Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications

Thatayaone Moakofi; Broderick Oluyede

doi:10.1515/ms-2023-0058

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Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications

Thatayaone Moakofi and Broderick Oluyede

Published/Copyright: June 15, 2023

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From the journal Mathematica Slovaca Volume 73 Issue 3

ABSTRACT

In this work, a new family of distributions referred to as type II exponentiated half-logistic-Gompertz-G (TIIEHL-Gom-G) family of distributions is introduced and studied. Some of the main statistical properties of these family of distributions are derived. The model parameters are estimated using the maximum likelihood estimation technique and consistency of maximum likelihood estimators is evaluated by performing a simulation study. The importance and versatility of the TIIEHL-Gom-G family of distributions is demonstrated in an application to two real data sets from different fields.

2020 Mathematics Subject Classification: 60E05; 62F10

Keywords: Exponentiated Half-Logistic; Gompertz Distribution; Maximum Likelihood Estimation

(Communicated by Gejza Wimmer)

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1. Appendix

1.1. Series expansion of density function

We present the series expansion of the TIIEHL-Gom-G density function. Using the generalized binomial and Taylor series expansions given by

(1+z)−β=∑k=0∞(−1)k(β+k−1k)zkfor |z|<1 and ez=∑i=0∞zii!,

we have

1.1 f(x;α,γ,ψ)=2αexp(αγ(1−[1−G(x;ψ)]−γ))[1−G(x;ψ)]−γ−1×(1+(1−exp(1γ(1−[1−G(x;ψ)]−γ))))−(α+1)g(x;ψ)=2α∑k=0∞(α+kk)(−1)kexp(αγ(1−[1−G(x;ψ)]−γ))[1−G(x;ψ)]−γ−1×(1−exp(1γ(1−[1−G(x;ψ)]−γ)))kg(x;ψ)=2α∑k,m=0∞(α+kk)(km)(−1)k+mexp(k+αγ(1−[1−G(x;ψ)]−γ))×[1−G(x;ψ)]−γ−1g(x;ψ)=2α∑k,m,l=0∞(α+kk)(km)(−1)k+m(k+αγ)l(1−[1−G(x;ψ)]−γ)ll!×[1−G(x;ψ)]−γ−1g(x;ψ)=2α∑k,m,l,i=0∞(α+kk)(km)(li)(−1)k+m+i(k+αγ)ll![1−G(x;ψ)]−γ(i+1)−1g(x;ψ)=2α∑k,m,l,i,q=0∞(α+kk)(km)(li)(γ(i+1)+qq)(−1)k+m+i+q(k+αγ)ll!×Gq(x;ψ)g(x;ψ)=∑q=0∞Cq+1gq+1(x;ψ),

where gq+1(x;ψ)=(q+1)[G(x;ψ)]qg(x;ψ) is the exponentiated-G (exp-G) pdf with the power parameter q + 1 > 0 and parameter vector ψ, where

