The paper introduces the hybrid odd exponential-Φ (HOE-Φ) family, a novel framework for generating a continuous distribution characterized by an additional parameter. The extensive statistical properties of this family are derived and explored in detail. Parameter estimation is performed using the maximum likelihood estimation technique. The efficacy and versatility of the proposed model are demonstrated through a comparative analysis involving two distinct real-world datasets.

In the last two decades, increasing interest has been in constructing new statistical distributions to fit newer and broader datasets. One of the modern themes in this regard is to add one or more parameters to an already defined distribution. This procedure proved very efficient in providing more flexibility to fit more datasets to these distributions.

Many statisticians and researchers have presented several methods and techniques in this direction. Marshal and Olkin1 presented the MO-G generator. They defined an extended two-parameter exponential distribution, which proved to be a good alternative to some well-known two-parameter models (e.g., Weibull, gamma, etc.). Eugene et al.2 defined and studied the beta-G family of distributions. They showed that the special submodel, beta-normal, has great flexibility to model symmetric heavy-tailed, skewed, and bimodal distributions. Cordeiro and de Castro3 used the Kumaraswamy distribution to build the Kumaraswamy-G family. The authors in Ref. 3 studied many interesting properties of this family. In particular, they showed that the moments of the distributions of the Kumaraswamy-G family are indeed linear functions of the probability-weighted moments of the parent distributions. Alzaatreh et al.4 introduced a new method called the T-X transformer to generate new continuous distributions. They showed that the resulting distributions are indeed weighted hazard functions of X. Bourguignon et al.5 Studied the mathematical properties of the wider Weibull-G family. They derived many explicit expressions for many quantities in this family such as quantile functions, ordinary and incomplete moments, generating functions, and order statistics. Khaleel et al.6 introduced and studied the Marshall–Olkin Topp Leone-G family. When used to fit real-world phenomena, this family proved to be effective in creating robust compound probability models. Many other families and methods were studied by many other researchers; for example, Refs. 7–9. Our goal is to create a new class of continuous distributions with greater flexibility than many of the standard and generalized distributions. It is also noticed that the new family provides high skewness, as we shall see in Sec. II.

Now, we are going to construct a family of distributions that generalizes the exponential distribution, called the hybrid odd exponential family, or the HOE-Φ family for short. Following Ref. 4, let r(t) = λ exp(−λt) where t, λ > 0, be the exponential probability density function. Let W(Φ(z, ξ)) be the function of the cumulative density function Φ(z, ξ) of a random variable Z that has the form
$W(Φ(z,ξ))=Φ(z,ξ)1−Φ(z,ξ)log11−Φ(z,ξ).$
(1.1)
One can easily check that W(Φ(z, ξ)) satisfies the conditions of Ref. 4, Eq. (2.1). In a work that is to appear, the second author and his collaborators chose to call this function the hybrid odd function, hence the name of our family. Now, integrating r(t) from zero to W(Φ(z, ξ)), we get the HOE-Φ cumulative density function Ψ(zλ, ξ),
$Ψ(z;λ,ξ)=∫0W(Φ(z,ξ))r(t)dt=∫0Φ(z,ξ)1−Φ(z,ξ)log11−Φ(z,ξ)λexp(−λt)dt=−exp(−λt)|0Φ(z,ξ)1−Φ(z,ξ)log11−Φ(z,ξ)=1−expλΦ(z,ξ)1−Φ(z,ξ)log(1−Φ(z,ξ)).$
(1.2)
If we differentiate with respect to z, we obtain the HOE-Φ probability density function ψ(z, λ, ξ),
$ψ(z;λ,ξ)=λΦ(z,ξ)−log(1−Φ(z,ξ))(1−Φ(z,ξ))2ϕ(z,ξ)×expλΦ(z,ξ)1−Φ(z,ξ)log(1−Φ(z,ξ)),$
(1.3)
where ϕ(z, ξ) is the probability density function of the random variable Z. The extra parameter λ is a shape parameter that, as we shall see later, provides more flexibility to the random variable Z. Hereafter, we denote a random variable Z with a pdf given in 1.3 by Z ∼ HOE − Φ(λ, ξ). Equations (1.2) and (1.3) can be written alternatively as
$Ψ(z;λ,ξ)=1−(1−Φ(z,ξ))λΦ(z,ξ)1−Φ(z,ξ),$
(1.4)
and
$ψ(z;λ,ξ)=λΦ(z,ξ)−log(1−Φ(z,ξ))(1−Φ(z,ξ))2ϕ(z,ξ)×1−(1−Φ(z,ξ))λΦ(z,ξ)1−Φ(z,ξ),$
(1.5)
respectively. The hazard rate function is easily computed to be
$h(z;λ,ξ)=λΦ(z,ξ)−log(1−Φ(z,ξ))(1−Φ(z,ξ))2ϕ(z,ξ).$
(1.6)

The remainder of the paper is structured as follows: In Sec. II, we examine three distributions resulting from the construction above. We derive some mathematical properties of the HOE-Φ family in Sec. III, including the quantile function, series representation of the density function, moments, moments generating function, and incomplete moments. We examine entropies and order statistics in Sec. IV. We estimate the HOE-Φ family parameters in Sec. V using the maximum likelihood estimation method. We provide a simulation study in Sec. VI. In Sec. VII, we study applications of the HOE-Φ family on two real datasets. We conclude the paper in Sec. VIII.

In this section, we give three special models of the HOE-Φ generator. We also give plots for their density functions. We include the plots of the cumulative density function, the survival function, and the hazard rate function for one of the special models that will be relevant for our study later.

