The Generalized Inverted Generalized Exponential Distribution with an Application to a Censored Data

Transcription

1 J. Stat. Appl. Pro. 4, No. 2, Joural of Statstcs Applcatos & Probablty A Iteratoal Joural The Geeralzed Iverted Geeralzed Expoetal Dstrbuto wth a Applcato to a Cesored Data P. E Ogutude 1, ad A. O. Adejumo 2 1 Departmet of Mathematcs, Coveat Uversty, Ogu State, Ngera 2 Departmet of Statstcs, Uversty of Ilor, Kwara State, Ngera Receved: 12 Ja. 2015, Revsed: 24 Apr. 2015, Accepted: 23 Apr Publshed ole: 1 Jul Abstract: We propose a two parameter Iverted Geeralzed Expoetal IGE ad a three parameter Geeralzed Iverted Geeralzed Expoetal GIGE probablty models as geeralzatos of the oe-parameter Expoetal dstrbuto ad some other dstrbutos the lterature. We explore the statstcal propertes of the GIGE dstrbuto ad ts parameters were estmated for both cesored ad ucesored cases usg the method of maxmum lkelhood estmato MLE. A applcato to a real data set s also provded to assess the flexblty of the GIGE dstrbuto over some of ts sub-models. Keywords: Cesored, Expoetal dstrbuto, Geeralzatos, Iverted Geeralzed Expoetal, Probablty models, Ucesored. 1 Itroducto [1] ad [2] proposed ad studed a geeralzato of the Expoetal dstrbuto called the Geeralzed Expoetal GE dstrbuto by troducg a shape parameter to the Expoetal dstrbuto. The cumulatve desty fucto cdf of the Geeralzed Expoetal GE dstrbuto s gve by; F GE x=1 e λ x α ;x>0, α > 0, λ > 0 1 The correspodg probablty desty fucto pdf s gve by; α s a shape parameter λ s the scale parameter f GE x=αλ e λ x 1 e λ x α ;x>0, α > 0, λ > 0 2 Besde, verted dstrbutos have bee studed by a umber of researchers. For stace, the Iverted Expoetal dstrbuto was troduced by [4] ad ts applcablty as a lfetme model has bee detfed by [6]. Practcally, f a radom varable X has a expoetal dstrbuto, the varable Y = X 1 wll have a Iverted Expoetal IE dstrbuto. Motvated by the work of [3], we shall frst troduce a two parameter probablty model kow as the Iverted Geeralzed Expoetal IGE dstrbuto whch s practcally the verse or recprocal of the Geeralzed Expoetal GE dstrbuto. Wth ths uderstadg, we shall further propose a three parameter model amed the Geeralzed Iverted Geeralzed Expoetal GIGE dstrbuto. However, the rest of ths artcle s orgazed as follows; I Secto 2, we preset both the IGE ad GIGE dstrbutos, Secto 3 deals wth some basc statstcal propertes of the proposed models coupled wth the estmato of model Correspodg author e-mal: peluemma@yahoo.com Natural Sceces Publshg Cor.

