Review Article Open Access
Further Developments on the (EG) Exponential-MIR Class of Distributions
Clement Boateng Ampadu*
31 Carrolton Road, Boston, MA 02132-6303, USA
*Corresponding author: MClement Boateng Ampadu, 31 Carrolton Road, Boston, MA 02132-6303, USA E-mail: @
Received: July 30, 2018; Accepted: August 27, 2018; Published: October 11, 2018
Citation: Ampadu CB (2018) Further Developments on the (EG) Exponential-MIR Class of Distributions. J Adv Res Biotech 3(2):1-5. DOI: http://dx.doi.org/10.15226/2475-4714/3/2/00137
Abstract
The Modified Inverse Rayleigh (MIR) distribution appeared in [Khan, M. S. (2014).Modified inverse Rayleigh distribution. International Journal of Computer Applications, 87(13):28–33] who got some theoretical properties of this distribution, and in[Nasiru, S., Mwita, P. N. and Ngesa, O. (2017). Exponentiated Generalized Exponential Dagum Distribution. Journal of King Saud University- Science, In Press] they introduced the (EG) Exponential-X class of distributions and obtained some theoretical properties with application. By assuming the random variable X follows the MIR distribution, some theoretical properties with application of the (EG) Exponential-MIR

Class of distributions appeared in [Nasiru, S., Mwita, P. N. and Ngesa, O. (2018). Discussion on Generalized Modified Inverse Rayleigh Distribution. Applied Mathematics and Information Sciences, 12(1):113-124]. In the present paper we propose some extensions of the (EG) Exponential-MIR class of distributions. The (EG) Exponential- MIR class of distributions is part of Chapter 5 [Nasiru, S. (2018). A New Generalization of Transformed-Transformer Family of Distributions. Doctor of Philosophy thesis in Mathematics (Statistics Option). Pan African University, Institute for Basic Sciences, Technology and Innovation, Kenya], where the naming convention “NEGMIR” is used