1.2 Cq+1= ∑k,m,l,i=0∞(α+kk)(km)(li)(γ(i+1)+qq)(−1)k+m+i+q(k+αγ)ll!(2αq+1).

1.2. Rényi entropy

Rényi entropy is defined to be

1.3 IR(v)=11−vlog(∫0∞[f(x;α,γ,ψ)]vdx),v≠1, v>0.

Note that

1.4 [f(x;α,γ,ψ)]v=(2α)vexp(vαγ(1−[1−G(x;ψ)]−γ))[1−G(x;ψ)]−vγ−v×(1+(1−exp(1γ(1−[1−G(x;ψ)]−γ))))−v(α+1)gv(x;ψ)=(2α)v∑k=0∞(v(α+1)+k−1k)(−1)k[1−G(x;ψ)]−vγ−v×(1−exp(1γ(1−[1−G(x;ψ)]−γ)))kgv(x;ψ)×exp(vαγ(1−[1−G(x;ψ)]−γ))=(2α)v∑k,m=0∞(v(α+1)+k−1k)(−1)k+m[1−G(x;ψ)]−vγ−v×exp(vα+kγ(1−[1−G(x;ψ)]−γ))gv(x;ψ)=(2α)v∑k,m,l=0∞(v(α+1)+k−1k)(−1)k+m(vα+kγ)l×(1−[1−G(x;ψ)]−γ)ll![1−G(x;ψ)]−vγ−vgv(x;ψ)=(2α)v∑k,m,l,i=0∞(v(α+1)+k−1k)(li)(−1)k+m+i(vα+kγ)ll!×[1−G(x;ψ)]−γ(v+i)−vgv(x;ψ)=(2α)v∑k,m,l,i,q=0∞(v(α+1)+k−1k)(li)(−1)k+m+i+q×(γ(v+i)+v+q−1q)(vα+kγ)ll!Gq(x;ψ)gv(x;ψ).

Now,

1.5 ∫0∞fv(x;α,γ,ψ)dx=(2α)v∑k,m,l,i,q=0∞(v(α+1)+k−1k)(li)(γ(v+i)+v+q−1q)×(−1)k+m+i+q(vα+kγ)ll!∫0∞Gq(x;ψ)gv(x;ψ)dx.

Consequently, Rényi entropy for the TIIEHL-Gom-G family of distributions is given by

1.6 IR(v)=11−vlog[∑k,m,l,i,q=0∞(v(α+1)+k−1k)(li)(γ(v+i)+v+q−1q)×(−1)k+m+i+q(vα+kγ)ll!(2α)v[1+qv]v∫0∞([1+qv](G(x;ψ))qv(g(x;ψ)))vdx]=11−vlog[∑q=0∞τqexp((1−v)IREG)]

for v > 0, v ≠ 1, where IREG=11−vlog⁡[∫0∞([1+qv](G(x;ψ))qv(g(x;ψ)))vdx] is the Rényi entropy of exponentiated-G (exp-G) distribution with power parameter (qv+1) and

τq=∑k,m,l,i=0∞(v(α+1)+k−1k)(li)(γ(v+i)+v+q−1q)×(−1)k+m+i+q(vα+kγ)ll!(2α)v[1+qv]v.

Therefore, Rényi entropy of the TIIEHL-Gom-G family of distributions can be obtained from those of the exponentiated-G (exp-G) family of distributions.

2. Applications

We employ two real data sets to compare the fits of TIIEHL-Gom-W distribution to its nested models and to the following non-nested models:

type II general inverse exponential Burr III (TIIGIE-BIII) distribution with the pdf

fTIIGIE−BIII(x;λ,θ,c,k)=λθckx−c−1(1+x−c)−k−1[1−(1+x−c)−k]θ−1(1−[1−(1+x−c)−k]θ)2×exp(−λ[1−(1+x−c)−k]θ1−[1−(1+x−c)−k]θ)

for λ, θ, c, k > 0 and x > 0, type II exponentiated half-logistic Topp-Leone-Weibull logarithmic (TIIEHL-TL-WL) distribution by with the pdf

fTIIEHL−TL−WL(x;a,b,θ,λ)=4abθλxλ−1exp(−xλ)[1−exp(−2xλ)]b−1exp(−xλ)(1+[1−exp(−2xλ)]b)a+1×(1−[1−exp(−2xλ)]b)a−1(1−θ[1−[1−exp(−2xλ)]b1+[1−exp(−2xλ)]b]a)−1−log(1−θ)

for a, b, θ, λ and x > 0, type II general inverse exponential Lomax (TIIGIE-Lx) distribution with the pdf

fTIIGIE−Lx(x;λ,α,a,b)= λαab(1+xb)−(a+1)(1+xb)a(α+1)exp⁡(λ(1−(1+xb)aα))

for λ, α, a, b > 0 and x > 0, exponentiated half logistic-power generalized Weibull-log-logistic with the pdf

