Georgia Souther Uiversity From the SelectedWors of Broderic O Oluyede April 26, 214 Geeralizatios of the Iverse Weibull ad Related Distributios with Applicatios Broderic O Oluyede, Georgia Souther Uiversity Tao Yag, Clemso Uiversity Available at: https://worsbepresscom/ broderic_oluyede/33/
Electroic Joural of Applied Statistical Aalysis EJASA, Electro J App Stat Aal http://siba-eseuisaletoit/idexphp/ejasa/idex e-issn: 27-5948 DOI: 11285/i275948v71p94 Geeralizatios of the iverse Weibull ad related distributios with applicatio By Oluyede, Yag April 26, 214 This wor is copyrighted by Uiversità del Saleto, ad is licesed uder a Creative Commos Attribuzioe - No commerciale - No opere derivate 3 Italia Licese For more iformatio see: http://creativecommosorg/liceses/by-c-d/3/it/
Electroic Joural of Applied Statistical Aalysis Vol 7, Issue 1, 214, 94-116 DOI: 11285/i275948v71p94 Geeralizatios of the iverse Weibull ad related distributios with applicatio Broderic O Oluyede a ad Tao Yag b a Departmet of Mathematical Scieces, Georgia Souther Uiversity, Georgia, Uited States b Departmet of Mathematical Scieces, Clemso Uiversity,Clemso, Uited States April 26, 214 I this paper, the geeralized iverse Weibull distributio icludig the expoetiated or proportioal reverse hazard ad Kumaraswamy geeralized iverse Weibull distributios are preseted Properties of these distributios icludig the behavior of the hazard ad reverse hazard fuctios, momets, coefficiets of variatio, sewess, ad urtosis, etropy, Fisher iformatio matrix are studied Estimates of the model parameters via method of maximum lielihood (ML, ad method of momets (MOM are preseted for complete ad cesored data Numerical examples are also preseted eywords: Iverse Weibull Distributio, Proportioal Iverse Weibull Distributio, Geeralized Distributio 1 Itroductio The iverse Weibull distributio ca be readily applied to a wide rage of situatios icludig applicatios i medicie, reliability ad ecology Keller et al (1985 obtaied the iverse Weibull model by ivestigatig failures of mechaical compoets subject to degradatio Calabria ad Pulcii (199 computed the maximum lielihood ad least squares estimates of the parameters of the iverse Weibull distributio They also obtaied the Bayes estimator of the model parameters as well as cofidece limits for reliability ad tolerace limits See Calabria ad Pulcii (1989, 1994, ad Johso et al (1984 for additioal details Kha et al (28 preseted some importat theoretical properties of the iverse Weibull distributio Samata ad Correspodig author: boluyede@georgiasoutheredu c Uiversità del Saleto ISSN: 27-5948 http://siba-eseuisaletoit/idexphp/ejasa/idex
Electroic Joural of Applied Statistical Aalysis 95 Bhowmic (21 preseted a determiistic ivetory system with Weibull distributio deterioratio ad ramp type demad rate The iverse Weibull (IW cumulative distributio fuctio (cdf is give by [ F(x;α, exp (α(x x, x x, α >, >, (1 where α, x ad are the scale, locatio ad shape parameters, respectively Ofte the parameter x is called the miimum life or guaratee time Whe α 1 ad x x + α, the F(α + x ;1; F(α + x ;1 e 1 3679 This value is i fact the characteristic life of the distributio I what follows, we assume that x, ad the iverse Weibull cdf becomes F(x;α, exp[ (αx, x, α >, > (2 Note that whe α 1, we have the Fréchet distributio fuctio The iverse Weibull probability desity fuctio (pdf is give by f (x;α, α x 1 exp[ (αx, x, α >, > (3 Whe 1 ad 2, the iverse Weibull distributio pdfs are referred to as the iverse expoetial ad iverse Raleigh pdfs, respectively The th raw or o cetral momets are give by E(X Γ(1 / α, for > (4 Note that E(X does ot exist whe See Johso et al (1984 This paper is orgaized as follows Sectio 2 cotais some basic utility otios I sectio 3, the expoetiated or proportioal iverse Weibull (PIW ad Kumaraswamy geeralized iverse Weibull distributios are preseted The mode, hazard fuctio ad reverse hazard fuctio are also preseted i sectio 3 Glaser s Lemma is applied to the PIW distributio to determie the behavior of the hazard fuctio I sectio 4, the momets, etropy measures ad Fisher iformatio are preseted