂ψ. In order to obtain the estimators for the parameters, we set (41), (42), and (43) to zero and solve the system numerically using methods such as the quasi-Newton algorithms since the equations do not have closed form. To obtain interval estimates of the parameters, a p×p observed information matrix can be estimated as J(ϑ)=∂2l/∂q∂r (for q,r=α,θ,ψ), whose elements are evaluated numerically. To compute the approximate confidence intervals of the parameters, the multivariate normal distribution Np(0,J(ϑ^)-1). Here, J(ϑ^) is the observed information evaluated at ϑ^. To investigate whether the EOF-G distributions are superior to the odd Fréchet family of distributions for given data sets, the likelihood ratio (LR) test can be performed using the following hypotheses: H0:α=1 versus Ha:H0 is false. The LR test statistic is given by LR=2l(ϑ^)-l(ϑ¯), where ϑ^ is the vector of unrestricted estimates under Ha and ϑ¯ is the vector of restricted maximum likelihood estimates under H0. The LR test statistic is asymptotically distributed as Chi-square random variable with degrees of freedom equal to the difference between the numbers of parameters of the two models. As a decision rule, the null hypothesis is rejected when the LR test statistic exceeds the upper 100(1-η)% quantile of the Chi-square distribution.


## 6. Simulation Study

In this section, Monte Carlo simulations are performed to assess the accuracy and consistency of the maximum likelihood estimators. For the purpose of illustration, the simulations are performed using the estimators of the parameters of the EOFNH distribution. The quantile function given in (9) is used to generate random observations from the EOFNH distribution. The simulations are repeated N=1,000 times each with sample size n=25,75,150,300,600,800 and parameter values I:α=0.5,β=0.5,λ=0.5,θ=0.5, II:α=3.3,β=0.8,λ=0.2,θ=0.8, and III:α=0.9,β=0.4,λ=0.2,θ=0.6. Table 1 presents the average bias (AB), the root mean square error (RMSE), and coverage probability (CP) of the 95% confidence intervals for the estimators of the parameters. The results indicated that the ABs and RMSEs decrease as the sample size increases. These results clearly show the accuracy and the consistency of the maximum likelihood estimators. Also, the CPs are quite close to the nominal value. Thus, the maximum likelihood technique works very well to estimate the parameters of the EOFNH distribution.Table 1
Monte Carlo simulation results: AB and RMSE and CP.
I II III Parameter n AB RMSE CP AB RMSE CP AB RMSE CP α 25 0.5406 1.9394 0.9780 24.9701 133.9291 0.9160 3.1057 17.0755 0.8780 75 0.1710 0.5134 0.9470 12.5503 76.6534 0.9030 1.1174 3.9980 0.8850 150 0.0919 0.3459 0.9120 5.6552 41.2149 0.9200 0.4664 1.3878 0.9020 300 0.0242 0.2526 0.8670 2.1667 5.1107 0.9540 0.2038 0.6941 0.9280 600 -0.0043 0.2076 0.8300 0.9715 2.8387 0.9640 0.1071 0.4109 0.9430 800 -0.0010 0.1794 0.8670 0.7123 2.3503 0.9560 0.0623 0.3374 0.9430  β 25 0.0040 0.2147 0.9620 0.4450 1.3048 0.9810 0.0212 0.1894 0.9920 75 0.0488 0.3523 0.9630 0.2138 0.5569 0.9880 0.0163 0.1195 0.9830 150 0.0740 0.4896 0.9620 0.1105 0.3330 0.9890 0.0117 0.0705 0.9680 300 0.0890 0.4095 0.975 0.0633 0.2147 0.9720 0.0068 0.0438 0.9620 600 0.066 0.2745 0.9670 0.0324 0.1275 0.9630 0.0032 0.0296 0.9520 800 0.0357 0.1378 0.9730 0.0264 0.1105 0.9620 0.0021 0.0255 0.9510  λ 25 12.5655 249.9064 0.9340 3.7073 32.1823 0.8930 4.9031 47.5403 0.8170 75 0.6886 2.7878 0.8530 1.2894 15.0985 0.8850 1.4279 12.6625 0.8460 150 0.3001 0.9320 0.8280 0.1493 2.7383 0.9200 0.2900 1.9911 0.8740 300 0.1265 0.5992 0.8020 0.0298 0.1599 0.9490 0.0726 0.2735 0.8950 600 0.0400 0.4286 0.7900 0.0075 0.0931 0.9800 0.0375 0.1450 0.9320 800 0.0351 0.3629 0.8400 0.0053 0.0856 0.9670 0.0227 0.1175 0.9300  θ 25 0.0856 0.3163 0.9999 0.4125 1.7492 0.8970 0.1846 0.5637 0.9810 75 0.0616 0.2515 0.9800 0.1192 0.8350 0.8370 0.0725 0.3927 0.9290 150 0.0599 0.2400 0.9500 0.1242 0.7001 0.8280 0.0599 0.3320 0.9310 300 0.0790 0.2422 0.9200 0.0602 0.4882 0.8550 0.0435 0.2675 0.9260 600 0.0824 0.2313 0.8840 0.0506 0.3537 0.8940 0.1230 0.1618 0.9400 800 0.0601 0.1957 0.9040 0.0396 0.3043 0.9130 0.0179 0.1502 0.9410