Keywords: T-X (W) family of distributions; Exponentiated Generalized distributions; Modified Inverse Rayleigh distribution; biological data; health data
Introduction
T-X (W) Family of Distributions
This family of distributions is a generalization of the betagenerated family of distribu-tions first proposed by Eugene et.al [Eugene, N, Lee, C, Famoye, F: The beta-normal dis-tribution and its applications. Communications in Statistics-Theory and Methods 31(4), 497–512 (2002)]. In particular, let r(t) be the PDF of the random variable T ∈ [a, b], −∞ ≤ a < b ≤ ∞, and let W (F (x)) be a monotonic and absolutely continuous function of the CDF F (x) of any random variable X. The CDF of a new family of distributions defined by Alzaatreh et.al [Alzaatreh, A, Lee, C, Famoye, F: A new method for generating families of continuous distributions. Metron 71(1), 63–79 (2013b)] is given by
G(x)= a W(F(x)) r(t)dt=R{W(F(x))} MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4raiaacI cacaWG4bGaaiykaiabg2da9maapedabaGaamOCaiaacIcacaWG0bGa aiykaiaadsgacaWG0bGaeyypa0JaamOuaiaacUhacaWGxbGaaiikai aadAeacaGGOaGaamiEaiaacMcacaGGPaGaaiyFaaWcbaGaamyyaaqa aiaadEfacaGGOaGaamOraiaacIcacaWG4bGaaiykaiaacMcaa0Gaey 4kIipaaaa@50E1@
Where R (•) is the CDF of the random variable T and a0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyyaiabgw MiZkaaicdaaaa@395C@
Remark 1.1
The PDF of the T-X (W) family of distributions is obtained by differentiating the CDF above.
Remark 1.2
When we set W(F(x)):=ln(1F(x)) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4vaiaacI cacaWGgbGaaiikaiaadIhacaGGPaGaaiykaiaacQdacqGH9aqpcqGH sislcaGGSbGaaiOBaiaacIcacaaIXaGaeyOeI0IaaiOraiaacIcaca GG4bGaaiykaiaacMcaaaa@45FF@ then we use the term “T-X Family of Distributions” to describe all sub-classes of the T-X (W) family of distributions induced by the weight function W(x)=ln(1x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4vaiaacI cacaWG4bGaaiykaiabg2da9iabgkHiTiaacYgacaGGUbGaaiikaiaa igdacqGHsislcaGG4bGaaiykaaaa@40FA@ x). A description of different weight functions that are appropriate given the support of the random variable T is discussed in [Alzaatreh, A, Lee, C, Famoye, F: A new method for generating families of continuous distributions. Metron 71(1), 63–79 (2013b)] A plethora of results studying properties and application of the T- X(W) family of distributions have appeared in the literature, and the research papers, assuming open access, can be easily obtained on the web via common search engines, like Google, etc.
The Exponentiated Generalized (EG) T-X family of distributions
This class of distributions appeared in [Suleman Nasiru, Peter N. Mwita and Oscar Ngesa, Exponentiated Generalized Transformed-Transformer Family of Distributions, Journal of Statistical and Econometric Methods, vol.6, no.4, 2017, 1-17] In particular the CDF Admits the following integral representation
G(x)= 0 log[1 (1 F ¯ (x) d ) c ] r(t)dt MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4raiaacI cacaWG4bGaaiykaiabg2da9maapedabaGaamOCaiaacIcacaWG0bGa aiykaiaadsgacaWG0baaleaacaaIWaaabaGaeyOeI0IaciiBaiaac+ gacaGGNbGaai4waiaaigdacqGHsislcaGGOaGaaGymaiabgkHiTiqa dAeagaqeaiaacIcacaWG4bGaaiykamaaCaaameqabaGaamizaaaali aacMcadaahaaadbeqaaiaadogaaaWccaGGDbaaniabgUIiYdaaaa@51CE@
Where c, d > 0 and F(x) =1− F(x) and F(x) is the CDF of a base distribution.
Remark 1.3
Note that if we set L(x):= (1 F ¯ (x) d ) c MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitaiaacI cacaWG4bGaaiykaiaacQdacqGH9aqpcaGGOaGaaGymaiabgkHiTiqa dAeagaqeaiaacIcacaWG4bGaaiykamaaCaaaleqabaGaamizaaaaki aacMcadaahaaWcbeqaaiaadogaaaaaaa@4350@ where c, d > 0 and F ¯ (x)=1F(x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGabmOrayaara GaaiikaiaadIhacaGGPaGaeyypa0JaaGymaiabgkHiTiaadAeacaGG OaGaamiEaiaacMcaaaa@3EFE@ , and F(x) is the CDF of a base distribution, then L(x) gives the CDF of the exponentiated generalized class of distributions [G.M. Cordeiro, E.M.M. Ortega and C.C.D. da Cunha, The exponentiated generalized class of distributions, Journal of Data Science,11(1), (2013),1-27]
The (EG) Exponential-MIR family of distributions
Here we assume the random variable X follows the MIR distribution with CDF
F α,θ(x) = e ( α x + θ x 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacqaHXoqycaGGSaGaeqiUdeNaaiikaiaadIhacaGGPaaabeaa kiabg2da9iaadwgadaahaaWcbeqaaiabgkHiTiaacIcadaWcaaqaai abeg7aHbqaaiaadIhaaaGaey4kaSYaaSaaaeaacqaH4oqCaeaacaWG 4bWaaWbaaWqabeaacaaIYaaaaaaaliaacMcaaaaaaa@48FB@
And the random variable T follows the exponential distribution with PDF
r λ (t)=λ e λt MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOCamaaBa aaleaacqaH7oaBaeqaaOGaaiikaiaadshacaGGPaGaeyypa0Jaeq4U dWMaamyzamaaCaaaleqabaGaeyOeI0Iaeq4UdWMaamiDaaaaaaa@4294@
and CDF R λ (t)=1 e λt MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOuamaaBa aaleaacqaH7oaBaeqaaOGaaiikaiaacshacaGGPaGaeyypa0JaaGym aiabgkHiTiaacwgadaahaaWcbeqaaiabgkHiTiabeU7aSjaadshaaa aaaa@4266@
Now put L (α,θ,d,c) (x):= (1 F ¯ (α,θ) (x) d ) c MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaWGKbGaaiil aiaadogacaGGPaaabeaakiaacIcacaWG4bGaaiykaiaacQdacqGH9a qpcaGGOaGaaGymaiabgkHiTiqadAeagaqeamaaBaaaleaacaGGOaGa eqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiikaiaacIhacaGGPa WaaWbaaSqabeaacaWGKbaaaOGaaiykamaaCaaaleqabaGaam4yaaaa aaa@51A8@