In Eq. (1.3), if we set ϕ(z, ξ) and Φ(z, ξ) to be the pdf and cdf of the exponential distribution exp(θ), respectively, where ξ = θ, we get the hybrid odd exponential exponential (HOEE) distribution density function (z, λ, θ > 0),
$ψ(z;λ,θ)=λθ1+θz−exp(−θz)exp(−θz)exp−λθz1−exp(−θz)exp(−θz).$
(2.1)
Similarly, the cumulative density function, the survival function, and the hazard rate function are given by
$Ψ(z;λ,θ)=1−exp−λθz1−exp(−θz)exp(−θz),$
(2.2)
$S(z;λ,θ)=exp−λθz1−exp(−θz)exp(−θz),$
(2.3)
and
$h(z;λ,θ)=λθ1+θz−exp(−θz)exp(−θz),$
(2.4)
respectively. Plots of the density function, the cdf, the survival function, and the hazard rate function for selected values of parameters are shown in Fig. 1.
FIG. 1.

Pdf, cdf, survival, and hazard rate functions of the HOEE distribution for selected values of parameters.

FIG. 1.

Pdf, cdf, survival, and hazard rate functions of the HOEE distribution for selected values of parameters.

Close modal
In Eq. (1.3), if we set ϕ(z, ξ) and Φ(z, ξ) to be the pdf and cdf of the Weibull distribution W(a, b), respectively, where ξ = (a,b)T, we get the hybrid odd exponential Weibull (HOEW) distribution density function (z, λ, a, b > 0),
$ψ(z;λ,a,b)=λab1+zba−e−zbae−zbae−λzba1−e−zbae−zba.$
(2.5)
Similarly, the cumulative density function, the survival function, and the hazard rate function are given by
$Ψ(z;λ,a,b)=1−e−λzba1−e−zbae−zba,$
(2.6)
$S(z;λ,a,b)=e−λzba1−e−zbae−zba,$
(2.7)
and
$h(z;λ,a,b)=λab1+zba−e−zbae−zba,$
(2.8)
respectively. Plots of the density function for selected values of parameters are shown in Fig. 2.
FIG. 2.

Pdf of the HOEW distribution for selected values of parameters.

FIG. 2.

Pdf of the HOEW distribution for selected values of parameters.

Close modal
In Eq. (1.3), if we set ϕ(z, ξ) and Φ(z, ξ) to be the pdf and cdf of the uniform distribution U(a, b), respectively, where ξ = (a,b)T, we get the hybrid odd exponential uniform (HOEU) distribution density function where λ > 0, − < a < b < and azb,
$ψ(z;λ,a,b)=λb−az−ab−a−log1−z−ab−a1−z−ab−a2×expλz−ab−a1−z−ab−alog1−z−ab−a.$
(2.9)
Similarly, the cumulative density function, the survival function and the hazard rate function are given by
$Ψ(z;λ,a,b)=1−expλz−ab−a1−z−ab−alog1−z−ab−a,$
(2.10)
$S(z;λ,a,b)=expλz−ab−a1−z−ab−alog1−z−ab−a,$
(2.11)
and
$h(z;λ,a,b)=λb−az−ab−a−log1−z−ab−a1−z−ab−a2,$
(2.12)
respectively. Plots of the density function for selected values of parameters are shown in Fig. 3.
FIG. 3.

Pdf of the HOEU distribution for selected values of parameters.

FIG. 3.

Pdf of the HOEU distribution for selected values of parameters.