## 7. Application

In this section, the application of the EOFNH and EOFW distributions is illustrated using a real data set. The data consists of the Fatigue time of 101 6061-T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set given in Table2 can be found in Birnbaum and Saunders [22]. The performance of the EOFNH and EOFW distributions is compared with that of the odd Fréchet Nadarajah-Haghighi (OFNH) and odd Fréchet Weibull (OFW) distributions using the Akaike information criterion (AIC) [23, 24] and Bayesian information criterion (BIC) [25]. The maximum likelihood estimates of the parameters of the fitted distributions are computed by maximizing the log-likelihood function via the subroutinemle2 uisng thebbmle package in the R software [26].Table 2
Fatigue time of 101 6061-T6 aluminum coupons.
70 90 96 97 99 100 103 104 104 105 107 108 108 108 109 109 112 112 113 114 114 114 116 119 120 120 120 121 121 123 124 124 124 124 124 128 128 129 129 130 130 130 131 131 131 131 131 132 132 132 133 134 134 134 134 134 136 136 137 138 138 138 139 139 141 141 142 142 142 142 142 142 144 144 145 146 148 148 149 151 151 152 155 156 157 157 157 157 158 159 162 163 163 164 166 166 168 170 174 196 212The PDFs of the OFNH and OFW distributions are, respectively, given by(44)fx=βλθ1+λxβ-1e1-1+λxβ1-1-e1-1+λxβθ-11-e1-1+λxβθ+1e-1-e1-1+λxβ-1-1θ,x>0,and(45)fx=βλθxβ-1e-λxβ1-1-e-λxβθ-11-e-λxθ+1e-1-e-λxβ-1-1θ,x>0.Table 3 displays the maximum likelihood estimates of the parameters of the EOFNH, EOFW, OFNH, and OFW distributions with their corresponding standard errors in bracket and the model selection criteria. The results revealed that the EOFNH distribution provided the best fit for the data since it has the least values of AIC and the BIC. The EOFW distribution also performed better than the OFNH and OFW distributions. The OFNH distribution is a submodel of the EOFNH distribution with α=1. Hence, testing H0:α=1 versus Ha:α≠1 using the LR test gave a test statistic of 6.4703 with corresponding p-value of 0.01097. This implies that there is enough evidence to reject H0 at the 5% significance level and conclude that the EOFNH distribution provides better fit to the data than the OFNH distribution. Similarly, the LR test was performed to compare the performances of the EOFW distribution and the OFW distribution. The analysis gave a test statistic of 5.1065 with a corresponding p-value of 0.0238. This implies that the EOFW distribution performs better than the OFW distribution at the 5% significance level.Table 3
Maximum likelihood estimates and goodness-of-fit statistics.
Model Parameter estimates - l AIC BIC EOFNH α ^ = 1.5838  ( 0.2023 ) β ^ = 1.1413  ( 0.4131 ) -470.3600 948.7255 959.1860 λ ^ = 0.0071  ( 0.0041 ) θ ^ = 2.7505  ( 0.1088 )  OFNH α ^ = 0.4650  ( 0.4204 ) λ ^ = 0.0174  ( 0.0263 ) -473.6000 953.1958 961.0412 θ ^ = 4.7456  ( 1.8584 )  EOFW α ^ = 2.2281  ( 1.0985 ) β ^ = 1.0205  ( 0.3152 ) -471.2600 950.5259 960.9864 λ ^ = 0.0099  ( 0.0166 ) θ ^ = 2.3066  ( 0.6716 )  OFW β ^ = 0.2785  ( 0.0181 ) λ ^ = 0.1823  ( 0.0156 ) -473.8200 953.6325 961.4778 θ ^ = 13.2247  ( 0.0007 )Figure5 displays the histogram of the data with the fitted densities and the empirical CDF with the fitted CDFs.Figure 5
Plots of histogram of data and fitted densities; and empirical CDF and fitted CDF.The P-P plots of the fitted distributions are displayed in Figure6.Figure 6
P-P plots of fitted distributions.


## 8. Conclusion

The development of new statistical distribution plays a critical role in parametric statistical inference. Because of this, researchers in the field of distribution theory attempt to develop generators for generalizing the existing distributions. In line with this, the study developed and studied a new class of distributions called the EOF-G family. The statistical properties including the moments, incomplete moments, generating function, entropies, and order statistics are derived. The maximum likelihood method is used to develop estimators for the parameters of the new family. The application of the special distributions developed using the EOF-G family is demonstrated using a real data set and the result compared with other existing distributions. From the application, it is evident that the special models developed from the EOF-G family can provide reasonable parametric fit to a given data set. Hence, it is hoped that the new class of distributions will attract wider applications in different fields of study.



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