fEHL−PGW−LLoG(x;α,β,δ,c)=2αβδ[1+(1−(1+xc)−1(1+xc)−1)α]β−1e(1−[1+(1−(1+xc)−1(1+xc)−1)α]β)×((1+xc)−1)−(α+3)(1+e(1−[1+(1−(1+xc)−1(1+xc)−1)α]β))−2×[1−e(1−[1+(1−(1+xc)−1(1+xc)−1)α]β)1+e(1−[1+(1−(1+xc)−1(1+xc)−1)α]β)]δ−1cxc−1(1−(1+xc)−1)α−1,

for α, β, δ, c > 0, odd exponentiated half logistic- Burr XII (OEHL-BXII) with the pdf

fOEHLBXII(x;α,λ,a,b)=2αλabxa−1exp⁡(λ[1−(1+xa)b])(1−exp⁡(λ[1−(1+xa)b]))α−1(1+xa)−b−1(1+exp⁡(λ[1−(1+xa)b]))α+1,

for α, λ, a, b > 0, generalized Weibull Gompertz (GWG) with the pdf

fGWG(x;a,b,c,d)=abxb−1exp(−axb(ecxd)+cxd)(1+cdbxd−e−cxd),

for a,b,c,d > 0, Harris extended Gompertz (HEG) with the pdf

fHEG(x;θ,β,c,k)=θ1kβcxexp⁡(−β(cx−1)log⁡(c))(1−(1−θ)exp⁡(−kβ(cx−1)log⁡(c)))1+1k,

for Θ,β,c,k > 0, Kumaraswamy inverse Gompertz (KuIG) with the pdf

fKIG(x;α,β,γ)=αγx2exp⁡(βx)exp⁡(−αβexp⁡(βx−1))(1−exp⁡(−αβexp⁡(βx−1)))γ−1,

for α,β,γ > 0, and generalized Gompertz (GG) with the pdf

fGG(x;λ,c,θ)=θλexp⁡(cx)exp⁡(−λcexp⁡(cx−1))(1−exp⁡(−λcexp⁡(cx−1)))θ−1,

for λ,c,θ > 0.

2.1. Elements of score vector

The first derivative of the log-likelihood function with respect to each component of the parameter vector Δ=(α,γ,ψ)T, that is, elements of the score vector U(Δ) are given by

∂ℓn∂α=nα+∑i=1n(1γ(1−[1−G(x;ψ)]−γ))−∑i=1nln(1+(1−exp(1γ(1−[1−G(x;ψ)]−γ)))),∂ℓn∂γ =∑i=1n−αγ−2(1−[1−G(x;ψ)]−γ)−αγ[1−G(x;ψ)]−γln[1−G(x;ψ)]−∑i=1nln[1−G(x;ψ)]−(α+1)∑i=1nexp(1γ(1−[1−G(x;ψ)]−γ))(1+(1−exp(1γ(1−[1−G(x;ψ)]−γ))))×(−γ−2(1−[1−G(x;ψ)]−γ)−1γ[1−G(x;ψ)]−γln[1−G(x;ψ)]),∂ℓ∂ψk=−∑i=1nα[1−G(xi;ψ)]−γ−1∂G(xi;ψ)∂ψk−(−γ−1)∑i=1n∂G(xi;ψ)∂ψk[1−G(xi;ψ)]+(α+1)∑i=1nexp(1γ(1−[1−G(xi;ψ)]−γ))[1−G(xi;ψ)]−γ−1∂G(xi;ψ)∂ψk(1+(1−exp(1γ(1−[1−G(xi;ψ)]−γ))))+∑i=1n∂g(xi;ψ)∂ψk(g(xi;ψ)).

Received: 2022-02-05

Accepted: 2022-06-25

Published Online: 2023-06-15

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https://doi.org/10.1515/ms-2023-0058

Keywords for this article

Exponentiated Half-Logistic; Gompertz Distribution; Maximum Likelihood Estimation