And observe the CDF of the (EG) Exponential-MIR family of distributions as proposed in [Nasiru, S, Mwita, P N and Ngesa, O.(2018) Discussion on Generalized Modified Inverse Rayleigh Distribution. Applied Mathematics and Information Sciences, 12(1):113-124; Nasiru, S. (2018). A New Generalization of Transformed-Transformer Family of Distributions. Doctor of Philosophy thesis in Mathematics (Statistics Option) Pan African University, Institute for Basic Sciences, Technology and Innovation, Kenya] is given by
G (α,θ,d,c,λ) (x)=1 (1 L (α,θ,d,c) (x)) λ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4ramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaWGKbGaaiil aiaadogacaGGSaGaeq4UdWMaaiykaaqabaGccaGGOaGaamiEaiaacM cacqGH9aqpcaaIXaGaeyOeI0IaaiikaiaaigdacqGHsislcaWGmbWa aSbaaSqaaiaacIcacqaHXoqycaGGSaGaeqiUdeNaaiilaiaadsgaca GGSaGaam4yaiaacMcaaeqaaOGaaiikaiaadIhacaGGPaGaaiykamaa CaaaleqabaGaeq4UdWgaaaaa@57BD@
Further Developments
In this section we present some new generalizations of the (EG) Exponential-MIR family of distributions which are induced by the other weight functions introduced in [Alzaatreh, A, Lee, C, Famoye, F: A new method for generating families of continuous distributions. Metron 71(1), 63–79 (2013b)], when the random variable T in the T-X(W) class of distributions has support [a,) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaai4waiaadg gacaGGSaGaeyOhIuQaaiykaaaa@3A89@ , where [a,) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaai4waiaadg gacaGGSaGaeyOhIuQaaiykaaaa@3A89@
Definition 2.1
A random variable Y1 will be called T − X distributed of type I if the weight function is given by
W(x)=log(1x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4vaiaacI cacaGG4bGaaiykaiabg2da9iabgkHiTiaacYgacaGGVbGaai4zaiaa cIcacaaIXaGaeyOeI0IaaiiEaiaacMcaaaa@41E5@
Definition 2.2
A random variable Y2 will be called T − X distributed of type II if the weight function is given by
W(x)= x 1x MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4vaiaacI cacaGG4bGaaiykaiabg2da9maalaaabaGaamiEaaqaaiaaigdacqGH sislcaWG4baaaaaa@3DDF@
Definition 2.3
A random variable Y3 will be called T − X distributed of type III if the weight function is given by
W(x)=log(1 x α ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4vaiaacI cacaGG4bGaaiykaiabg2da9iabgkHiTiGacYgacaGGVbGaai4zaiaa cIcacaaIXaGaeyOeI0IaamiEamaaCaaaleqabaGaeqySdegaaOGaai ykaaaa@43BE@
For some α>0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaeqySdeMaey Opa4JaaGimaaaa@3957@
Type II
Definition 2.5
The CDF of the (EG) Exponential-MIR class of distributions of type II has the following integral representation for a0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyyaiabgw MiZkaaicdaaaa@395C@

K (α,θ,d,c,λ) (x)= a L (α,θ,d,c) (x) 1 L (α,θ,d,c) (x) r λ (t)dt MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4samaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaWGKbGaaiil aiaadogacaGGSaGaeq4UdWMaaiykaaqabaGccaGGOaGaamiEaiaacM cacqGH9aqpdaWdXaqaaiaadkhadaWgaaWcbaGaeq4UdWgabeaakiaa cIcacaWG0bGaaiykaiaadsgacaWG0baaleaacaWGHbaabaWaaSaaae aacaWGmbWaaSbaaWqaaiaacIcacqaHXoqycaGGSaGaeqiUdeNaaiil aiaadsgacaGGSaGaam4yaiaacMcaaeqaaSGaaiikaiaadIhacaGGPa aabaGaaGymaiabgkHiTiaadYeadaWgaaadbaGaaiikaiabeg7aHjaa cYcacqaH4oqCcaGGSaGaamizaiaacYcacaWGJbGaaiykaaqabaWcca GGOaGaamiEaiaacMcaaaaaniabgUIiYdaaaa@6903@

Note that L (α,,θ,1,c) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiaacYcacqaH4oqCcaGGSaGaaGym aiaacYcacaWGJbGaaiykaaqabaGccaGGOaGaamiEaiaacMcaaaa@4264@ implies the following from Definition 2.5
Corollary 2.6
The CDF of the exponentiated Exponential-MIR class of distributions of type II is given by
K (α,θ,c,λ) * (x)= R λ ( F (α,θ) c (x) 1 F (α,θ) c (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4samaaDa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaWGJbGaaiil aiabeU7aSjaacMcaaeaacaGGQaaaaOGaaiikaiaadIhacaGGPaGaey ypa0JaamOuamaaBaaaleaacqaH7oaBaeqaaOGaaiikamaalaaabaGa amOramaaDaaaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaae aacaWGJbaaaOGaaiikaiaadIhacaGGPaaabaGaaGymaiabgkHiTiaa dAeadaqhaaWcbaGaaiikaiabeg7aHjaacYcacqaH4oqCcaGGPaaaba Gaam4yaaaakiaacIcacaWG4bGaaiykaaaaaaa@5CC2@
Where R λ (.)=1 e λ(.) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOuamaaBa aaleaacqaH7oaBaeqaaOGaaiikaiaac6cacaGGPaGaeyypa0JaaGym aiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiabeU7aSjaacIcaca GGUaGaaiykaaaaaaa@4333@

and F (α,θ) (x)= e ( α x + θ x 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaacIhacaGGPaGaeyypa0JaaiyzamaaCaaaleqabaGaeyOeI0Iaai ikamaalaaabaGaeqySdegabaGaamiEaaaacqGHRaWkdaWcaaqaaiab eI7aXbqaaiaadIhadaahaaadbeqaaiaaikdaaaaaaSGaaiykaaaaaa a@4A52@

