The Odd Power Lindley Generator of Probability Distributions: Properties, Characterizations and Regression Modeling

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1 The Odd Power Lidley Geerator of Probability Distributios: Properties, Characterizatios ad Regressio Modelig Mustafa Ç. Korkmaz 1, Emrah Altu 2, Haitham M. Yousof 3 & G.G. Hamedai 4 1 Departmet of Measuremet ad Evaluatio, Artvi Çoruh Uiversity, Artvi, Turkey 2 Departmet of Statistics, Barti Uiversity, Barti, Turkey 3 Departmet of Statistics, Mathematics ad Isurace, Beha Uiversity, Beha, Egypt 4 Departmet of Mathematics, Statistics ad Computer Sciece, Marquette Uiversity, USA Correspodece: Mustafa Ç. Korkmaz, Departmet of Measuremet ad Evaluatio, Artvi Çoruh Uiversity, 08000, Artvi, Turkey. Received: October 30, 2018 Accepted: December 17, 2018 Olie Published: Jauary 25, 2019 doi: /ijsp.v82p70 Abstract URL: I this study, a ew fleible family of distributios is proposed with its statistical properties as well as some useful characterizatios. The maimum likelihood method is used to estimate the ukow model parameters by meas of two simulatio studies. A ew regressio model is proposed based o a special member of the proposed family called, the log odd power Lidley Weibull distributio. Residual aalysis is coducted to evaluate the model assumptios. Four applicatios to real data sets are give to demostrate the usefuless of the proposed model. Keywords: miture family, power Lidley distributio, regressio modelig, Q-Q plot, sesitivity aalysis 1. Itroductio As of late, there has bee a etraordiary ethusiasm for itroducig more fleible distributios through etedig the classical distributios by icorporatig additioal shape parameters to the baselie model. May geeralized families of distributios have bee proposed ad studied over the last two decades for modelig data i may applied areas. So, several classes of distributios have bee costructed by etedig commo families of cotiuous distributios. These geeralized distributios give more fleibility via addig oe or more) shape parameters to the baselie model. They were pioeered by Gupta et al. 1998) who proposed the epoetiated-g Ep-G) class, which cosists of raisig the cumulative distributio fuctio cdf) to a positive power parameter. May other classes ca be cited such as the T-X family by Alzaatreh et al. 2013), the Loma-G by Cordeiro et al. 2014), Burr X geerator by Yousof et al. 2016), the geeralized two-sided class of distributios by Korkmaz ad Geç 2017), the Burr XII geerator by Cordeiro et al. 2018), amog others. Recetly, Ghitay et al. 2013) itroduced the power Lidley PL) distributio with the followig cdf ad probability desity fuctio pdf) F PL ; α, β) = 1 1+β α / β + 1) ] e βα, 0, 1) ad f PL ; α, β) = αβ α ) α 1 e βα / β + 1), > 0, 2) respectively, where α > 0 is a shape parameter ad β > 0 is a scale parameter. Usig the T- X idea, we defie the ew family by takig W G ; ψ )] = G,ψ ) 1 G,ψ ) ad rt) = f PL t; α, β), where G ; ψ ) is the baselie cdf depedig o the vector parameter ψ. The cdf of the ew family is give by F) = F ; α, β,ψ ) = β G,ψ ) β G,ψ ) α ep β G,ψ ) 1 G,ψ ), R. 3) α The pdf correspodig to 3) is give by 70

2 F) = f ; α, β,ψ ) = αβ 2 g,ψ ) G,ψ ) α 1 β + 1 )] α+1 4) 1 G,ψ 1 + G,ψ ) 1 G,ψ ) α ep β G,ψ ) 1 G,ψ ), R, α } {{ } A where g ;ψ ) is the baselie pdf ad α > 0 ad β > 0 are two etra parameters. The hazard rate fuctio hrf) is give by h ; α, β, ψ ) = αβ 2 g ; ψ ) G ; ψ ) α 1 β + 1) 1 G ; ψ )] α β β+1 G ;ψ ) ] α ) G ;ψ ) G ;ψ ) ] α ) 1 G ;ψ ), R. 5) We call this ew family the odd power Lidley-G family ad deote it by OPL Gα, β, ψ). For α = 1, the OPL G family is reduced to the OL G family which was itroduced by Silva et al. 2017). Let T be a PL radom variable i 1) ad 2). The OPL-G radom variable havig cdf 3) ca be derived as follows P X ) = Pr { T G ; ψ ) / 1 G ; ψ )]}. Hece, the radom variable X = G 1 T/ T + 1) ; ψ ] has the OPL-G distributio, where G 1 ; ψ ) is the iverse of the baselie cdf. We ca also motivate the OPL-G family with the miture family structure as follows. Let F 1 ; α, β, ψ ) be the cdf of the ) odd Weibull-G family OW-G) Bourguigo et al., 2014) ad F 2 ; α, β, ψ be the cdf of a geeralized gamma-g family GG-G). We ote that the cdfs of the OW-G ad GG-G families are give by ) { G ;ψ ) ] α } F 1 ; α, β, ψ = 1 ep β 1 G ;ψ ), R, ad ) F 2 ; α, β, ψ = β respectively. The, the OPL-G family ca be epressed as G ;ψ ) ] α ) 1 G ;ψ ) ep { β G ;ψ ) ] α } 1 G ;ψ ), R, OPL G ; α, β, ψ ) = p OW-G ; α, β, ψ )] + 1 p) GG-G ; α, β, ψ )], 6) where p = β/ β + 1) is the miig proportio. Hece, we ca say that the OPL-G family is a miture family. The rest of the paper is orgaized as follows. Useful epasios for pdf ad cdf of OPL-G family are preseted i Sectio 2. Some of its special cases are take up i Sectio 3. I Sectio 4, we derive some of its mathematical properties. Sectio 5 deals with some characterizatios of the ew family. Sectio 6 offers the maimum likelihood estimatio method. Two Mote Carlo simulatio studies performed i Sectio 7. A ew log-locatio regressio model as well as residual aalysis are preseted i Sectio 8. Sectio 9 is devoted to applicatios to real data sets to illustrate empirically the importace of the ew model. Fially, some coclusios ad future work are give i Sectio Useful Epasios for Desity of OPL-G Family By Epadig the quatity A i power series, the OPL-G pdf i 4) ca be epressed as f ) = j=0 + αβ 2+ j 1) j G ) α j+α 1 g ) β + 1 j! 1 G )] } {{ α j+α+1 } B 1 j=0 αβ 2+ j 1) j G ) α j+2α 1 g ). β + 1 j! 1 G )] } {{ α j+2α+1 } B 2 71