Close modal
The quantile function of the HOE-Φ family can be found by inverting Eq. (1.2). If we denote this inverse by Q(u), where z = Q(u) = Ψ−1(u), the quantile function associated with Eq. (1.2) is given by
$z=Q(u)=QΦ11−λlog(1−u)W−11λlog11−u,$
(3.1)
where QΦ(u) = Φ−1(z) is the quantile function associated with Φ(z, ξ), and Wr is the r-Lambert function (see Ref. 10, Theorem 3). Equation (3.1) can be utilized both theoretically to derive many properties such as median, or in applications such as simulation studies.
In this subsection, we will derive useful expansions for the cdf and the pdf given in Eqs. (1.2) and (1.3), respectively. For an arbitrary cumulative density function Φ(z), we can obtain a new distribution called the exponentiated-Φ distribution whose cdf and pdf are Φc(z) and c Φc−1(z)ϕ(z), respectively, where c > 0 is the exponent and ϕ(z) is the pdf of Φ(z). We will denote the cdf and the pdf of the exponentiated-Φ distribution with exponent c > 0, by Tc(z) and tc(z), respectively. Using the Taylor expansion of the exponential function in Eq. (1.3), we get
$ψ(z;λ,ξ)=∑l=0∞λl+1l!Φl+1(z,ξ)ϕ(z,ξ)(1−Φ(z,ξ))l+2[log(1−Φ(z,ξ))]l−λl+1l!Φl(z,ξ)ϕ(z,ξ)(1−Φ(z,ξ))l+2[log(1−Φ(z,ξ))]l+1.$
(3.2)
Using the expansion $log(1−z)=−∑j=0∞zj+1j+1$ combined with the result in Ref. 11, 0.314 and the expansion $1(1−z)s=∑j=0∞s+j−1jzj$ with some algebraic simplification, we obtain the following form for ψ(zλ, ξ):
$ψ(z;λ,ξ)=∑l,m,n=0∞λl+1l![(−1)lcm+(−1)l+2dm]l+n+1n×Φ2l+m+n+1(z,ξ)ϕ(z,ξ),$
(3.3)
where ci and di are determined by the recurrence relations, c0 = d0 = 1, $ci=1i∑k=1i(k(l+1)−i)1k+1ci−k$ and $di=1i∑k=1i(k(l+2)−i)1k+1di−k$. Equation (3.3) can be written as
$ψ(z;λ,ξ)=∑l,m,n=0∞νl,m,nt2l+m+n+2(z,ξ),$
(3.4)
where $νi,j,k=λi+1i!(2i+j+k+2)[(−1)icj+(−1)i+2dj]i+k+1k$. Integrating Eq. (3.4) gives a series expansion of Eq. (1.2) as
$Ψ(z;λ,ξ)=∑l,m,n=0∞νl,m,nT2l+m+n+2(z,ξ).$
(3.5)
This shows that the HOE-Φ pdf function is a triple linear combination of the exponentiated-Φ densities. Equations (3.4) and (3.5) will be very useful in deriving more properties of the HOE-Φ family.
With the aid of 3.4, the r-moment of the HOE-Φ family is computed as follows:
$μr=E[Zr]=∑l,m,n=0∞νl,m,nE[Y2l+m+n+2r],$
(3.6)
where $E[Yjr]=j∫−∞∞zrΦj−1(z,ξ)ϕ(z,ξ)dz$ is the moment of the exponentiated-Φ distribution with exponent j of order r. This integral can be evaluated numerically via the equation $E[Yjr]=j∫01uj−1QΦr(u,ξ)du$, where QΦ(u, ξ) is the quantile function of the baseline distribution. The moments generating function M(t) of a random variable Z is M(t) = E(etZ). In our case, we get
$M(t)=∑l,m,n=0∞νl,m,nM2l+m+n+2(t),$
(3.7)
where Mj(t) is the moments generating function of the exponentiated-Φ distribution with exponent j, given by $Mj(t)=j∫−∞∞etzΦj−1(z,ξ)ϕ(z,ξ)dz=j∫01uj−1etQΦ(u,ξ)du$. In most cases, the integrand will have a complicated form and will require a numerical method to be carried out.
The rth incomplete moment mr(w) of the HOE-Φ density function 1.3 is given by
$mr(w)=∫−∞wzrψ(z;λ,ξ)dz.$
(3.8)
Substituting Eq. (3.4) in Eq. (3.8), we get
$mr(w)=∑l,m,n=0∞νl,m,nmr2l+m+n+2(w),$
(3.9)
where $mrj(w)=j∫−∞wzrΦj−1(z,ξ)ϕ(z,ξ)dz$ is the incomplete moment of the exponentiated-Φ distribution with exponent j of order r. This integral can be evaluated numerically via the equation $mrj(w)=j∫0Φ(w,ξ)uj−1QΦr(u,ξ)du$.

An entropy is a tool to estimate the uncertainty in a random variable Z. In this section, we will derive formulas for the Rényi entropy12 and the Shannon entropy.13

The Rényi entropy of a random variable Z with pdf ψ(z) is computed via the formula
$IR(b)=11−blog∫0∞ψb(z)dz,$
(4.1)
where b > 0 and b ≠ 0. Before we give the formula for the Rényi entropy, we state the following.

Proposition 4.1.
Given a random variable Z and a pdf provided by Eq. (1.3), the expansion of ψb(zλ, ξ) for b > 0 may be found to be
$ψb(z;λ,ξ)=∑l,m,n=0∞ηl,m,nΦ2l+m+n+b(z,ξ)ϕb(z,ξ),$
(4.2)
where
$ηl,m,n=∑p=0∞(−1)2p+lλl+bblcml!bpl+n+2b−1n,$
(4.3)
and the ci’s are given by the recurrence relation (c0 = 1)
$ci=1i∑k=1i(k(m+l+1)−i)1k+1ci−k.$
(4.4)

Now, if we substitute Eq. (4.2) in Eq. (4.1), we obtain the following formula for the Rényi entropy of the HOE-Φ family:
$IR(b)=11−blog∑l,m,n=0∞ηl,m,nIl,m,n(b,ξ),$
(4.5)
where $Il,m,n(b,ξ)=∫0∞Φ2l+m+n+b(z,ξ)ϕb(z,ξ)dz$. For the majority of parent distributions, the last integral can be computed numerically.
The Shannon entropy of a random variable Z with pdf ψ(z) is, by definition, equal to E{−log[ψ(Z)]}. The Shannon entropy becomes the Rényi entropy in the limit b ↑ 1. For the HOE-Φ family, the formula of the Shannon entropy becomes
$E{−log[ψ(Z;λ,ξ)]}=log1λ−E{log[Φ(Z,ξ)]}+2E{log[1−Φ(Z,ξ)]}−Elog1−log[1−Φ(Z,ξ)]Φ(Z,ξ)−E{log[ϕ(Z,ξ)]}−λEΦ(Z,ξ)1−Φ(Z,ξ)log[1−Φ(Z,ξ)].$
(4.6)
Before we continue the derivation, let us define the quantity B(b1, b2, b3) to be
$B(b1,b2,b3)=∫01yb1(1−y)b2[log(1−y)]b3dy.$
(4.7)
We have the following.

Proposition 4.2.
Let Z be a random variable such that ZHOE-Φ(λ, ξ). Then, the Shannon entropy of Z is given by the formula
$E{−log[ψ(Z;λ,ξ)]}=log1λ−E{log[ϕ(Z,ξ)]}+∑l=0∞αl2B(l+1,l+2,l+1)−2B(l,l+2,l+2)−λB(l+2,l+3,l+1)+λB(l+1,l+3,l+2)+∑l,m=0∞βl,m[B(l,l−m+2,l+1)−B(l+1,l−m+2,l)]+∑l,m,n=0∞γl,m,nB(l−2m−n+1,l+2,l+2m+n)−B(l−2m−n+2,l+2,l+2m+n−1),$
(4.8)
where $αi=λi+1i!,βi,j=(−1)2j+1jαi$, and $γi,j,k=j+k−1kβi,j$.