Note that L (α,,θ,1,1) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiaacYcacqaH4oqCcaGGSaGaaGym aiaacYcacaaIXaGaaiykaaqabaGccaGGOaGaamiEaiaacMcaaaa@4237@ implies the following from Definition 2.5
Corollary 2.7
The CDF of the Exponential-MIR class of distributions of type II K (α,θ,λ) ** (x)= R λ ( F (α,θ) (x) 1 F (α,θ) (x) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4samaaDa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacqaH7oaBcaGG PaaabaGaaiOkaiaacQcaaaGccaGGOaGaamiEaiaacMcacqGH9aqpca GGsbWaaSbaaSqaaiabeU7aSbqabaGccaGGOaWaaSaaaeaacaWGgbWa aSbaaSqaaiaacIcacqaHXoqycaGGSaGaeqiUdeNaaiykaaqabaGcca GGOaGaamiEaiaacMcaaeaacaaIXaGaeyOeI0IaamOramaaBaaaleaa caGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiikaiaadI hacaGGPaaaaiaacMcaaaa@5AB2@

Where R λ (.)=1 e λ(.) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOuamaaBa aaleaacqaH7oaBaeqaaOGaaiikaiaac6cacaGGPaGaeyypa0JaaGym aiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiabeU7aSjaacIcaca GGUaGaaiykaaaaaaa@4333@

and F (α,θ) (x)= e ( α x + θ x 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaacIhacaGGPaGaeyypa0JaaiyzamaaCaaaleqabaGaeyOeI0Iaai ikamaalaaabaGaeqySdegabaGaamiEaaaacqGHRaWkdaWcaaqaaiab eI7aXbqaaiaadIhadaahaaadbeqaaiaaikdaaaaaaSGaaiykaaaaaa a@4A52@
Type III
Definition 2.8
The CDF of the (EG) Exponential-MIR class of distributions of type III has the following integral representation for a0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyyaiabgw MiZkaaicdaaaa@395C@ and ξ>0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaeqOVdGNaey Opa4JaaGimaaaa@397B@

Z (α,θ,d,c,λ,ξ) (x)= a log(1 L (α,θ,d,c) ξ (x)) r λ (t)dt MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOwamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaWGKbGaaiil aiaadogacaGGSaGaeq4UdWMaaiilaiabe67a4jaacMcaaeqaaOGaai ikaiaadIhacaGGPaGaeyypa0Zaa8qmaeaacaWGYbWaaSbaaSqaaiab eU7aSbqabaGccaGGOaGaamiDaiaacMcacaWGKbGaamiDaaWcbaGaam yyaaqaaiabgkHiTiGacYgacaGGVbGaai4zaiaacIcacaaIXaGaeyOe I0IaamitamaaDaaameaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacY cacaWGKbGaaiilaiaadogacaGGPaaabaGaeqOVdGhaaSGaaiikaiaa dIhacaGGPaGaaiykaaqdcqGHRiI8aaaa@6661@

Note that L (α,θ,1,c) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaaIXaGaaiil aiaacogacaGGPaaabeaakiaacIcacaWG4bGaaiykaaaa@41B3@ implies the following from Definition 2.8
Corollary 2.9
The CDF of the exponentiated Exponential-MIR class of distributions of type III is given by
Z * (α,θ,c,λ,ξ) (x)= R λ (log(1 F (α,θ) (x) cξ (x))) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOwamaaCa aaleqabaGaaiOkaaaakmaaBaaaleaacaGGOaGaeqySdeMaaiilaiab eI7aXjaacYcacaWGJbGaaiilaiabeU7aSjaacYcacqaH+oaEcaGGPa aabeaakiaacIcacaWG4bGaaiykaiabg2da9iaadkfadaWgaaWcbaGa eq4UdWgabeaakiaacIcacqGHsislciGGSbGaai4BaiaacEgacaGGOa GaaGymaiabgkHiTiaadAeadaWgaaWcbaGaaiikaiabeg7aHjaacYca cqaH4oqCcaGGPaaabeaakiaacIcacaWG4bGaaiykamaaCaaaleqaba Gaam4yaiabe67a4baakiaacIcacaWG4bGaaiykaiaacMcacaGGPaaa aa@5FDE@
Where R λ (.)=1 e λ(.) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOuamaaBa aaleaacqaH7oaBaeqaaOGaaiikaiaac6cacaGGPaGaeyypa0JaaGym aiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiabeU7aSjaacIcaca GGUaGaaiykaaaaaaa@4333@
and F (α,θ) (x)= e ( α x + θ x 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaacIhacaGGPaGaeyypa0JaaiyzamaaCaaaleqabaGaeyOeI0Iaai ikamaalaaabaGaeqySdegabaGaamiEaaaacqGHRaWkdaWcaaqaaiab eI7aXbqaaiaadIhadaahaaadbeqaaiaaikdaaaaaaSGaaiykaaaaaa a@4A52@