2 224 P. E Ogutude, A. O. Adejumo: The Geeralzed Iverted Geeralzed Expoetal... parameters; Secto 4 provdes the estmato of the model parameters usg the method of maxmum lkelhood estmato MLE for both the Cesored ad Ucesored cases, Secto 5 dscusses the applcato of the GIGE dstrbuto usg a real data set followed by a cocludg remark. 2 The Geeralzed Iverted Geeralzed Expoetal Dstrbuto We shall start by proposg the Iverted Geeralzed Expoetal IGE dstrbuto. Let X deote a o-egatve cotuous radom varable ad gve that the Geeralzed Expoetal dstrbuto s as defed Equato 1 ad Equato 2, therefore, the cdf ad the pdf of the Iverted Geeralzed Expoetal dstrbuto are respectvely gve by; F IGE x=1 e λ x α ;x,α,λ > 0 3 α s a shape parameter λ s the scale parameter f IGE x=αλ e λ x x 2 1 e λ x α ;x,α,λ > 0 4 We ca otherwse ame the Iverted Geeralzed Expoetal dstrbuto as the Complemetary or Recprocal Geeralzed Expoetal dstrbuto. We preset the Relablty fucto ad the Falure rate respectvely as; For x>0, α > 0 ad λ > 0 For x>0, α > 0 ad λ > 0 S IGE x=1 e λ x α 5 h IGE x=αλ e λ x x 2 1 e λ x 6 Wth ths uderstadg, we ca cofdetly propose the Geeralzed Iverted Geeralzed Expoetal GIGE dstrbuto by geeralzg the IGE dstrbuto as follows; F GIGE x=1 e λ x α ;x,α,λ, > 0 7 Dfferetatg Equato 7 wth respect to x gves the pdf of the propose GIGE dstrbuto as; α ad are shape parameters λ s the scale parameter For otato purposes, we wrte; X GIGEλ, α, f GIGE x=αλ x 2 e λ x 1 e λ x α ;x,α,λ, > 0 8 The plot for the pdf ad cdf of the GIGE dstrbuto at varous parameter values are gve Fgure 1 ad 2 respectvely. As show Fgure 1, the proposed GIGE dstrbuto s postvely skewed ad the shape of the model s umodal. Specal Cases: Some kow dstrbutos the lterature are foud to be sub-models of the proposed GIGE dstrbuto. For stace, For = 1, we get the Iverted Geeralzed Expoetal IGE dstrbuto. For α = = 1, we get the Iverse Expoetal dstrbuto. Observatos from the GIGE dstrbuto wth parameters α, λ, ca be smulated usg the followg trasformato; where U s a radom varable uformly dstrbuted o0, 1. X = λ [log1 U 1 α] 9 Natural Sceces Publshg Cor.

3 J. Stat. Appl. Pro. 4, No. 2, / Fg. 1: Plot for the pdf of the GIGE dstrbuto at α = 1, λ = 3, = 2 Fg. 2: Plot for the cdf of the GIGE dstrbuto at α = 1, λ = 3, = 2 3 Statstcal Propertes of the GIGE Dstrbuto Ths secto provdes some basc statstcal propertes of the proposed Geeralzed Iverted Geeralzed Expoetal Dstrbuto. 3.1 Relablty Aalyss The relablty survval fucto s gve by; Sx=1 Fx 10 Therefore, the relablty fucto for the GIGE dstrbuto s gve by; S GIGE x=1 e x λ α 11 For x>0,α > 0,λ > 0, > 0 The correspodg plot for the relablty fucto of the GIGE dstrbuto s as show Fgure 3; The probablty that a system havg age x uts of tme wll survve up to x+ t uts of tme for x>0,α > 0,λ > 0, > 0 ad t > 0 s gve by; S GIGE t x= S GIGEx+ t 12 S GIGE x Natural Sceces Publshg Cor.

4 226 P. E Ogutude, A. O. Adejumo: The Geeralzed Iverted Geeralzed Expoetal... Fg. 3: Plot for the Survval fucto of the GIGE dstrbuto at α = 1, λ = 3, = 2 Fg. 4: Plot for the Falure rate of the GIGE dstrbuto at α = 1, λ = 3, = 2 Hazard fucto s gve by; S GIGE t x= 1 e λ x+t α 1 e λ x α 13 hx= fx 1 Fx Therefore, the correspodg falure rate hazard fucto s gve by; x>0,α > 0,λ > 0, > 0 14 h GIGE x=αλ x 2 e λ x 1 e λ x 15 The plot for the falure rate at dfferet parameter values s provded Fgure Momets The r-th momet of a cotuous radom varable X s gve by; µ r = E[X r ]= 0 x r fxdx 16 Natural Sceces Publshg Cor.