The term E{log[ϕ(Z, ξ)]} can be computed numerically via some suitable program.

Order statistics is an essential notion in almost all branches of statistical theory. Let Z1, Z2, …, Zs be a random sample from the HOE-Φ family. The density function of the ith-order statistic Zi:s is
$ψi:s(z;λ,ξ)=s!(i−1)!(s−i)!∑j=0s−i(−1)js−ijψ(z;λ,ξ)×Ψj+i−1(z;λ,ξ).$
(4.9)
Using the binomial expansion, the term ψ(zλ, ξj+i−1(zλ, ξ) can be written as
$ψ(z;λ,ξ)Ψj+i−1(z;λ,ξ)=∑k=0j+i−1(−1)kj+i−1kλΦ(z,ξ)−log(1−Φ(z,ξ))(1−Φ(z,ξ))2ϕ(z,ξ)×expλ(i+j)Φ(z,ξ)1−Φ(z,ξ)log(1−Φ(z,ξ)).$
(4.10)
Some expansions and algebraic simplifications will give
$ψ(z;λ,ξ)Ψj+i−1(z;λ,ξ)=∑k=0j+i−1∑l,m,n=0∞βi,j,k,l,m,nt2l+m+n+2(z,ξ),$
(4.11)
where
$βi,j,k,l,m,n=(−1)k(i+j)ll!(2l+m+n+2)[(−1)lcm+(−1)l+2dm]×j+i−1kl+n+1n,$
(4.12)
and the cp’s, dp’s are those of Eq. (3.3). Putting everything together, we have
$ψi:s(z;λ,ξ)=R∑j=0s−i∑k=0j+i−1∑l,m,n=0∞γi,j,k,l,m,nt2l+m+n+2(z,ξ),$
(4.13)
where $R=s!(i−1)!(s−i)!$ and $γi,j,k,l,m,n=(−1)js−ijβi,j,k,l,m,n$. Equation (4.13) shows that the density function of the ith-order statistic is a (multi) linear combination of exponentiated-Φ densities. This fact facilitates many calculations involving the order statistics, as they are basically calculations involving the exponentiated-Φ densities.
We can use many ways to estimate the parameters of a distribution. One such method is the maximum likelihood estimation method. This method is heavily used in statistics because it enjoys desirable properties, such as providing approximations for relatively large samples. There are many statistical measures that estimate the efficiency of fitting a model, including the maximum log-likelihood $(l̂max)$, Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Anderson–Darling (A*), and Cramér–von Mises (W*). In general, smaller values of the measures above indicate that the model is better at fitting the dataset compared with other models with larger values for the same set of measures. Let Z1, Z2, …, Zm be a random sample of size m taken from the HOE-Φ distribution. Let θ = (λ,ξ)T be the parameter vector, where $ξ=(ξ1,ξ2,…,ξn)T$ is the parameter vector of the parent distribution. The log-likelihood function of the HOE-Φ distribution is
$lm(θ)=mlog(λ)+∑i=1mlog[Φ(zi,ξ)−log(1−Φ(zi,ξ))]−2∑i=1mlog[1−Φ(zi,ξ)]+∑i=1mlog[ϕ(zi,ξ)]+λ∑i=1mΦ(zi,ξ)1−Φ(zi,ξ)log[1−Φ(zi,ξ)].$
(5.1)
The score function’s components $Um(θ)=(∂lm(θ)∂λ,∂lm(θ)∂ξ1,∂lm(θ)∂ξ2,…,∂lm(θ)∂ξn)T$ are given by (k = 1, 2, …, n)
$∂lm(θ)∂λ=mλ+∑i=1mΦ(zi,ξ)1−Φ(zi,ξ)log[1−Φ(zi,ξ)],$
(5.2)
and
$∂lm(θ)∂ξk=∑i=1m2−Φ(zi,ξ)[1−Φ(zi,ξ)]{Φ(zi,ξ)−log[1−Φ(zi,ξ)]}+2∑i=1mΦξk(z,ξ)1−Φ(z,ξ)+∑i=1mϕξk(z,ξ)ϕ(z,ξ)−λ∑i=1mΦ(z,ξ)Φξk(z,ξ)[1−Φ(z,ξ)]2+λ∑i=1mΦξk(z,ξ)−log[1−Φ(z,ξ)][1−Φ(z,ξ)]2,$
(5.3)
where $Φξk(z,ξ)=∂Φ(z,ξ)∂ξk$ and $ϕξk(z,ξ)=∂ϕ(z,ξ)∂ξk$. Setting Eqs. (5.2) and (5.3) equal to zero and solving the resulting system produces the desired estimators. The resulting system will have a rather complicated non-linear form for most parent distributions, so a computer program (MATLAB is a good candidate) is necessary to solve these equations.

A Monte-Carlo simulation procedure is used in this section to estimate the parameters of the HOEE distribution using the MLE method and with the aid of the R programming language under the square error loss function. For Monte-Carlo experiments, HOEE data are used to generate 3000 random samples, where z is the HOEE lifetime for various actual parameter values and sample sizes n as (30, 50, 100, 200). The most effective estimation methods are those that minimize the estimators’ bias and root mean squared error (RMSE). Tables I and II summarize the simulation results of the MLE method. We include the mean, the RMSE, and the bias as measurements of the efficiency of these estimations. These tables show that RMSE and bias go smaller as we increase the sample size n. We also notice that for specific choices of the parameters, RMSE and bias are identically zero. The parameters mean we now have obvious behaviors.

TABLE I.

MLE estimation for the parameters of the HOEE distribution using a Monte Carlo method.