3 Usig the geeralized biomial epasio for the quatities B 1 ad B 2, we ca write f ) = = j.k=0 + αβ 2+ j β + 1 j,k=0 αβ 2+ j β + 1 1) j Γ α j + α + k + 1) g ) G ) α j+α+k 1 j!k!γ α j + α + 1) 1) j Γ α j + 2α + k + 1) g ) G ) α j+2α 1 j!k!γ α j + 2α + 1) a j,k α j + α + k) g ) G ) α j+α+k 1 j.k=0 + b j,k α j + 2α + k) g ) G ) α j+2α+k 1, j,k=0 fially where f ) = a j,k π α j+1)+k ) +b j,k π α j+2)+k ) ], 7) j,k=0 a j,k = b j,k = αβ 2+ j 1) j Γ α j + α + k + 1) j!k! β + 1) Γ α j + α + 1) α j + α + k), αβ 2+ j 1) j Γ α j + 2α + k + 1) j!k! β + 1) Γ α j + 2α + 1) α j + 2α + k), ad π δ ) = δg,ψ) G,ψ) δ 1 is the pdf of the Ep-G distributio with power parameter δ. The correspodig OPL-G cdf ca be writte as F ) = a j,k Π α j+1)+k ) +b j,k Π α j+2)+k ) ], 8) j.k=0 where Π δ ) = G,ψ) δ is the cdf of Ep-G with power parameter δ. Equatio 7) ad 8) reveal that pdf of OPL-G is a liear combiatio of Ep-G desities. Thereby, some properties of the proposed family such as momets ad geeratig fuctio ca be determied by meas of Ep-G distributio. The properties of Ep-G distributios have bee studied by may authors i recet years, see Shirke ad Kakade 2006) for epoetiated log-ormal ad Nadarajah ad Gupta 2007) for epoetiated gamma distributios, amog others. 3. Some Members of the OPL-G Family The OPL-G family ca produce great fleible models by usig ay baselie models. Here, we preset three special cases of this family sice they eted some widely-kow distributios. We give pdf ad cdf of the ew distributios. The hrfs of these distributios ca be obtaied from Equatio 5). 3.1 The OPL-Normal OPL-N) Distributio The ormal distributio, symmetrical uimodal pdf shaped ad icreasig hrf shaped, is well-kow i statistics ad other areas. So, the OPL-N distributio is defied from 3) ad 4) by takig G) ad g) to be the cdf ad pdf of the ormal distributio. Its pdf ad cdf are give by ad f OPL N ; Θ) = αβ2 ϕ µ F OPL N ; Θ) = β β + 1 ) Φ µ ) ] α 1 β + 1) 1 Φ )] µ α µ ) ep β Φ µ ) α, R, 1 Φ µ ) Φ µ ) 1 Φ α ep β µ ) Φ µ ) 1 Φ µ ) Φ µ ) 1 Φ α α, R, respectively, where Θ = α, β, µ, ) T is the parameter vector, µ R, α, β, > 0, ϕ ) ad Φ ) are pdf ad cdf of the stadard ormal distributio respectively. We deote it by OPL N Θ). For α = 1, the OL-ormal OLN) distributio is 72

4 obtaied. Plots of the OPL-N desity ad hazard fuctios for selected parameter values are displayed i Figure 1. From this Figure, we see that we obtai bi-modal shaped, firstly ui-modal shaped ad the icreasig shaped, left skewed ad right skewed distributios. Also, the plots idicate that the hrf of the OPL-N distributio is icreasig ad the bathtube shaped. Desity OPL N0.1,2,0,1) OPL N0.1,1.5,0,1) OPL N0.1,1,0,1) OPL N0.1,0.75,0,1) OPL N0.1,0.5,0,1) Desity OPL N0.05,25,12,1) OPL N1,1,0,1) OPL N1.1,1,2,1) OPL N0.75,0.75,3,1) OPL N0.5,1, 2,1) Desity OPL N0.01,0.25,0,1) OPL N0.05,0.5,0,1) OPL N0.01,1,0,1) OPL N0.01,0.5,0,1) OPL N0.05,0.25,0,1) hrf OPL N0.01,2,0,1) OPL N0.1,0.5, 5,1) OPL N2,3, 1,5) OPL N0.02,0.5, 15,1) OPL N0.5,0.5, 15,5) Figure 1. The pdf ad hrf of the OPL-N distributio for selected parameter values 3.2 The OPL-Weibull OPL-W) Distributio The Weibull cdf with the shape γ > 0 ad the scale parameters θ > 0 is G ; θ, γ) = 1 ep θ) γ ) for > 0). The pdf ad cdf of a radom variable X with OPL-W distributio, say X OPL W Θ) are, respectively, give by f OPL W ; Θ) = αγθγ β 2 γ 1 1 e ) α 1 θ)γ 1 + e θ)γ 1 ) α] { ep β e θ) γ 1 ) α}, > 0, β + 1) e αθ)γ ad F OPL W ; Θ) = β e θ) γ 1 ) ] α ep { β e θ)γ 1 ) α}, 0 β + 1 where Θ = α, β, θ, γ) T is the parameter vector ad α, β, θ, γ > 0. For α = 1, the OL-Weibull OLW) distributio Silva et al., 2017) is obtaied. Plots of the OPL-W desity ad hazard fuctios for selected parameter values are displayed i Figure 2. From this Figure, we ca say that the OPL-W distributio has very fleible pdf shapes such as ui-modal, decreasig, U-shaped,firstly U-shaped ad the decreasig shaped. Also, its hrf ca be icreasig, decreasig ad bathtube shaped. 73