5 J. Stat. Appl. Pro. 4, No. 2, / If a cotuous radom varable X s such that; X GIGEα, λ,,the rth momet s gve by; Let θ = x λ Equato 17 ad by followg [3], the; E[X r ]= αλ x 2 e x λ 1 e x λ α dx 17 0 E[X r ] = αλ r r θ r e θ 1 e θ α dθ 18 Therefore; 0 = αλ r r θ r e θ 1+ 0 =1 a e θ dθ a = α 1α 2...α! E[X r ]=αλ r r Γ1 r 1+ =1 a +1 1 r The geeral expresso for the momet geeratg fucto mgf of the GIGE dstrbuto s gve by; M X t=α r=0 t r r! λ r r Γ1 r 1+ =1 a +1 1 r Parameter Estmato Ths secto provdes the estmato of the proposed GIGE dstrbuto both for the Cesored ad Ucesored cases usg the method of maxmum lkelhood estmato MLE as follows; 4.1 For Cesored Case: Cesorg s a stuato whch the value of a observato s oly partally kow. It s a form of mssg data problem ad t s commo survval aalyss. Followg [3], let X ad C be depedet radom varables, where X deotes the lfetme of th dvdual ad C deotes the cesorg tme ad t = mx,c for =1,2,3,...,. Each X s dstrbuted accordg to Equato 8 wth parameters α, λ ad. Let m deote the umber of falures ad let C ad L deote the sets of cesored ad ucesored observatos respectvely. The lkelhood fucto for the cesored case s gve by; L= F ft St C ft ad St are the pdf ad relablty fucto of the GIGE dstrbuto as gve Equatos 8 ad 11 respectvely. Let l = logl; l = mlogα+ log+ logλ 2 F logt λ F t +α 1 log F 1 e λ t + α C log 1 e t λ 22 Natural Sceces Publshg Cor.

6 228 P. E Ogutude, A. O. Adejumo: The Geeralzed Iverted Geeralzed Expoetal... Dfferetatg l wth respect to α, λ ad respectvely gves; α = m α + log 1 e t λ F λ = m λ F e t +α 1 t t λ F 1 e t λ e + α t t λ C 1 e t λ = m e λ t +α 1λ t t λ F F 1 e t λ The 3 3 observed formato matrx for hypothess testg ad terval estmato for parameters α, λ ad s gve by; Jθ= J λ,λ J λ,α J λ, J α,α J α, J λ,λ = 2 l λ 2 = m λ 2 +α 1λ λ 1 e t λ t J λ, = 2 l λ = 1 F t F J, e t t λ +α 1 1 e t λ + αλ J λ,α = 2 l λ α = F C t e λ t 1 e t λ λ 1 e t λ t e λ t t 1 e t λ e +α 1 t t λ F 1 e t λ λ e + α 1 e t λ t t t λ C 1 e t λ 1 λ 1 e t λ t J, = 2 l 2 = m 2 α 1λ 2 F e + t t λ C 1 e t λ J α,α = 2 l α 2 = m α 2 26 t 2 e t λ αλ 2 t 2 e t λ 1 e t λ 2 C 1 e t λ For Ucesored Case: Let X 1,X 2,...X be a radom sample of depedetly ad detcally dstrbuted radom varables each havg a Geeralzed Iverted Geeralzed Expoetal dstrbuto defed Equato 8, the lkelhood fucto L s gve by; L X α,λ, = =1 αλ x 2 e λ x 1 e λ x α 28 Natural Sceces Publshg Cor.