Estimatio of the parameters of the PIW distributio via the methods of momets ad maximum lielihood as well as umerical examples for complete ad right cesored data are preseted i sectio 5 Sectio 6 deals with Kumaraswamy geeralized iverse Weibull distributio The mode, hazard fuctio, reverse hazard fuctio, momets, Shao etropy ad estimates of the model parameters for cesored data are preseted 2 Basic Utility Notios I this sectio, some basic utility otios ad defiitios are preseted Suppose the distributio of a cotiuous radom variable X has the parameter set θ {θ 1,θ 2,,θ } Let the pdf of the radom variable X be give by f (x;θ The hazard fuctio of X ca be iterpreted as the istataeous failure rate or the coditioal probability desity of failure at time x, give that
96 Oluyede, Yag the uit has survived util time x, see Shaed ad Shathiumar (1994 The hazard fuctio h(x;θ is defied to be h(x;θ P(x X x + x lim x x[1 F(x;θ F (x;θ F(x;θ f (x;θ 1 F(x;θ, (5 where F(x;θ is the survival or reliability fuctio The cocept of reverse hazard rate was itroduced as the hazard rate i the egative directio ad received miimal attetio, if ay, i the literature Keilso ad Sumita (1982 demostrated the importace of the reverse hazard rate ad reverse hazard orderigs Shaed ad Shathiumar (1994 preseted results o reverse hazard rate See Ross (1983, Chadra ad Roy (21, Bloc ad Savits (1998 for additioal details We preset a formal defiitio of the reverse hazard fuctio of a distributio fuctio F The reverse Hazard fuctio ca be iterpreted as a approximate probability of a failure i [x,x + dx, give that the failure had occurred i [,x Defiitio 21 Let (a, b, a < b <, be a iterval of support for F The the reverse hazard fuctio of X (or F at t > a is deoted by τ F (t ad is defied as τ(t;θ d dt logf(t;θ f (t;θ F(t;θ (6 Some useful fuctios that are employed i subsequet sectios are give below The gamma ad digamma fuctios are give by Γ(x tx 1 e t dt ad Ψ(x Γ (x Γ(x respectively, where Γ (x tx 1 (logte t dt is the first derivative of the gamma fuctio Defiitio 22 The th -order derivative formula of gamma fuctio is give by: Γ ( (s z s 1 (logz exp( zdz (7 This derivative will be used frequetly i this paper The lower icomplete gamma fuctio ad the upper icomplete gamma fuctio are respectively x γ(s,x t s 1 e t dt ad Γ(s,x t s 1 e t dt, (8 x 3 Geeralized Iverse Weibull Distributios The proportioal iverse Weibull (PIW distributio has a cdf give by G(x;α,,γ [F(x γ exp[ γ(αx, for α >, >, γ >, ad x (9 The correspodig pdf is give by g(x;α,,γ αγ(αx 1 exp[ γ(αx, (1
Electroic Joural of Applied Statistical Aalysis 97 for α >, >, γ >, ad x Joes (29 explored the bacgroud ad geesis of the Kumaraswamy distributio (Kumaraswamy (198 ad, more importatly, made clear some similarities ad differeces betwee the beta ad Kumaraswamy distributios Amog the advatages are: the ormalizig costat is very simple; the distributio ad quatile fuctios have simple explicit formula which do ot ivolve special fuctios; explicit formula for momets of order statistics ad L-momets However, compared to Kumaraswamy distributio, the beta distributio has the followig advatages: simpler formula for momets ad momet geeratig fuctio (mgf; a oe-parameter sub-family of symmetric distributios; simpler momet estimatio ad more ways of geeratig the distributio via physical processes Cordeiro ad Ortega (21 amog others studied the Kumaraswamy Weibull distributio ad applied it to failure time data Kumaraswamy (198 i his paper proposed a two-parameter distributio (Kumaraswamy distributio defied i (, 1 Here we will refer to it as Kum distributio Its cdf is give by: F(x;a;b 1 (1 x a b, x (,1, a >, b > (11 The parameters a ad b are the shape parameters The Kum distributio has the pdf give by: f (x;a,b abx a 1 (1 x a b 1, x (,1, a >, b > (12 The Kum-Geeralized Iverse Weibull (KGIW cdf is give by G(x;α,,λ,ϕ 1 (1 F λ (x;α, ϕ 1 {1 exp[ λ(αx } ϕ, for x >, α >, >, λ >, ad ϕ > The correspodig KGIW pdf is give by g(x;α,,λ,ϕ αλϕ(αx 1 exp[ λ(αx {1 exp[ λ(αx } for x >, α >, >, λ >, ad ϕ > 31 Mode of the Proportioal Iverse Weibull Distributio Cosider the PIW distributio Note that, lg(x;α,,γ l(αγ (1 + l(αx γ(αx Differetiatig lg(x;α,,γ with respect to x, we obtai lg(x;α,,γ x 1 + + αγ(αx 1 x 1 [ α γ x x (1 + Now, set lg(x;α,,γ x equal ad solve for x, to get x ( α 1 γ 1 +