θλnMeanRMSEBias
2.5 0.005 30 θ 0.0084 0.0095 0.0034
λ 2.3926 0.2909 −0.1073
50 θ 0.0070 0.0065 0.0020
λ 2.4172 0.2368 −0.0827
100 θ 0.0059 0.0030 0.0009
λ 2.4274 0.1672 −0.0725
200 θ 0.0053 0.0017 0.0003
λ 2.4530 0.1156 −0.0469
2.5 0.03 30 θ 0.0583 0.1086 0.0283
λ 2.4216 0.5247 −0.0783
50 θ 0.0447 0.0449 0.0147
λ 2.4538 0.4305 −0.0461
100 θ 0.0374 0.0233 0.0074
λ 2.4713 0.3216 −0.0286
200 θ 0.0336 0.0139 0.0036
λ 2.4863 0.2290 −0.0136
2.5 0.08 30 θ 0.2850 5.4634 0.2050
λ 2.4795 0.6802 −0.0204
50 θ 0.1225 0.1655 0.0425
λ 2.5091 0.5568 0.0091
100 θ 0.0992 0.0702 0.0192
λ 2.4984 0.4055 −0.0015
200 θ 0.0885 0.0401 0.0085
λ 2.5046 0.2890 0.0046
2.5 0.1 30 θ 0.4259 8.8161 0.3259
λ 2.4724 0.7149 −0.0275
50 θ 0.1573 0.2408 0.0573
λ 2.5186 0.5965 0.0186
100 θ 0.1255 0.0927 0.0255
λ 2.4965 0.4225 −0.0034
200 θ 0.1113 0.0516 0.0113
λ 2.5036 0.3026 0.0036
1.1 0.005 30 θ 0.0070 0.0057 0.0020
λ 1.1023 0.0777 0.0023
50 θ 0.0055 0.0016 0.0005
λ 1.1143 0.0443 0.0143
100 θ 0.0057 0.0016 0.0007
λ 1.0982 0.0280 −0.0017
200 θ 0.0050
λ 1.1000
1.1 0.03 30 θ 0.0551 0.1136 0.0251
λ 1.0842 0.2304 −0.0157
50 θ 0.0412 0.0371 0.0112
λ 1.0985 0.1910 −0.0014
100 θ 0.0356 0.0228 0.0056
λ 1.1018 0.1453 0.0018
200 θ 0.0330 0.0139 0.0030
λ 1.0985 0.1030 −0.0014
θλnMeanRMSEBias
2.5 0.005 30 θ 0.0084 0.0095 0.0034
λ 2.3926 0.2909 −0.1073
50 θ 0.0070 0.0065 0.0020
λ 2.4172 0.2368 −0.0827
100 θ 0.0059 0.0030 0.0009
λ 2.4274 0.1672 −0.0725
200 θ 0.0053 0.0017 0.0003
λ 2.4530 0.1156 −0.0469
2.5 0.03 30 θ 0.0583 0.1086 0.0283
λ 2.4216 0.5247 −0.0783
50 θ 0.0447 0.0449 0.0147
λ 2.4538 0.4305 −0.0461
100 θ 0.0374 0.0233 0.0074
λ 2.4713 0.3216 −0.0286
200 θ 0.0336 0.0139 0.0036
λ 2.4863 0.2290 −0.0136
2.5 0.08 30 θ 0.2850 5.4634 0.2050
λ 2.4795 0.6802 −0.0204
50 θ 0.1225 0.1655 0.0425
λ 2.5091 0.5568 0.0091
100 θ 0.0992 0.0702 0.0192
λ 2.4984 0.4055 −0.0015
200 θ 0.0885 0.0401 0.0085
λ 2.5046 0.2890 0.0046
2.5 0.1 30 θ 0.4259 8.8161 0.3259
λ 2.4724 0.7149 −0.0275
50 θ 0.1573 0.2408 0.0573
λ 2.5186 0.5965 0.0186
100 θ 0.1255 0.0927 0.0255
λ 2.4965 0.4225 −0.0034
200 θ 0.1113 0.0516 0.0113
λ 2.5036 0.3026 0.0036
1.1 0.005 30 θ 0.0070 0.0057 0.0020
λ 1.1023 0.0777 0.0023
50 θ 0.0055 0.0016 0.0005
λ 1.1143 0.0443 0.0143
100 θ 0.0057 0.0016 0.0007
λ 1.0982 0.0280 −0.0017
200 θ 0.0050
λ 1.1000
1.1 0.03 30 θ 0.0551 0.1136 0.0251
λ 1.0842 0.2304 −0.0157
50 θ 0.0412 0.0371 0.0112
λ 1.0985 0.1910 −0.0014
100 θ 0.0356 0.0228 0.0056
λ 1.1018 0.1453 0.0018
200 θ 0.0330 0.0139 0.0030
λ 1.0985 0.1030 −0.0014
TABLE II.

Table I continued.

 1.1 0.08 30 θ 0.3133 6.0228 0.2333 λ 1.0959 0.2874 −0.0040 50 θ 0.1144 0.1241 0.0344 λ 1.1142 0.2453 0.0142 100 θ 0.0958 0.0698 0.0158 λ 1.1106 0.1795 0.0106 200 θ 0.0871 0.0404 0.0071 λ 1.1079 0.1316 0.0079 1.1 0.1 30 θ 0.4713 9.8935 0.3713 λ 1.0975 0.3098 −0.0024 50 θ 0.149 0.1729 0.0490 λ 1.1080 0.2538 0.0080 100 θ 0.1205 0.0929 0.0205 λ 1.1160 0.1953 0.0160 200 θ 0.1097 0.0523 0.0097 λ 1.1075 0.1383 0.0075
 1.1 0.08 30 θ 0.3133 6.0228 0.2333 λ 1.0959 0.2874 −0.0040 50 θ 0.1144 0.1241 0.0344 λ 1.1142 0.2453 0.0142 100 θ 0.0958 0.0698 0.0158 λ 1.1106 0.1795 0.0106 200 θ 0.0871 0.0404 0.0071 λ 1.1079 0.1316 0.0079 1.1 0.1 30 θ 0.4713 9.8935 0.3713 λ 1.0975 0.3098 −0.0024 50 θ 0.149 0.1729 0.0490 λ 1.1080 0.2538 0.0080 100 θ 0.1205 0.0929 0.0205 λ 1.1160 0.1953 0.0160 200 θ 0.1097 0.0523 0.0097 λ 1.1075 0.1383 0.0075