Note that L (α,θ,1,1) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaaIXaGaaiil aiaaigdacaGGPaaabeaakiaacIcacaWG4bGaaiykaaaa@4187@ implies the following from Definition 2.8
Corollary 2.10
The CDF of the Exponential-MIR class of distributions of type III is given by
Z ** (α,θ,λ,ξ) (x)= R λ (log(1 F (α,θ) (x) ξ (x))) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOwamaaCa aaleqabaGaaiOkaiaacQcaaaGcdaWgaaWcbaGaaiikaiabeg7aHjaa cYcacqaH4oqCcaGGSaGaeq4UdWMaaiilaiabe67a4jaacMcaaeqaaO GaaiikaiaadIhacaGGPaGaeyypa0JaamOuamaaBaaaleaacqaH7oaB aeqaaOGaaiikaiabgkHiTiGacYgacaGGVbGaai4zaiaacIcacaaIXa GaeyOeI0IaamOramaaBaaaleaacaGGOaGaeqySdeMaaiilaiabeI7a XjaacMcaaeqaaOGaaiikaiaadIhacaGGPaWaaWbaaSqabeaacqaH+o aEaaGccaGGOaGaamiEaiaacMcacaGGPaGaaiykaaaa@5E0C@
Where R λ (.)=1 e λ(.) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOuamaaBa aaleaacqaH7oaBaeqaaOGaaiikaiaac6cacaGGPaGaeyypa0JaaGym aiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiabeU7aSjaacIcaca GGUaGaaiykaaaaaaa@4333@
and F (α,θ) (x)= e ( α x + θ x 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaacIhacaGGPaGaeyypa0JaaiyzamaaCaaaleqabaGaeyOeI0Iaai ikamaalaaabaGaeqySdegabaGaamiEaaaacqGHRaWkdaWcaaqaaiab eI7aXbqaaiaadIhadaahaaadbeqaaiaaikdaaaaaaSGaaiykaaaaaa a@4A52@
Type IV
Definition 2.11
The CDF of the (EG) Exponential-MIR class of distributions of type IV has the following integral representation for ξ>0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaeqOVdGNaey Opa4JaaGimaaaa@397B@ and a0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyyaiabgw MiZkaaicdaaaa@395C@
Q (α,θ,d,c,λ,ξ) (x)= a L (α,θ,d,c) ξ (x) 1 L (α,θ,d,c) ξ (x) r λ (t)dt MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyuamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaWGKbGaaiil aiaadogacaGGSaGaeq4UdWMaaiilaiabe67a4jaacMcaaeqaaOGaai ikaiaadIhacaGGPaGaeyypa0Zaa8qmaeaacaWGYbWaaSbaaSqaaiab eU7aSbqabaGccaGGOaGaamiDaiaacMcacaWGKbGaamiDaaWcbaGaam yyaaqaamaalaaabaGaamitamaaDaaameaacaGGOaGaeqySdeMaaiil aiabeI7aXjaacYcacaGGKbGaaiilaiaacogacaGGPaaabaGaeqOVdG haaSGaaiikaiaadIhacaGGPaaabaGaaGymaiabgkHiTiaadYeadaqh aaadbaGaaiikaiabeg7aHjaacYcacqaH4oqCcaGGSaGaamizaiaacY cacaWGJbGaaiykaaqaaiabe67a4baaliaacIcacaWG4bGaaiykaaaa a0Gaey4kIipaaaa@6F02@
Note that L (α,θ,1,c) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaaIXaGaaiil aiaacogacaGGPaaabeaakiaacIcacaWG4bGaaiykaaaa@41B3@ implies the following from Definition 2.11
Corollary 2.12
The CDF of the exponentiated Exponential-MIR class of distributions of type IV is given by
Q * (α,θ,c,λ,ξ) (x)= R λ ( F (α,θ) (x) ξ (x) 1 F (α,θ) (x) ξ (x) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyuamaaCa aaleqabaGaaiOkaaaakmaaBaaaleaacaGGOaGaeqySdeMaaiilaiab eI7aXjaacYcacaWGJbGaaiilaiabeU7aSjaacYcacqaH+oaEcaGGPa aabeaakiaacIcacaWG4bGaaiykaiabg2da9iaadkfadaWgaaWcbaGa eq4UdWgabeaakiaacIcadaWcaaqaaiaadAeadaWgaaWcbaGaaiikai abeg7aHjaacYcacqaH4oqCcaGGPaaabeaakiaacIcacaWG4bGaaiyk amaaCaaaleqabaGaeqOVdGhaaOGaaiikaiaadIhacaGGPaaabaGaaG ymaiabgkHiTiaadAeadaWgaaWcbaGaaiikaiabeg7aHjaacYcacqaH 4oqCcaGGPaaabeaakiaacIcacaWG4bGaaiykamaaCaaaleqabaGaeq OVdGhaaOGaaiikaiaadIhacaGGPaaaaiaacMcaaaa@66EC@
Where R λ (.)=1 e λ(.) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOuamaaBa aaleaacqaH7oaBaeqaaOGaaiikaiaac6cacaGGPaGaeyypa0JaaGym aiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiabeU7aSjaacIcaca GGUaGaaiykaaaaaaa@4333@