5 Desity OPL W0.005,1,1,0.5) OPL W0.005,1,1,2) OPL W0.005,1,1,3) OPL W0.005,0.5,1,2) OPL W0.005,0.5,1,3) Desity OPL W0.5,1,1,0.5) OPL W1,1,1,1) OPL W2,1,1,0.5) OPL W3,1,1,0.5) OPL W4,1,1,0.5) Desity OPL W0.2,0.5,1,1) OPL W0.2,0.5,1,2) OPL W0.5,1,1,2) OPL W0.1,0.3,1,3) OPL W0.09,2,1,5) hrf OPL W0.03,1,1,3) OPL W5,1,1,1) OPL W0.5,1,1,0.25) OPL W2,0.6,1,2) OPL W0.1,0.2,1,2.5) Figure 2. The pdf ad hrf of the OPL-W distributio for selected parameter values 3.3 The OPL-Gamma OPL-Ga) Distributio Cosider the gamma distributio with the shape parameter γ > 0 ad the scale parameter θ > 0, where the pdf ad cdf for > 0) are give by G ; λ, θ) = γ λ, θ) Γ 1 λ), where γ λ, θ) = θ t λ 1 e t dt is the upper icomplete gamma fuctio 0 ad Γ ) is complete gamma fuctio. The pdf ad cdf of OPL-Ga are give by ad f OPL Ga ; Θ) = F OPL Ga ; Θ) = β β + 1 αβ 2 θ λ λ 1 e θ γ λ, θ) Γ 1 λ) ] α 1 β + 1) Γ λ) 1 γ λ, θ) Γ 1 λ) ] α+1 ep { β ) γλ,θ) α } Γλ) γλ,θ) γλ,θ) Γλ) γλ,θ) 1 + γλ,θ) Γλ) γλ,θ) ) ] α ep { β α } Γλ) γλ,θ)), respectively, where Θ = α, β, θ, λ) T is the parameter vector ad α, β, θ, λ > 0. We deote it by OPL Ga Θ). For α = 1, the OL-gamma OLGa) distributio is obtaied. Plots of the OPL-Ga desity ad hazard fuctios for selected parameter values are displayed i Figure 3. From this Figure, we observe decreasig, ui-modal shaped, U-shaped ad firstly U- shaped ad the decreasig shaped distributios. Also, the plots poit out that the OPL-Ga distributio has decreasig, icreasig, bathtube shaped. ) α ] 74

6 Desity OPL Ga0.1,0.3,1.5,1) OPL Ga0.3,0.1,0.9,0.1) OPL Ga0.5,0.3,0.5,0.3) OPL Ga1.1,0.3,0.5,2.5) OPL Ga1,1,0.2,0.5) hrf OPL Ga0.1,0.3,1.5,1) OPL Ga0.3,0.1,0.9,0.1) OPL Ga1.1,0.3,0.5,2.5) OPL Ga0.05,1,0.5,1) OPL Ga0.5,0.3,0.5,0.3) Figure 3. The pdf ad hrf of the OPL-Ga distributio for selected parameter values 4. Mathematical Properties 4.1 Momets, Icomplete Momets ad Geeratig Fuctio The r th ordiary momet of X is give by µ r = EX r ) = r f ) d. The we obtai µ r = a j,k Yα r j+1)+k +b ] j,kyα r j+2)+k. 9) j,k=0 Heceforth, Y δ deotes the Ep-G model with power parameter δ. For δ > 0, we have E ) Yδ r = δ r g ; ψ ) G ; ψ ) δ 1 d, ) which ca be computed umerically i terms of the baselie quatile fuctio qf) Q G u; ψ = G 1 u; ψ ) as E ) Yδ = δ 1 ) Q 0 G u; ψ u δ 1 du. Settig r = 1 i 9), we have the mea of X. The last itegratio ca be computed umerically for most paret distributios. The skewess ad kurtosis measures ca be calculated from the ordiary momets usig well-kow relatioships. The th cetral momet of X, say M, is ) M = EX µ) = 1) h µ 1 h ) µ h. h=0 The r th icomplete momet, say I r t), of X ca be epressed from 6) as t { t I r t) = r f ) d = a j,k r π α j+1)+k ) +b j,k r π α j+2)+k ) ] } d. 10) j,k=0 The mea deviatios about the mea Υ 1 = E X µ 1 )] ad about the media Υ 2 = E X M )] of X are give by Υ 1 = 2µ 1 Fµ 1 ) 2I 1µ 1 ) ad Υ 2 = µ 1 2I 1 M), respectively, where µ 1 = E X), M = MediaX) = Q0.5) is the media, Fµ 1 ) is easily calculated from 3) ad I 1 t) is the first icomplete momet give by 10) with r = 1. A geeral equatio for I 1 t) ca be derived from 10) as I 1 t) = a j,k J α j+1)+k ) + b j,k J α j+2)+k ) ], j,k=0 where J δ ) = t π δ ) d is the first icomplete momet of the Ep-G model. The momet geeratig fuctio mgf) M X t) = E e t X) of X ca be derived usig equatio 9) as M X t) = a j,k M α j+1)+k t) +b j,k M α j+2)+k t) ], j,k=0 where M δ t) is the mgf of Y δ. Hece, M X t) ca be determied from the Ep-G geeratig fuctio. 75

7 4.2 Quatile Fuctio qf) ad Radom Number Geeratio The OPL-G family ca easily be simulated by ivertig 3) as follows: if U U0, 1), the the radom variable X U ca be obtaied from the baselie qf, say Q G u) = G 1 u). I fact, the radom variable 1 X U = G 1 β W 1β+1)U 1)e β 1 ) β β W 1β+1)U 1)e β 1 ) β 1 has cdf 3), where G 1 ) is the iverse of the baselie cdf ad W 1 ) deotes the egative brach of the Lambert W fuctio. X U ca be used as a radom umber geerator for OPL-G distributio. Also, we ca obtai radom umber from OPL-G by usig miture structure i 6). We ca give this procedure with the followig a algorithm. AlgorithmMiture Form) 1. Geerate U Ui f orm0, 1); 2. Geerate Y OW Gα, β, ψ); 3. Geerate Z GG Gα, β, ψ); 4. if U β β+1, the X = Y otherwise, set X = Z. By usig packet programme, the radom variates from the OLP-G distributio ca be geerated by the trasformatio method. For eample, we first geerate a radom variate Y from the PL distributio by usig the rplidley fuctio i the LidleyR package i R program, the set X = G 1 Y Y+1). 5. Characterizatios This sectio deals with various characterizatios of OPL-G distributio. These characterizatios are preseted i two directios: i) based o a simple relatioship betwee two trucated momets ad ii) i terms of the hazard fuctio. It should be poited out that due to the ature of the OPL-G distributio our characterizatios may be the oly possible oes for this distributio. We preset our characterizatios i) ad ii) i two subsectios. 5.1 Characterizatios Based o Trucated Momets We employ a theorem due to Gläzel 1987), see Theorem 1 of Appedi A.The result, however, holds also whe the iterval H is ot closed sice the coditio of Theorem 1 is o the iterior of H. We like to metio that this kid of characterizatio based o a trucated momet is stable i the sese of weak covergece see, Gläzel 1990). α ) 1 Propositio 5.1. Let X : Ω R be a cotiuous radom variable ad let q 1 ) = G,ψ)] ad q2 ) = q 1 ) ep { β G,ψ) 1 G,ψ) i Theorem1 has the form ] α } for R. The radom variable X belogs to the family 4) if ad oly if the fuctio η defied Proof. Let X be a radom variable with pdf 4), the ad ) 1/α ) 1/α η ) = 1 2 ep β G, ψ) α 1 G, ψ), R. 1 F )) E q 1 X) X ] = β α β + 1 ep β G, ψ) 1 G, ψ) 1 F )) E q 2 X) X ] β = 2 β + 1) ep 2β G, ψ) α 1 G, ψ). 76