In this section, we demonstrate the potentiality of the HOE-Φ family by providing applications to two real datasets. We compare the results of the hybrid odd exponential exponential (HOEE) distribution given in Eq. (2.1) with other models such as [0,1] truncated exponentiated exponential exponential distribution ([0,1] TEEE),14 Beta exponential distribution (BeE),15 Kumaraswamy exponential distribution (KuE),3 Exponentiated Generalized exponential distribution (EGE),16 Weibull exponential distribution (WeE),5 Gompertz exponential distribution (GoE),17 Marshal Olkin exponential distribution (MOE),1 and the exponential distribution (E). The efficiency of fitting the datasets was judged by the Akaike information criterion (AIC), the consistent Akaike information criterion (CAIC), the Bayesian information criterion (BIC), the Hannan–Quinn information criterion (HQIC), the Cramér–von Mises criterion (W*), and the Anderson–Darling criterion (A*).

This dataset is obtained by workers at the UK National Physical Laboratory, and it demonstrates the strengths of 1.5 cm glass fibers. This dataset was previously used by Smith and Naylor,18 Bourguignon,19 Khaleel,6 and Oguntunde.20,21 The observations are
$0.55,0.74,0.77,0.81,0.84,1.24,0.93,1.04,1.11,1.13,1.30,1.25,1.27,1.28,1.29,1.48,1.36,1.39,1.42,1.48,1.51,1.49,1.49,1.50,1.50,1.55,1.52,1.53,1.54,1.55,1.61,1.58,1.59,1.60,1.61,1.63,1.61,1.61,1.62,1.62,1.67,1.64,1.66,1.66,1.66,1.70,1.68,1.68,1.69,1.70,1.78,1.73,1.76,1.76,1.77,1.89,1.81,1.82,1.84,1.84,2.00,2.01,2.24.$
Table III shows the performance of the HOEE distribution compared to the other models mentioned earlier. As mentioned earlier, the model with the lowest values of AIC, CAIC, BIC, and HQIC is considered to be superior to the other models. The first row in Table III confirms that the HOEE model is the best of the competing models to fit the dataset above. Figure 4 shows the plots of the empirical histogram of the observed data, the plot of the empirical cdf, the P–P plot of the HOEE distribution, and the Q–Q plot of the HOEE distribution for the glass fibers dataset.
TABLE III.

The performance of the HOEE distribution compared to the other models for the glass fibers dataset.

DistributionsEstimatesAICCAICBICHQICW*A*
HOEE λ = 0.003 34.198 34.398 38.484 35.883 0.168 0.950
θ = 2.602
γ = (−)
[0,1] TEEE λ = 70.744 69.996 70.402 76.425 72.524 0.804 4.383
θ = 33.677
γ = 0.038
BeE λ = 17.526 54.257 54.663 60.686 56.785 0.574 3.148
θ = 22.625
γ = 0.386
KuE λ = 9.096 41.012 41.419 47.442 43.541 0.339 1.860
θ = 60.145
γ = 0.627
EGE λ = 1.616 68.766 69.173 75.196 71.295 0.786 4.287
θ = 31.368
γ = 1.616
WeE λ = 5.780 36.413 36.820 42.843 38.942 0.237 1.303
θ = 0.832
γ = 0.511
GoE λ = 0.021 35.617 36.024 42.046 38.146 0.144 0.842
θ = 8.577
γ = 0.423
MOE λ = 168.723 55.363 55.563 59.649 57.049 0.340 1.874
θ = 3.530
γ = (−)
λ = 0.663 179.660 179.726 182.803 180.503 0.570 3.127
θ = (−)
γ = (−)
DistributionsEstimatesAICCAICBICHQICW*A*
HOEE λ = 0.003 34.198 34.398 38.484 35.883 0.168 0.950
θ = 2.602
γ = (−)
[0,1] TEEE λ = 70.744 69.996 70.402 76.425 72.524 0.804 4.383
θ = 33.677
γ = 0.038
BeE λ = 17.526 54.257 54.663 60.686 56.785 0.574 3.148
θ = 22.625
γ = 0.386
KuE λ = 9.096 41.012 41.419 47.442 43.541 0.339 1.860
θ = 60.145
γ = 0.627
EGE λ = 1.616 68.766 69.173 75.196 71.295 0.786 4.287
θ = 31.368
γ = 1.616
WeE λ = 5.780 36.413 36.820 42.843 38.942 0.237 1.303
θ = 0.832
γ = 0.511
GoE λ = 0.021 35.617 36.024 42.046 38.146 0.144 0.842
θ = 8.577
γ = 0.423
MOE λ = 168.723 55.363 55.563 59.649 57.049 0.340 1.874
θ = 3.530
γ = (−)
λ = 0.663 179.660 179.726 182.803 180.503 0.570 3.127
θ = (−)
γ = (−)
FIG. 4.

The analysis of the glass fibers dataset. (a) Empirical histogram of the observed data. (b) The plot of the empirical cdf. (c) The P–P plot of the HOEE distribution. (d) The Q–Q plot of the HOEE distribution.