and F (α,θ) (x)= e ( α x + θ x 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaacIhacaGGPaGaeyypa0JaaiyzamaaCaaaleqabaGaeyOeI0Iaai ikamaalaaabaGaeqySdegabaGaamiEaaaacqGHRaWkdaWcaaqaaiab eI7aXbqaaiaadIhadaahaaadbeqaaiaaikdaaaaaaSGaaiykaaaaaa a@4A52@
Note that L (α,β,θ,1,1) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabek7aIjaacYcacqaH4oqCcaGG SaGaaGymaiaacYcacaaIXaGaaiykaaqabaGccaGGOaGaamiEaiaacM caaaa@43D8@ implies the following from Definition 2.11
Corollary 2.13
The CDF of the Exponential-MIR class of distributions of type IV is given by
Q ** (α,θ,λ,ξ) (x)= R λ ( F (α,θ) (x) ξ (x) 1 F (α,θ) (x) ξ (x) ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyuamaaCa aaleqabaGaaiOkaiaacQcaaaGcdaWgaaWcbaGaaiikaiabeg7aHjaa cYcacqaH4oqCcaGGSaGaeq4UdWMaaiilaiabe67a4jaacMcaaeqaaO GaaiikaiaadIhacaGGPaGaeyypa0JaamOuamaaBaaaleaacqaH7oaB aeqaaOGaaiikamaalaaabaGaamOramaaBaaaleaacaGGOaGaeqySde MaaiilaiabeI7aXjaacMcaaeqaaOGaaiikaiaadIhacaGGPaWaaWba aSqabeaacqaH+oaEaaGccaGGOaGaamiEaiaacMcaaeaacaaIXaGaey OeI0IaamOramaaBaaaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaa cMcaaeqaaOGaaiikaiaadIhacaGGPaWaaWbaaSqabeaacqaH+oaEaa GccaGGOaGaamiEaiaacMcaaaGaaiykaaaa@6602@
Where R λ (.)=1 e λ(.) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOuamaaBa aaleaacqaH7oaBaeqaaOGaaiikaiaac6cacaGGPaGaeyypa0JaaGym aiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiabeU7aSjaacIcaca GGUaGaaiykaaaaaaa@4333@
and F (α,θ) (x)= e ( α x + θ x 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaacIhacaGGPaGaeyypa0JaaiyzamaaCaaaleqabaGaeyOeI0Iaai ikamaalaaabaGaeqySdegabaGaamiEaaaacqGHRaWkdaWcaaqaaiab eI7aXbqaaiaadIhadaahaaadbeqaaiaaikdaaaaaaSGaaiykaaaaaa a@4A52@
Application
In this section we compare the Exponential-Weibull class of distributions of type II and the Exponential-Weibull class of distributions of type IV in modeling the aircraft data, Table 1 [Suleman Nasiru, Peter N. Mwita and Oscar Ngesa, Discussion on Generalized Modified Inverse Rayleigh, Appl. Math. Inf. Sci. 12, No. 1, 113-124 (2018)]
Remark 3.1
When a random variable X follows the Exponential-Weibull class of distributions of type II we write XEWII(a,b,c) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamiwaiablY Ji6iaadweacaWGxbGaamysaiaadMeacaGGOaGaamyyaiaacYcacaWG IbGaaiilaiaadogacaGGPaaaaa@40AC@
Remark 3.2
When a random variable X follows the Exponential-Weibull class of distributions of type IV we write XEWIV(a,b,c,d) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamiwaiablY Ji6iaadweacaWGxbGaamysaiaadAfacaGGOaGaamyyaiaacYcacaWG IbGaaiilaiaadogacaGGSaGaamizaiaacMcaaaa@4252@ In this section we assume the CDF of the Weibull distribution is given by F (b,c) (x)=1 e ( x c ) b MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaamOyaiaacYcacaWGJbGaaiykaaqabaGccaGGOaGa amiEaiaacMcacqGH9aqpcaaIXaGaeyOeI0IaamyzamaaCaaaleqaba GaeyOeI0IaaiikamaalaaabaGaamiEaaqaaiaadogaaaGaaiykamaa CaaameqabaGaamOyaaaaaaaaaa@463A@