8 Further, Coversely, if η is give as above, the η ) q 1 ) q 2 ) = q 1 ) 2 ep β G, ψ) α 1 G, ψ) < 0 f or R. s η ) q 1 ) ) = η ) q 1 ) q 2 ) = αβg ) ) α 1, ψ G, ψ )] α+1, R, 1 G, ψ ad hece α G, ψ) s ) = β 1 G, ψ), R. Now, accordig to Theorem 1, X has desity 4). Corollary 5.1. Let X : Ω R be a cotiuous radom variable ad let q 1 be as i Propositio 5.1. The, X has pdf 4) if ad oly if there eist fuctios q 2 ad η defied i Theorem 1 satisfyig the differetial equatio η ) q 1 ) η ) q 1 ) q ) = αβg ) ) α 1, ψ G, ψ )] α+1, R. 1 G, ψ The geeral solutio of the differetial equatio i Corollary 5.1 is { η ) = ep G,ψ) 1 G,ψ) ] α } β αβg,ψ ) G,ψ ) α 1 )] α+1 ep 1 G,ψ { G,ψ) ] α } ] β 1 G,ψ) q 1 )) 1 q 2 ) d + D, where D is a costat. Note that a set of fuctios satisfyig the above differetial equatio is give i Propositio 5.1 with D = 0. Clearly, there eist other triplet of fuctios q 1, q 2, η) satisfyig the coditios of Theorem Characterizatio Based o Hazard Fuctio It is kow that the hazard fuctio, h F, of a twice differetiable distributio fuctio, F, satisfies the first order differetial equatio f ) f ) = h F ) h F ) h F). For may uivariate cotiuous distributios, this is the oly characterizatio available i terms of the hazard fuctio. The followig characterizatio establish a o-trivial characterizatio of OPL-G distributio, for α = 1, i terms of the hazard fuctio, which is ot of the above trivial form. Propositio 5.2. Let X : Ω R be a cotiuous radom variable. For α = 1,the pdf of X is 4) if ad oly if its hazard fuctio h F ) satisfies the differetial equatio h F ) g, ψ ) g, ψ ) h F ) = β 2 g, ψ ) d d )] 1 β + 1 G, ψ 1 G, ψ )] 2, R, with the iitial coditio lim h F ) = β2 β+1 lim g, ψ ). Proof. If X has pdf 4), the clearly the above differetial equatio holds. Now, if the differetial equatio holds, the 77

9 or d { g, ψ ) } 1 hf ) = β 2 d d d )] 1 β + 1 G, ψ 1 G, ψ )] 2 h F ) = β 2 g, ψ ) )] 1 β + 1 G, ψ )] 2 1 G, ψ,, which is the hazard fuctio of the OPL-G distributio for α = 1. Remark 5.1. It is easy to see that )] 1 d β + 1 G, ψ d )] 2 1 G, ψ = g, ψ ) )] 2 β + 1 G, ψ )] 2 1 G, ψ 3 + 2β 1 G, ψ ). 6. Estimatio ad Iferece Here, we cosider estimatio of the ukow parameters of the OPL-G distributio by the maimum likelihood method. Let 1,..., be a radom sample from the OPL-G distributio with a q + 2) 1 parameter vector Ψ =α, β, ψ), where ψ is a q 1 baselie parameter vector. The log-likelihood fuctio for Ψ is give by lψ) = log α 2 log β log β + 1) + α + 1) log 1 G i ; ψ )] + log g i ; ψ ) + α 1) log ) 1 + η α i β η α i, where η i = G i ; ψ ) / 1 G i ; ψ )]. The compoets of the score vector, U Ψ) = l Ψ = ad for r = 1,..., q) where l α = α + log G i ; ψ) l ψ r = log 1 G i ; ψ )] + l β = 2 β η α i log η i ) 1 + η α β i β + 1 η α i, g i ; ψ)/ ψ r g ] ) + α 1) Gi ; ψ)/ ψ i ; ψ r G i ; ψ)/ ψ + α + 1) r 1 G i ; ψ ) +α ζ i η α 1 i 1 + η α αβ i ζ i = G i ; ψ)/ ψ r ] 1 G i ; ψ )] 2. log G i ; ψ) l α, l β, l η α i log η i), ζ i η α 1 i, ψ ), are give as Settig the oliear system of equatios U α = U β = U ψr = 0 for r = 1,..., q) ad solvig them simultaeously yields the MLEs Ψ = α, β, ψ ). To solve these equatios, it is more coveiet to use oliear optimizatio methods such as the quasi-newto algorithm to umerically maimize lψ). For iterval estimatio of the parameters, we ca evaluate 78