FIG. 4.

The analysis of the glass fibers dataset. (a) Empirical histogram of the observed data. (b) The plot of the empirical cdf. (c) The P–P plot of the HOEE distribution. (d) The Q–Q plot of the HOEE distribution.

Close modal
This dataset represents the mortality rate of COVID-19 in Italy (see https://covid19.who.int/) during a period of 59 days, which was recorded from 27 February to 27 April 2020. This dataset was used previously by Almongy22 and EL-Sagheer.23 The observations are
$4.571,7.201,3.606,8.479,11.410,8.961,10.919,10.908,6.503,18.474,11.010,17.337,16.561,13.226,15.137,8.697,15.787,13.333,11.822,14.242,11.273,14.330,16.046,11.950,10.282,11.775,10.138,9.037,12.396,10.644,8.646,8.905,8.906,7.407,7.445,7.214,6.194,4.640,5.452,5.073,4.416,4.859,4.408,4.639,3.148,4.040,4.253,4.011,3.564,3.827,3.134,2.780,2.881,3.341,2.686,2.814,2.508,2.450,1.518.$
Table IV shows the performance of the HOEE distribution compared to the other models mentioned earlier. The first row in Table IV confirms that the HOEE model is the best of the competing models to fit the dataset above. Figure 5 shows the plots of the empirical histogram of the observed data, the plot of the empirical cdf, the P–P plot of the HOEE distribution, and the Q–Q plot of the HOEE distribution for the COVID-19 dataset.
TABLE IV.

The performance of the HOEE distribution compared to the other models for the COVID-19 dataset.

DistributionsEstimatesAICCAICBICHQICW*A*
HOEE λ = 22.682 340.010 340.224 344.165 341.632 0.130 0.799
θ = 0.020
γ = (−)
[0,1] TEEE λ = 0.778 341.957 342.393 348.189 344.390 0.160 0.908
θ = 4.469
γ = 0.228
BeE λ = 3.076 341.487 341.923 347.719 343.920 0.148 0.853
θ = 6.461
γ = 0.050
KuE λ = 5.415 342.378 342.814 348.610 344.811 0.168 0.946
θ = 0.454
γ = 0.452
EGE λ = 0.981 342.114 342.551 348.347 344.547 0.159 0.904
θ = 3.488
γ = 0.244
WeE λ = 1.928 341.402 341.838 347.634 343.835 0.133 0.804
θ = 0.374
γ = 0.040
GoE λ = 0.484 345.170 345.606 351.402 347.603 0.141 0.941
θ = 1.623
γ = 0.092
MOE λ = 10.274 344.604 344.818 348.759 346.226 0.163 1.017
θ = 0.315
γ = (−)
λ = 0.122 367.655 367.725 369.732 368.466 0.147 0.848
θ = (−)
γ = (−)
DistributionsEstimatesAICCAICBICHQICW*A*
HOEE λ = 22.682 340.010 340.224 344.165 341.632 0.130 0.799
θ = 0.020
γ = (−)
[0,1] TEEE λ = 0.778 341.957 342.393 348.189 344.390 0.160 0.908
θ = 4.469
γ = 0.228
BeE λ = 3.076 341.487 341.923 347.719 343.920 0.148 0.853
θ = 6.461
γ = 0.050
KuE λ = 5.415 342.378 342.814 348.610 344.811 0.168 0.946
θ = 0.454
γ = 0.452
EGE λ = 0.981 342.114 342.551 348.347 344.547 0.159 0.904
θ = 3.488
γ = 0.244
WeE λ = 1.928 341.402 341.838 347.634 343.835 0.133 0.804
θ = 0.374
γ = 0.040
GoE λ = 0.484 345.170 345.606 351.402 347.603 0.141 0.941
θ = 1.623
γ = 0.092
MOE λ = 10.274 344.604 344.818 348.759 346.226 0.163 1.017
θ = 0.315
γ = (−)
λ = 0.122 367.655 367.725 369.732 368.466 0.147 0.848
θ = (−)
γ = (−)
FIG. 5.

The analysis of the COVID-19 dataset. (a) Empirical histogram of the observed data. (b) The plot of the empirical cdf. (c) The P–P plot of the HOEE distribution. (d) The Q–Q plot of the HOEE distribution.

FIG. 5.

The analysis of the COVID-19 dataset. (a) Empirical histogram of the observed data. (b) The plot of the empirical cdf. (c) The P–P plot of the HOEE distribution. (d) The Q–Q plot of the HOEE distribution.

Close modal

Over the past ten years, there has been a growing interest in generating new distributions from old ones. We suggest the hybrid odd exponential-Φ family (HOE-Φ) as a family of distributions that extends existing continuous distributions. We provide two special models of this family: the hybrid odd exponential exponential (HOEE) distribution and the hybrid odd exponential Weibull (HOEW) distribution. We derive some mathematical properties such as quantile function, expansion of the pdf and cdf, moments, moments generating function, incomplete moments, Rényi and Shannon entropies, and order statistics. We discuss the estimation of the family parameters using the MLE method. We provide applications of the HOE-Φ family on two real datasets.

This study was funded by the Researchers Supporting Project number (RSPD2024R969), King Saud University, Riyadh, Saudi Arabia.

The authors have no conflicts to disclose.

Authors have worked equally to write and review the paper.

Ghanam A. Mahdi: Conceptualization (equal); Data curation (equal); Investigation (equal); Writing – original draft (equal). Mundher A. Khaleel: Data curation (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal). Ahmed M. Gemeay: Conceptualization (equal); Resources (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Nagy: Formal analysis (equal); Validation (equal); Writing – original draft (equal). A. H. Mansi: Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). Md Moyazzem Hossain: Resources (equal); Writing – original draft (equal). Eslam Hussam: Formal analysis (equal); Methodology (equal); Validation (equal).