and the CDF of the Exponential distribution is given by
R (a) (x)=1 e ax MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOuamaaBa aaleaacaGGOaGaamyyaiaacMcaaeqaaOGaaiikaiaadIhacaGGPaGa eyypa0JaaGymaiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiaadg gacaWG4baaaaaa@422D@
Theorem 3.3
The CDF of the Exponential-Weibull class of distributions of type II is given by
K (a,b,c) ** (x)=1 e ae ( x c ) b (1 e ( x c ) b ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4samaaDa aaleaacaGGOaGaamyyaiaacYcacaWGIbGaaiilaiaadogacaGGPaaa baGaaiOkaiaacQcaaaGccaGGOaGaamiEaiaacMcacqGH9aqpcaaIXa GaeyOeI0IaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadwgacaGG OaWaaSaaaeaacaWG4baabaGaam4yaaaacaGGPaWaaWbaaWqabeaaca WGIbaaaSGaaiikaiaaigdacqGHsislcaWGLbWaaWbaaWqabeaacqGH sislcaGGOaWaaSaaaeaacaWG4baabaGaam4yaaaacaGGPaWaaWbaae qabaGaamOyaaaaaaWccaGGPaaaaaaa@5475@
Proof in Corollary 2.7 λ:=a MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaeq4UdWMaai Ooaiabg2da9iaadggaaaa@3A54@ , let and F (α,θ) (x):= F (b,c) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaadIhacaGGPaGaaiOoaiabg2da9iaadAeadaWgaaWcbaGaaiikai aadkgacaGGSaGaam4yaiaacMcaaeqaaOGaaiikaiaadIhacaGGPaaa aa@479E@
Theorem 3.4
The CDF of the Exponential-Weibull class of distributions of type IV is given by
Q (a,b,c,d) ** (x)=1 e a (1 e ( x c ) b ) d 1 (1 e ( x c ) b ) d MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyuamaaDa aaleaacaGGOaGaamyyaiaacYcacaWGIbGaaiilaiaadogacaGGSaGa amizaiaacMcaaeaacaGGQaGaaiOkaaaakiaacIcacaWG4bGaaiykai abg2da9iaaigdacqGHsislcaWGLbWaaWbaaSqabeaacqGHsisldaWc aaqaaiaadggacaGGOaGaaGymaiabgkHiTiaadwgadaahaaadbeqaai abgkHiTiaacIcadaWcaaqaaiaadIhaaeaacaWGJbaaaiaacMcadaah aaqabeaacaWGIbaaaaaaliaacMcadaahaaadbeqaaiaadsgaaaaale aacaaIXaGaeyOeI0IaaiikaiaaigdacqGHsislcaWGLbWaaWbaaWqa beaacqGHsislcaGGOaWaaSaaaeaacaWG4baabaGaam4yaaaacaGGPa WaaWbaaeqabaGaamOyaaaaaaWccaGGPaWaaWbaaWqabeaacaWGKbaa aaaaaaaaaa@5E15@
Proof In Corollary 2.13, let ξ:=d MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaeqOVdGNaai OoaiaaygW7cqGH9aqpcaWGKbaaaa@3BF0@ λ:=a MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaeq4UdWMaai Ooaiabg2da9iaadggaaaa@3A54@ , and F (α,θ) (x):= F (b,c) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaadIhacaGGPaGaaiOoaiabg2da9iaadAeadaWgaaWcbaGaaiikai aadkgacaGGSaGaam4yaiaacMcaaeqaaOGaaiikaiaadIhacaGGPaaa aa@479E@ In order to compare the two distribution models, we used the following criteria: -2(Log likelihood) and AIC (Akaike information criterion) , AICC (corrected Akaike information criterion), and BIC (Bayesian information criterion) for the data set. The better distribution corresponds to the smaller -2(Log-likelihood) AIC, AICC, and BIC values:

AIC = 2k − 2l
AICC=AIC+ 2k(k+1) nk1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamyqaiaadM eacaWGdbGaam4qaiabg2da9iaadgeacaWGjbGaam4qaiabgUcaRmaa laaabaGaaGOmaiaadUgacaGGOaGaam4AaiabgUcaRiaaigdacaGGPa aabaGaamOBaiabgkHiTiaadUgacqGHsislcaaIXaaaaaaa@4778@
BIC = k log (n) − 2l

Where k is the number of parameters in the statistical model, n is the sample size, and l is the maximized value of the log-likelihood function under the considered model. From Table 1 above, it is clear the EWII (3975.82, 1.0863, 121474) distribution has the smallest AICC and BIC values, whilst the EWIV (216660, 1.47149, 1.21237_107, 0.681738) distribution has the smallest -2(Log-likelihood) and AIC values. When we compared the CDF’s of the two distributions we obtained the following

On the other hand when we compared the PDF’s of the two distributions we obtained the Following
The results from Table 1 and the Figures above, suggest the EWIV (216660, 1.47149, 1.21237_107, 0.681738) distribution is slightly better than the EWII (3975.82, 1.0863, 121474) distribution in modeling the aircraft data, Table 3 [Suleman Nasiru, Peter N. Mwita and Oscar Ngesa, Discussion on Generalized Modified Inverse Rayleigh, Appl. Math. Inf. Sci. 12, No. 1, 113-124 (2018)]
Concluding Remarks
Our hope is that the researchers will further develop the properties and applications of the new class of distributions presented in this paper. Finally we hope the new developments have practical significance in modeling biological data, health data, etc.

Note that L (α,θ,1,c) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaaIXaGaaiil aiaadogacaGGPaaabeaakiaacIcacaWG4bGaaiykaaaa@41B4@ implies the following from Section 1.3
Figure 1: The CDF of EWII(3975.82, 1.0863, 121474) (red) and EWIV (216660, 1.47149, 1.21237 _ 107, 0.681738) (blue) fitted to the empirical distribution of the aircraft data, Table 3 [Suleman Nasiru, Peter N. Mwita and Oscar Ngesa, Discussion on Generalized Modified Inverse Rayleigh, Appl.Math. Inf. Sci. 12, No. 1, 113-124 (2018)]
Figure 2: TThe PDF of EWII (3975.82, 1.0863, 121474) (red) and EWIV (216660, 1.47149, 1.21237 _ 107, 0.681738) (green) fitted to the empirical distribution of the aircraft data, Table 3 [Suleman Nasiru, Peter N. Mwita and Oscar Ngesa, Discussion on Generalized Modified Inverse Rayleigh, Appl.
Math. Inf. Sci. 12, No. 1, 113-124 (2018)]
Table 1: Criteria for Comparison

Model

-2(Log-likelihood)

AIC

AICC

BIC

EWII(3975.82, 1.0863, 121474)

307.351

313.351

314.274

317.554

EWIV (216660, 1.47149, 1.21237*107, 0.681738)

305.338

313.338

314.938

318.943

Theorem 4.1
The CDF of the exponentiated Exponential-MIR class of distributions of type I is given by G (α,θ,c,λ) * (x)=1 (1 F (α,θ) (x) c (x)) λ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4ramaaDa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaGGJbGaaiil aiabeU7aSjaacMcaaeaacaGGQaaaaOGaaiikaiaadIhacaGGPaGaey ypa0JaaGymaiabgkHiTiaacIcacaaIXaGaeyOeI0IaamOramaaBaaa leaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiikai aadIhacaGGPaWaaWbaaSqabeaacaWGJbaaaOGaaiikaiaadIhacaGG PaGaaiykamaaCaaaleqabaGaeq4UdWgaaaaa@5710@

Where F (α,θ) (x)= e ( α x + θ x 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamOramaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacMcaaeqaaOGaaiik aiaadIhacaGGPaGaeyypa0JaamyzamaaCaaaleqabaGaeyOeI0Iaai ikamaalaaabaGaeqySdegabaGaamiEaaaacqGHRaWkdaWcaaqaaiab eI7aXbqaaiaadIhadaahaaadbeqaaiaaikdaaaaaaSGaaiykaaaaaa a@4A54@

Similarly, L (α,θ,1,1) (x) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaamitamaaBa aaleaacaGGOaGaeqySdeMaaiilaiabeI7aXjaacYcacaaIXaGaaiil aiaaigdacaGGPaaabeaakiaacIcacaWG4bGaaiykaaaa@4187@ implies the Exponential-MIR class of distributions of type I from Section 1.3. Consequently several Corollaries can be deduced from Chapter 5[4], where they obtained several statistical/mathematical properties with application. For example, we have the following from Section 5.2 of Chapter 5[4]
Corollary 4.2
The survival function of the exponentiated Exponential-MIR class of distributions of type I is given by
S * (x)= {1 F (α,θ) c (x)} λ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeaaciGaciaabaqaamaabaabaaGcbaGaam4uamaaCa aaleqabaGaaiOkaaaakiaacIcacaWG4bGaaiykaiabg2da9iaacUha caaIXaGaeyOeI0IaamOramaaDaaaleaacaGGOaGaeqySdeMaaiilai abeI7aXjaacMcaaeaacaWGJbaaaOGaaiikaiaadIhacaGGPaGaaiyF amaaCaaaleqabaGaeq4UdWgaaaaa@4A36@

Proof Let d = 1 in eqn (5.8) contained in Section 5.2 of Chapter 5[4]
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