10 umerically the elemets of the q + 2) q + 2) observed iformatio matri JΨ) = { 2 l θ r θ s }. Uder stadard regularity coditios whe, the distributio of Ψ ca be approimated by a multivariate ormal N p 0, J Ψ) 1 ) distributio to costruct approimate cofidece itervals for the parameters. Here, J Ψ) is the total observed iformatio matri evaluated at Ψ. The method of the re-samplig bootstrap ca be used for correctig the biases of the MLEs of the model parameters. Good iterval estimates may also be obtaied usig the bootstrap percetile method. We ca compute the maimum values of the urestricted ad restricted log-likelihoods to obtai likelihood ratio LR) statistics for testig sub-model of the OPL-G distributio. Hypothesis tests of the type H 0 : ω = ω 0 versus H 1 : ω ω 0, where ω is a vector formed with some compoets of Ψ ad ω 0 is a specified vector, ca be performed usig LR statistics. For eample, the test of H 0 : α = 1 versus H 1 : H 0 is ot true is equivalet to comparig the OPL-G ad PL-G distributios ad the LR statistic is give by w = 2{l α, β, ψ) l1, β, ψ)}, where α, β ad ψ are the MLEs uder H ad ψ is the estimate uder H 0. We ca compute the maimum values of the urestricted ad restricted log-likelihoods to obtai likelihood ratio LR) statistics for testig some sub-models of the OPL-G distributio. 7. Log-OPL-W Regressio Model Let X be a radom variable havig the OPL-W desity fuctio with four parameters α > 0, β > 0, θ > 0 ad γ > 0, give i Sectio 3. The desity fuctio of Y = logx), replacig γ = 1/ ad θ = ep µ), is give by for y R) f y) = 1 + ep αβ2 y µ ) ep y µ )]{1 ep ep y µ )]} α 1 β+1) 2 ep ep y µ )]] α+1 {1 ep ep y µ { {1 ep ep y µ )]} 2 ep ep y µ )] ] α ) ep β )]} 2 ep ep y µ )] ] α }, 11) where µ R is the locatio parameter, > 0 is the scale parameter ad α > 0, β > 0 are the shape parameters. LOPLW distributio is deoted as Y LOPLWα, β,, µ). Figure 5 displays desity plots of LOPLW distributio for some parameter values. As see from Figure 4, LOPLW distributio ca be very fleible for modelig left skewed data. fy) α=0.5,β=0.9,µ=0,=1 α=0.5,β=0.7,µ=0,=1 α=0.5,β=0.5,µ=0,=1 α=0.5,β=0.3,µ=0,=1 α=0.5,β=0.1,µ=0,= The correspodig survival fuctio is Figure 4. Plots of the LOPLW desity for selected parameter values S y) = 1 + β β + 1 { 1 ep ep y µ 2 ep ep y µ )]} { 1 ep ep y µ )]} )] α α ep β 2 ep ep )] y µ. 12) The stadardized radom variable Z = Y µ)/ has desity fuctio f z) = αβ2 { ep β epz epz)]{1 ep epz)]} α 1 β+1) 2 ep epz)]] α+1 {1 ep epz)]} 2 ep epz)] ] α }. 1 + ] {1 ep epz)]} α ) 2 ep epz)] 13) 79

11 Based o the LOPLW desity, give i 11), the log-liear locatio-scale regressio model is proposed by likig the respose variable y i ad the eplaatory variable vector v i = ) v i1,..., v ip by y i = v i β + z i, i = 1,...,, 14) where the radom error z i has desity fuctio 13), β = β 1,..., β p ), > 0, α > 0 ad β > 0 are ukow parameters. The parameter µ i = v i β is the locatio of y i. The locatio parameter vector µ = µ 1,..., µ ) is represeted by a liear model µ = Vβ, where V = v 1,..., v ) is a kow model matri. Let y 1, v 1 ),..., y, v ) be a sample of idepedet observatios, where each radom respose is defied by y i = mi{log i ), logc i )}. Let F ad C be the sets of idividuals for which y i is the log-lifetime or log-cesorig, respectively. The log-likelihood fuctio for the vector of parameters τ = α, β,, β ) from model 14) has the form lτ) = l i τ) + i F τ), where l i τ) = log f y i )], l c) τ) = logs y i )], f y i ) is the desity 11) ad S y i ) is the survival fuctio 12) of i C l c) i i Y i. The, the total log-likelihood fuctio for τ is give by ) l τ) = r log αβ 2 β+1) + z i u i ) + α 1) log { 1 ep u i ] } i F i F α + 1) log { 2 ep u i ] } i F + ] {1 ep ui ]} α ) log ep u i F i ] β ] {1 ep ui ]} α 15) 2 ep u i F i ] + ] log 1 + β {1 ep ui ]} α ) β+1 2 ep u i C i ] β ] {1 ep ui ]} α, 2 ep u i C i ] where u i = epz i ), z i = y i v i β)/ ad r is the umber of ucesored observatios ad c is the umber of cesored observatios. The MLE τ of the vector of ukow parameters ca be evaluated by maimizig the log-likelihood 15). The optim fuctio of R software is used to miimize the egative log-likelihood fuctio, give i 15). The asymptotic distributio of τ τ) is multivariate ormal N p+2 0, Kτ) 1 ), where Kτ) is the epected iformatio matri. The asymptotic covariace matri Kτ) 1 of τ ca be approimated by the iverse of the p+2) p+2) observed iformatio matri Łτ), whose elemets ca be evaluated umerically. The approimate multivariate ormal distributio N p+2 0, Łτ) 1 ) for τ ca be used to costruct approimate cofidece itervals for the parameters of τ. 8. Simulatio Studies I this Sectio, we perform two simulatio studies by usig the OPL-W ad OPL-N distributios to illustrate the performace of MLEs for the parameters of these distributio. The radom umbers geeratio is obtaied with rplidley fuctio i the LidleyR package i R program. This method is give by qf subsectio. MLEs results were obtaied usig optim-cg routie i the R programme. I the first simulatio study, we obtai the graphical results ad geerate N = 1000 samples of size = 20, 30,..., 1000 from OPL-W distributio with true parameters values α = 0.5, β = 10, θ = 1 ad γ = 2. I this simulatio study, we calculate the empirical mea, stadard deviatios sd), bias ad mea square error MSE) of the MLEs. The bias ad 1000 MSE are calculated by Bias h = ĥi h ) ad MS E h = 1 give results of this simulatio study i Figure ĥi h ) 2, respectively for h = α, β, θ, γ. We I the secod simulatio study, we geerate 1, 000 samples of sizes 20,50 ad 100 from selected OPL-N distributios. For this simulatio study, we obtai the empirical meas ad sd s of the parameters.the results of this simulatio study are reported i Table 1. From Figure 5 ad Table 1, we observe that whe the sample size icreases, the empirical meas approach the true parameter value for both distributios. For the same case, the stadard deviatios, biases ad MSEs decrease i all the cases. The above results are as epected. 80