The data that support the findings of this study are available within the article.

1.
A. W.
Marshall
and
I.
Olkin
, “
A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families
,”
Biometrika
84
(
3
),
641
652
(
1997
).
2.
N.
Eugene
,
C.
Lee
, and
F.
Famoye
, “
Beta-normal distribution and its applications
,”
Commun. Stat. -Theory Methods
31
(
4
),
497
512
(
2002
).
3.
G. M.
Cordeiro
and
M.
de Castro
, “
A new family of generalized distributions
,”
J. Stat. Comput. Simul.
81
(
7
),
883
898
(
2011
).
4.
A.
Alzaatreh
,
C.
Lee
, and
F.
Famoye
, “
A new method for generating families of continuous distributions
,”
Metron
71
(
1
),
63
79
(
2013
).
5.
M.
Bourguignon
,
R. B.
Silva
, and
G. M.
Cordeiro
, “
The Weibull-G family of probability distributions
,”
J. Data Sci.
12
(
1
),
53
68
(
2021
).
6.
M. A.
Khaleel
,
P. E.
Oguntunde
,
J. N. A.
Abbasi
,
N. A.
Ibrahim
, and
M. H.
, “
The Marshall-Olkin Topp Leone-G family of distributions: A family for generalizing probability models
,”
Sci. Afr.
8
,
e00470
(
2020
).
7.
G. M.
Cordeiro
,
M.
, and
E. M.
Ortega
, “
The exponentiated half-logistic family of distributions: Properties and applications
,”
J. Probab. Stat.
2014
,
864396
.
8.
M. H.
Tahir
,
G. M.
Cordeiro
,
A.
Alzaatreh
,
M.
Mansoor
, and
M.
Zubair
, “
The logistic-X family of distributions and its applications
,”
Commun. Stat. -Theory Methods
45
(
24
),
7326
7349
(
2016
).
9.
G. M.
Cordeiro
,
E. M.
Ortega
,
B. V.
Popović
, and
R. R.
Pescim
, “
The Lomax generator of distributions: Properties, minification process and regression model
,”
Appl. Math. Comput.
247
,
465
486
(
2014
).
10.
I.
Mezö
and
Á.
Baricz
, “
On the generalization of the Lambert W function
,”
Trans. Am. Math. Soc.
369
(
11
),
7917
7934
(
2017
).
11.
I. S.
and
I. M.
Ryzhik
,
Table of Integrals, Series, and Products
(
,
2014
).
12.
A.
Rényi
, “
On measures of entropy and information
,” in
Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics
(
University of California Press
,
1961
), Vol.
4
, pp.
547
562
.
13.
C. E.
Shannon
, “
Prediction and entropy of printed English
,”
Bell Syst. Tech. J.
30
(
1
),
50
64
(
1951
).
14.
A. A.
Ismael
, “
[0, 1] truncated exponentiated exponential exponential distribution with application
,”
Entrepreneurship J. Finance Bus.
2
(
4
),
223
231
(
2021
).
15.
S.
and
S.
Kotz
, “
The beta exponential distribution
,”
Reliab. Eng. Syst. Saf.
91
(
6
),
689
697
(
2006
).
16.
H. I.
Okagbue
,
P. E.
Oguntunde
,
P. O.
Ugwoke
,
A. A.
Opanuga
, and
E. C.
Erondu
, “
Exponentiated generalized exponential distribution: Ordinary differential equations
,” in
Transactions on Engineering Technologies: World Congress on Engineering and Computer Science 2017
(
Springer
,
Singapore
,
2019
), pp.
341
352
.
17.
S.
Bashir
and
A.
Qureshi
, “
Gompertz-exponential distribution: Record value theory and applications in reliability
,”
Istatistik J. Turk. Stat. Assoc.
14
(
1
),
27
37
(
2022
).
18.
R. L.
Smith
and
J.
Naylor
, “
A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution
,”
Appl. Stat.
36
(
3
),
358
369
(
1987
).
19.
M. A.
Khaleel
,
N. H.
Al-Noor
, and
M. K.
Abdal-Hameed
, “
Marshall Olkin exponential Gompertz distribution: Properties and applications
,”
Period. Eng. Nat. Sci.
8
(
1
),
298
312
(
2020
).
20.
P. E.
Oguntunde
,
M. A.
Khaleel
,
A. O.
,
H. I.
Okagbue
,
A. A.
Opanuga
, and
F. O.
Owolabi
, “
The Gompertz inverse exponential (GoIE) distribution with applications
,”
Cogent Math. Stat.
5
(
1
),
1507122
(
2018
).
21.
P. E.
Oguntunde
,
M. A.
Khaleel
,
M. T.
Ahmed
,
A. O.
, and
O. A.
Odetunmibi
, “
A new generalization of the Lomax distribution with increasing, decreasing, and constant failure rate
,”
Modell. Simul. Mater. Sci. Eng.
2017
,
6043169
.
22.
H. M.
Almongy
,
E. M.
Almetwally
,
H. M.
Aljohani
,
A. S.
Alghamdi
, and
E. H.
Hafez
, “
A new extended Rayleigh distribution with applications of COVID-19 data
,”
Results Phys.
23
,
104012
(
2021
).
23.
R. M.
EL-Sagheer
,
M. S.
Eliwa
,
K. M.
Alqahtani
, and
M.
EL-Morshedy
, “
Asymmetric randomly censored mortality distribution: Bayesian framework and parametric bootstrap with application to COVID-19 data
,”
J. Math.
2022
,
8300753
.