7 J. Stat. Appl. Pro. 4, No. 2, / Followg Ogutude et al 2014, Let l = logl X α,λ,, l = logα+ log+ logλ 2 =1 logx Dfferetatg l wth respect to α, λ ad respectvely gves; =1 λ x +α 1 =1 log 1 e λ x 29 α = α + log 1 e x λ 30 =1 λ = λ λ +α 1 =1 x =1 x λ e x λ 1 e x λ 31 Settg α = 0, λ parameters α, λ ad. = 0, = λ +α 1 =1 x =1 x λ e x λ 1 e x λ 32 = 0 ad solvg the resultg olear equatos gves the maxmum lkelhood estmates of 5 Applcato To assess the flexblty of the proposed GIGE dstrbuto, we make use of a cesored data gve by [5], [8] ad [3]. The data cosst of death tmes weeks of patets wth cacer of togue wth aeuplod DNA profle. The observatos are; 1, 3, 3, 4, 10, 13, 13, 16, 16, 24, 26, 27, 28, 30, 30, 32, 41, 51, 61, 65, 67, 70, 72, 73, 74, 77, 79, 80, 81, 87, 87, 88, 89, 91, 93, 93, 96, 97, 100, 101, 104, 104, 108, 109, 120, 131, 150, 157, 167, 231, 240, ad 400, where the twety-oe observatos wrtte bold deote cesored observatos. The aalyss Table 1 s performed wth the ad of R software. The Log-lkelhood ad Akake Iformato Crtera AIC for the GIGE, IGE ad IE dstrbutos are provded. Table 1: Descrptve Statstcs o Death Tmes M Q1 Q2 Mea Q3 Max Var Skewess Kurtoss Table 2: Performace of the GIGE, IGE ad IE dstrbutos Dstrbuto Parameters LOG-Lkelhood AIC Rak by AIC GIGE Proposed α = e = e 03 λ = e+03 IGE Proposed α = λ = IE [4] λ = It s good to ote that the model wth the lowest AIC s raked the best. Natural Sceces Publshg Cor.

8 230 P. E Ogutude, A. O. Adejumo: The Geeralzed Iverted Geeralzed Expoetal... 6 Cocluso A two-parameter Iverted Geeralzed Expoetal IGE dstrbuto ad a three-parameter Geeralzed Iverted Geeralzed Expoetal GIGE dstrbuto are defed ths artcle. The basc statstcal propertes of the GIGE dstrbuto are rgorously studed. The model s postvely skewed ad ts shape s umodal. A umber of dstrbutos are foud to be sub-models of the GIGE dstrbuto. We estmated the parameters of the GIGE dstrbuto for both the Cesored ad Ucesored cases usg the method of maxmum lkelhood estmato MLE. The result based o the data used, shows that the two-parameter Iverted Geeralzed Expoetal IGE dstrbuto s better tha the three-parameter Geeralzed Iverted Geeralzed Expoetal GIGE dstrbuto. Ths tur meas that, geeralzg the IGE dstrbuto by troducg aother shape parameter may ot be eedful except f the data set s more skewed. Ackowledgemet The authors are grateful to the aoymous referee for a careful checkg of the detals ad for helpful commets that mproved ths paper. Refereces [1] Gupta, R. D ad Kudu, D., Geeralzed Expoetal Dstrbutos, Australa ad New Zealad Joural of Statstcs, 41 2, , [2] Gupta, R. D ad Kudu, D., Geeralzed Expoetal Dstrbuto: Exstg results ad some recet developmets, Joural of Statstcal Plag ad Iferece, Vol. 137, Issue 11, , [3] Ja, K., Sgla N. ad Sharma S. K., The Geeralzed Iverse Geeralzed Webull Dstrbuto ad Its Propertes, Joural of Probablty, Volume 2014, Artcle ID , 11 pages, [4] Keller, A. Z ad Kamath, A. R., Relablty aalyss of CNC Mache Tools. Relablty Egeerg. Vol. 3, pp , [5] Kle J. P. ad Moeschberger, M. L., SurvvalAalyss: Techques for Cesoredad Trucated Data, Sprger, New York,NY,USA, [6] L, C. T, Dura, B. S ad Lews, T. O., Iverted Gamma as lfe dstrbuto. Mcroelectro Relablty, Vol. 29 4, , [7] Ogutude, P. E, Adejumo A. O & Balogu O. S., Statstcal Propertes of the Expoetated Geeralzed Iverted Expoetal Dstrbuto, Appled Mathematcs, 4 2: 47-55, [8] Sckle-Sataello, B. J., Farrar, W. B., Keyha-Rofagha, S., A reproducble System of flow cytometrc DNA aalyss of paraff embedded sold tumors: techcal mprovemets ad statstcal aalyss, Cytometry, vol. 9, pp , Natural Sceces Publshg Cor.