12 mea of alpha sd of alpha 0.05 Bias of alpha MSE of alpha mea of beta sd of beta 0.05 Bias of beta MSE of beta mea of theta sd of theta Bias of theta MSE of theta mea of gamma sd of gamma Bias of gamma MSE of gamma Figure 5. Simulatio results of the special OPL-W distributio Table 1. Empirical meas ad stadard deviatios i paretheses) for the special OPL-N distributios Parameters = 20 = 50 = 100 α, β, µ, α β µ α β µ α β µ 1,5,0, ) ) ) ) ) ) ) ) ) ) ) ) 0.5,5,0, ) ) ) ) ) ) ) ) ) ) ) ) 0.5,5,-1, ) ) ) ) ) ) ) ) ) ) ) ) 2,10,1, ) ) ) ) ) ) ) ) ) ) ) ) 0.5,5,5, ) ) ) ) ) ) ) ) ) ) ) ) 10,3,-5, ) ) ) ) ) ) ) ) ) ) ) ) 1,3,0.5, ) ) ) ) ) ) ) ) ) ) ) ) 0.5,5,-0.5, ) ) ) ) ) ) ) ) ) ) ) ) 9. Real Data Applicatios I this sectio, we preset three applicatios to real data sets to illustrate the usefuless of the OPL-N, OPL-W ad OPL- Ga distributios. We compare these distributio model with the odd Burr ormal OBN) Alizadeh et al., 2017), power ormal PN) Gupta ad Gupta, 2008), OLN, ormal N), Loma-Weibull LW) Cordeiro et al., 2014), OLW, Weibull W), epoetiated geeralized gamma EGGa) Cordeiro et al., 2011), OLGa, gamma Ga) ad PL models. The pdfs of the OBN, PN, LW ad EGGa are give by 81

13 f OBN ; α, β, µ, ) = αβϕ µ { Φ µ ) Φ µ ) α 1 1 Φ µ ) α + 1 Φ µ )] αβ 1 ) a ]} β+1,, µ R, α, β, > 0, f PN ; β, µ, ) = β 1 ϕ µ ) Φ µ )] β 1,, µ R, β, > 0, f LW ; α, β, θ, γ) = αβ α γθ γ γ 1 β + θ) γ] α 1, > 0, α, β, θ, γ > 0 ad f EGGa ; α, β, θ, λ) = αλθ θ) λ 1 Γ 1 β) γ β, θ) λ) Γ 1 β) ] α 1 e θ) λ, > 0, α, β, θ, λ > 0, respectively. To determie the best model, we also computed the estimated log-likelihood values ˆl, Akaike Iformatio Criteria AIC), corrected Akaike iformatio criterio CAIC), Bayesia iformatio criterio BIC), Haa Qui iformatio criterio HQIC), Kolmogorov-Smirov K-S), Cramer vo Mises W ) ad Aderso-Darlig A ) goodess of-fit statistics for all distributio models. We ote that the statistics W ad A are described i detail i Che ad Balakrisha 1995). I geeral, it ca be chose as the best model which has the smaller the values of the AIC, CAIC, BIC, HQIC, K-S, W ad A statistics ad the larger the values of ˆl ad p-values. All computatios are performed by the malike routie i the R programme. The details are the followigs. 9.1 Data Modellig for Three Sub-models The first real data set is breakig stregths of 100 yar give by Duca 1974). The data are: 66, 117, 132, 111, 107, 85, 89, 79, 91, 97, 138, 103, 111, 86, 78, 96, 93, 101, 102, 110, 95, 96, 88, 122, 115, 92, 137, 91, 84, 96, 97, 100, 105, 104, 137, 80, 104, 104, 106, 84, 92, 86, 104, 132, 94, 99, 102, 101, 104, 107, 99, 85, 95, 89, 102, 100, 98, 97, 104, 114, 111, 98, 99, 102, 91, 95, 111, 104, 97, 98, 102, 109, 88, 91, 103, 94, 105, 103, 96, 100, 101, 98, 97, 97, 101, 102, 98, 94, 100, 98, 99, 92, 102, 87, 99, 62, 92, 100, 96, 98. This data also aalyzed by Tahir et al. 2017). The secod data, studied by Meeker ad Escobar 1998, p. 383), gives the times of failure ad ruig times for a sample of devices from a field-trackig study of a larger system. The data are: 275, 13, 147, 23, 181, 30, 65, 10, 300, 173, 106, 300, 300, 212, 300, 300, 300, 2, 261, 293, 88, 247, 28, 143, 300, 23, 300, 80, 245, 266. This data also aalyzed by Cordeiro et al. 2010) ad Aleader et al. 2012). The third data set refers to the lifetimes of 50 idustrial devices put o life test at time zero give by Aarset 1987). The data are: 0.1, 0.2, 1.0, 1.0, 1.0, 1.0, 1.0, 2.0, 3.0, 6.0, 7.0, 11.0, 12.0, 18.0, 18.0, 18.0, 18.0, 18.0, 21.0, 32.0, 36.0, 40.0, 45.0, 45.0, 47.0, 50.0, 55.0, 60.0, 63.0, 63.0, 67.0, 67.0, 67.0, 67.0, 72.0, 75.0, 79.0, 82.0, 82.0, 83.0, 84.0, 84.0, 84.0, 85.0, 85.0, 85.0, 85.0, 85.0, 86.0, These data are also aalyzed by Cordeiro et al. 2017). The results are reported i Table 2 ad 3. These Tables clearly show that the distributio models of the OPL-G family model have the smallest values of the AIC, CAIC, BIC, HQIC, K-S, W ad A statistics ad have the largest values of the ˆl ad all p-values amog the fitted models. The, OPL-N, OPL-W ad OPL-Ga models could be chose as the best models for the three data sets uder the above criteria. The histograms of the all data sets ad the plots of the fitted pdfs ad cdfs for all models are show i Figures 6-8. From these Figures, we see that the OPL-N model fits to first data set as bi-modal shaped. For other data sets, the OPL-W ad OPL-Ga models fit to the histograms of the data sets with a more adequate fittig tha other models. The results of LR statistics are show i Table 4 for three data sets. We ca say that the additioal parameter of the all OPL-G distributio is essetial because we reject the ull hypotheses of all three LR tests i favour of the OPL-N, OPL-W ad OPL-Ga distributios. Hece, these models provide a better represetatio of the data sets tha the OLN, OLW ad OLGa models based o the LR test at the 5% sigificace level. 82

14 Desity OPL N OBN OLN PN N CDF OPL N OBN OLN PN N Emprical Figure 6. The fitted pdfs ad cdfs for the first data sets Desity OPL W LW OLW PL W CDF OPL W LW OLW PL W Emprical Figure 7. The fitted pdfs ad cdfs for the secod data sets Desity OPL Ga EGGa OLGa PL Ga CDF OPL Ga EGGa OLGa PL Ga Emprical Figure 8. The fitted pdfs ad cdfs for the third data sets 83

15 Table 2. MLEs, stadard erros of the estimates i paretheses), estimated log-likelihood values ˆl ) ad, K-S statistics ] deotes its p-value) Data Set Model α β µ ˆl K S I OPL-N ) ) ) ) ] OBN ) ) ) ) ] OLN ) ) ) ] PN ) ) ) ] N ) ) ] θ γ II OPL-W ) ) 7e-07) 3e-05) ] LW ) ) ) ) ] OLW ) ) ) ] PL ) ) ] W ) ) ] θ λ III OPL-Ga ) ) ) ) ] EGGa ) ) ) ) ] OLGa ) ) ) ] PL ) ) ] Ga ) ) ] 84

16 Table 3. Iformatio criteria results, A ad W statistics ] ad { } deote their p-values) Data Set Model AIC CAIC BIC HQIC A W I OPL-N ] {0.3859} OBN ] {0.2382} OLN ] {0.0039} PN ] {0.0908} N ] {0.0557} II OPL-W ] {0.2593} LW ] {0.1101} OLW ] {0.1591} PL ] {0.0978} W ] {0.1101} III OPL-Ga ] {0.3577} EGGa ] {0.0199} OLGa ] {0.1043} PL ] {0.0321} Ga ] {0.0282} Table 4. LR statistics for the data sets Data Set Model Hypothesis w p-value I OPL N vs OLN H 0 : α = 1, H 1 : H 0 false e-12 II OPL W vs OLW H 0 : α = 1, H 1 : H 0 false III OPL Ga vs OLGa H 0 : α = 1, H 1 : H 0 false e Regressio Modellig: Staford Heart Trasplat Data Recetly, Brito et al. 2017) itroduced the Log-Topp-Leoe odd log-logistic-weibull Log-TLOLL-W) regressio model. Brito et al. 2017) used the Staford heart trasplat data set to prove the usefuless of Log-TLOLL-W regressio model. Here, we use the same data set to demostrate the fleibility of LOPLW regressio model agaist to Log-TLOLL-W regressio model. These data set is available i p3state.msm package of R software. The sample size is = 103, the percetage of cesored observatios is 27%. The aim of this study is to relate the survival times t) of patiets with the followig eplaatory variables: 1 - year of acceptace to the program; 2 - age of patiet i years); 3 - previous surgery status 1 = yes, 0 = o); 4 - trasplat idicator 1 = yes, 0 = o); c i - cesorig idicator 0 =cesorig, 1 =lifetime observed). The regressio model fitted to the voltage data set is give by y i = β 0 + β 1 i1 + β 2 i2 + β 3 i3 + β 4 i4 + z i, 16) respectively, where the radom variable y i follows the LOPLW distributio give i 11). 85

17 The results for above regressio models are preseted i Table 5. The MLEs of the model parameters ad their SEs, p values ad l, AIC ad BIC statistics are listed i Table 5. Based o the figures i Table 5, LOPLW model has the lowest values of the l, AIC ad BIC statistics. Therefore, it is clear that LOPLW regressio model outperforms amog others for these data set. Accordig to results of LOPLW regressio model, β 1 ad β 2 are statistically sigificat at 5% level. Table 5. MLEs of the parameters to Staford Heart Trasplat Data for Log-Weibull, Log-TLOLL-W ad LOPLW regressio models with correspodig SEs, p-values ad l, AIC ad BIC statistics Parameters Models Log-Weibull Log-TLOLL-W LOPLW Estimate S.E. p-value Estimate S.E. p-value Estimate S.E. p-value α β β β β < <0.001 β β < < l AIC BIC Residual Aalysis of LOPLW model for Staford heart trasplat data set Residual aalysis is coducted to evaluate the adequacy of the fitted model. For this goal, two type residual are cosidered: martigale ad modified deviace residuals. The martigale residuals is defied i coutig process ad takes values betwee +1 ad. The martigale residuals for LOPLW model is, r Mi = log 1 + log 1 + β 1 + β β+1 β+1 ] {1 ep ui ]} α ) 2 ep u i ] ] α ) ep {1 ep ui ]} 2 ep u i ] { ep { β ] {1 ep ui ]} α } β 2 ep u i ] ifi F, ] {1 ep ui ]} α } ifi C, 2 ep u i ] 17) where u i = epz i ), z i = y i v i β)/. The modified deviace residual, proposed by Thereau et al. 1990), is give by { ) { sig rmi 2 rmi + log )]} 1 r 1/2 r Di = Mi, ifi F sig ) { } r 1/2 Mi 2rMi, ifi C, 18) where ˆr Mi is the martigale residual. Figure 9 displays the ide plot of the modified deviace residuals ad its Q-Q plot agaist to N0, 1) quatiles for Staford heart trasplat data set. Based o Figure 9, we coclude that oe of observed values appears as possible outliers. Therefore, the fitted model is appropriate for these data set. 86

18 a) b) Modified Deviace Residual Ide N0,1) quatiles Figure 9. a) Ide plot of the modified deviace residual ad b) Q-Q plot for modified deviace residual 10. Coclusios I this paper, we proposed a ew fleible class of distributios ad provided a comprehesive treatmet its mathematical properties as well as some useful characterizatios. The maimum likelihood method is used to estimate the model parameters, we assess the performace of the maimum likelihood estimators by meas of two simulatio studies. Also we itroduce a ew regressio model based o a special member of the ew family called the log odd power Lidley Weibull distributio. We show that the ew log locatio-scale regressio model for lifetime data ca be very useful i aalysig real data ad provide more realistic fits tha other regressio models. Ide plot of the modified deviace residual ad Q-Q plot for modified deviace residual are preseted to illustrate that our ew model is more appropriate to Staford heart trasplat data set tha other competitive models like log-weibull ad log-topp-leoe odd log-logistic- Weibull model. We hope that the results give i this paper will be useful for practitioers ad researchers. Appedi A Theorem 1. Let Ω, F, P be a give probability space ad let H = a, b] be a iterval for some d < b a =, b = might as well be allowed. Let X : Ω H be a cotiuous radom variable with the distributio fuctio F ad let q 1 ad q 2 be two real fuctios defied o H such that E q 2 X) X ] = E q 1 X) X ] η ), H, is defied with some real fuctio η. Assume that q 1, q 2 C 1 H), η C 2 H) ad F is twice cotiuously differetiable ad strictly mootoe fuctio o the set H. Fially, assume that the equatio ηq 1 = q 2 has o real solutio i the iterior of H. The F is uiquely determied by the fuctios q 1, q 2 ad η, particularly F ) = a C η u) η u) q 1 u) q 2 u) ep s u)) du, where the fuctio s is a solutio of the differetial equatio s = η q 1 df = 1. H ηq 1 q 2 ad C is the ormalizatio costat, such that We like to metio that this kid of characterizatio based o the ratio of trucated momets is stable i the sese of weak covergece see, Gläzel 2]), i particular, let us assume that there is a sequece {X } of radom variables with distributio fuctios {F } such that the fuctios q 1, q 2 ad η N) satisfy the coditios of Theorem 1 ad let q 1 q 1, q 2 q 2 for some cotiuously differetiable real fuctios q 1 ad q 2. Let, fially, X be a radom variable with distributio F. Uder the coditio that q 1 X) ad q 2 X) are uiformly itegrable ad the family {F } is relatively compact, the sequece X coverges to X i distributio if ad oly if η coverges to η, where 87

19 η ) = E q 2 X) X ] E q 1 X) X ]. This stability theorem makes sure that the covergece of distributio fuctios is reflected by correspodig covergece of the fuctios q 1, q 2 ad η, respectively. It guaratees, for istace, the covergece of characterizatio of the Wald distributio to that of the Lé vy-smirov distributio if α. A further cosequece of the stability property of Theorem 1 is the applicatio of this theorem to special tasks i statistical practice such as the estimatio of the parameters of discrete distributios. For such purpose, the fuctios q 1, q 2 ad, specially, η should be as simple as possible. Sice the fuctio triplet is ot uiquely determied it is ofte possible to choose ξη as a liear fuctio. Therefore, it is worth aalyzig some special cases which helps to fid ew characterizatios reflectig the relatioship betwee idividual cotiuous uivariate distributios ad appropriate i other areas of statistics. Refereces Aarset, M. V. 1987). How to idetify a bathtub hazard rate. IEEE Trasactios o Reliability, 361), Aleader, C., Cordeiro, G. M., Ortega, E. M., & Sarabia, J. M. 2012). Geeralized beta-geerated distributios. Computatioal Statistics & Data Aalysis, 566), Alizadeh, M., Cordeiro, G. M., Nascimeto, A. D., Lima, M. D. C. S., & Ortega, E. M. 2017). Odd-Burr geeralized family of distributios with some applicatios. Joural of Statistical Computatio ad Simulatio, 872), Alzaatreh, A., Lee, C., & Famoye, F. 2013). A ew method for geeratig families of cotiuous distributios. Metro, 71, Bourguigo, M., Silva, R. B., & Cordeiro, G. M. 2014). The Weibull-G family of probability distributios. Joural of Data Sciece, 121), Brito, E., Cordeiro, G. M., Yousof, H. M., Alizadeh, M., & Silva, G. O. 2017). The Topp-Leoe odd log-logistic family of distributios. Joural of Statistical Computatio ad Simulatio, Che, G., & Balakrisha, N. 1995). A geeral purpose approimate goodess-of-fit test, J. Qual. Techol., 27, Cordeiro, G. M., Alizadeh, M., Ozel, G., Hosseii, B., Ortega, E. M. M., & Altu, E. 2017). The geeralized odd loglogistic family of distributios: properties, regressio models ad applicatios. Joural of Statistical Computatio ad Simulatio, 875), Cordeiro, G. M., Ortega, E. M., & Nadarajah, S. 2010). The Kumaraswamy Weibull distributio with applicatio to failure data. Joural of the Frakli Istitute, 3478), Cordeiro, G. M., Ortega, E. M. M., Popovic, B. V., & Pescim, R. R. 2014). The Loma geerator of distributios: Properties, miificatio process & regressio model, Applied Mathematics ad Computatio, 247, Cordeiro, G. M., Ortega, E. M., & Silva, G. O. 2011). The epoetiated geeralized gamma distributio with applicatio to lifetime data. Joural of statistical computatio ad simulatio, 817), Cordeiro, G. M., Yousof, H. M., Ramires, T. G., & Ortega, E. M. M. 2018). The Burr XII system of desities: properties, regressio model ad applicatios. Joural of Statistical Computatio ad Simulatio, 883), Duca, A. J. 1974). Quality Cotrol ad Idustrial Statistics, Irwi Homewood, USA. Ghitay, M. E., Al-Mutairi, D. K., Balakrisha, N., & Al-Eezi, L. J. 2013). Power Lidley distributio ad associated iferece. Computatioal Statistics & Data Aalysis, 64, Gläzel, W. 1987). A characterizatio theorem based o trucated momets ad its applicatio to some distributio families, Mathematical Statistics ad Probability Theory Bad Tatzmasdorf, 1986), Vol. B, Reidel, Dordrecht,

20 Gläzel, W. 1990). Some cosequeces of a characterizatio theorem based o trucated momets, Statistics: A Joural of Theoretical ad Applied Statistics, 214), Gupta, R. C., Gupta, P. L., & Gupta, R. D. 1998). Modelig failure time data by Lehma alteratives. Commu. Stat. Theory Methods, 27, Gupta, R. D., & Gupta, R. C. 2008). Aalyzig skewed data by power ormal model. Test, 171), Korkmaz, M. Ç., & Geç, A. I. 2017). A ew geeralized two-sided class of distributios with a emphasis o two-sided geeralized ormal distributio. Commuicatios i Statistics - Simulatio ad Computatio, 462), Nadarajah, S., & Gupta, A. K. 2007). The epoetiated gamma distributio with applicatio to drought data. Calcutta Statistical Associatio Bulleti, 59, Silva, F. S., Percotii, A., de Brito, E., Ramos, M. W., Vea cio, R., & Cordeiro, G. M. 2017). The Odd Lidley-G Family of Distributios. Austria Joural of Statistics, 461), Shirke, D. T., & Kakade, C. S. 2006). O epoetiated logormal distributio. Iteratioal Joural of Agricultural ad Statistical Scieces, 2, Thereau, T. M., Grambsch, P. M., & Flemig, T. R. 1990). Martigale-based residuals for survival models. Biometrika, 771), Tahir, M. H., Zubair, M., Cordeiro, G. M., Alzaatreh, A., & Masoor, M. 2017). The Weibull-Power Cauchy Distributio: Model, Properties ad Applicatios. Hacettepe Joural of Mathematics ad Statistics. Yousof, H. M., Afify, A. Z., Hamedai, G. G., & Aryal, G. 2016). the Burr X geerator of distributios for lifetime data. Joural of Statistical Theory ad Applicatios, 16, Copyrights Copyright for this article is retaied by the authors), with first publicatio rights grated to the joural. This is a ope-access article distributed uder the terms ad coditios of the Creative Commos Attributio licese 89