98 Oluyede, Yag Obviously, whe < x < ( α γ 1+ 1, lg(x;α,,γ x >, so g(x;α,,γ is icreasig, ad whe x > ( α γ 1+ 1, g(x;α,,γ is decreasig, so g(x;α,,γ attais a maximum whe x ( α γ 1+ 1 Note that x is the mode of PIW distributio 32 Hazard Fuctio The hazard fuctio of the PIW distributio is give by λ G (x;α,,γ g(x;α,,γ Ḡ(x;α,,γ αγ(αx 1 exp[ γ(αx 1 exp[ γ(αx, for γ >, α >, >, x We study the behavior of the hazard fuctio of the PIW distributios via Glaser (198 lemma Note that η G (x;α,,γ η G (x g (x;α,,γ g(x;α,,γ α(1 + (αx 1 αγ(αx 1, ad η G(x α 2 γ( + 1(αx 2 α 2 (1 + (αx 2 Let η G (x, we get x 1 α (γ 1 So whe < x < x, η G (x >, η G (x ad whe x > x, η G (x < So the hazard fuctio is upside dow bathtub shape 33 Reverse Hazard Fuctio The reverse hazard fuctio for the PIW distributio is give by for γ >, α >, >, x τ G (x;α,,γ g(x;α,,γ G(x;α,,γ αγ(αx 1, 4 Momets, Etropy ad Fisher Iformatio I this sectio, we preset the momets ad related fuctios for the proportioal iverse Weibull distributio The cocept of etropy plays a vital role i iformatio theory The etropy of a radom variable is defied i terms of its probability distributio ad ca be show to be a good measure of radomess or ucertaity We preset Shao etropy ad etropy for the PIW distributio Also, preseted is Fisher iformatio matrix for the PIW distributio
Electroic Joural of Applied Statistical Aalysis 99 41 Momets The momets of the PIW distributio are give by E(X c x c g(x;α,,γdx αγ x c (αx 1 exp[ γ(αx dx ( γ c c α c Γ, where > c The variace is give by σ 2 E(X 2 E 2 (X γ 2 α 2[ ( 2 Γ Γ 2( 1 The coefficiet of variatio (CV is give by CV σ µ γ 1 α Γ( 1 2 Γ2 ( 1 γ 1 α 1 Γ( 1 The coefficiet of Sewess (CS is give by CS 2Γ3 ( 1 The coefficiet of Kurtosis (CK is give by CK Γ( 4 4Γ( 1 Γ( 2 3Γ( 1 Γ2 ( 1 Γ( 1 2 Γ( [Γ( 2 Γ2 ( 1 3 2 3 Γ( + 6Γ2 ( 1 + Γ( 3 Γ( 2 3Γ4 ( 1 [Γ( 2 Γ2 ( 1 2 The graphs of the mea agaist for values of α ad γ shows a decreasig mea for icreasig values of Table 1 shows the mode, mea, stadard deviatio (STD, CV, CS ad CK for some values of the parameters α, ad γ From the table 1, we ca see that as icreases, the Mea, STD, coefficiet of variatio, sewess ad urtosis are all decreasig Graph of the mea agaist for some values of the parameters α, ad γ shows a decreasig tred 42 Shao Etropy Shao etropy for PIW distributio is give by H(g E[ log g(x [log g(xg(x dx αγ[a + B +C,
1 Oluyede, Yag Figure 1: Mea of PIW Distributio Table 1: Mode, Mea, STD, Coefficiets of Variatio, Sewess ad Kurtosis α γ Mode Mea STD CV CS CK 1 5 1 9641925 116423 3657341 3141425 353572 489151 1 6 2 193991 126721 3173964 25561 285566 2467812 1 8 3 113436 12552 2238499 179725 218927 1416589 1 11 4 1125375 123964 1514753 1258139 18264 11796 2 12 5 5679638 635245 691668 114552 174989 946844 3 13 6 384178 419915 4223783 15714 1692484 8979469 4 14 7 2858669 36762 291897 97817 164594 8593978 5 15 8 2287533 2396567 2162252 922286 165245 8282494 where A, B ad C are obtaied below: A log(αγ, αγ log(αγ(αx 1 exp[ γ(αx dx
Electroic Joural of Applied Statistical Aalysis 11 B (1 + log(αx(αx 1 exp[ γ(αx dx 1 + α 1 + α 2 γ [ log(xexp[ γx dx logxexp( xdx logγ exp[ x dx usig the fact that Γ (t log (xx t 1 exp( xdx, we obtai B 1 + α 2 γ logγ,, ad C γ (αx 2 1 exp[ γ(αx dαx α 1 αγ Fially, Shao etropy reduces to H(g 1 + 2 log(αγ 43 - Etropy -etropy is a oe parameter geeralizatio of the Shao etropy - etropy is defied by H (g 1 [ 1 g (xdx, for 1 1 -etropy for PIW distributio is give by H (g 1 1 Let t γ (αx ( 1, so x 1 t α γ, the we have 1 H (g 1 α 1 1 γ 1 1 1 1 [ 1 1 [ 1 α γ (αx (+1 exp[ γ (αx [ 1 α 1 1 1 γ 1 α 1 1 1 γ ( 1 t γ 1 t + 1 dx exp[ t dt + 1 1 exp[ tdt ( 1 + 1 Γ
12 Oluyede, Yag 44 Fisher Iformatio The iformatio (or Fisher iformatio that a radom variable X cotais about the parameter θ is give by [ 2 I(θ E log( f (X,θ θ Now, if log( f (X,θ is twice differetiable with respect to θ, ad uder certai regularity coditios (Lehma (1998, Fisher Iformatio is give by [ 2 I(θ E θ log( f (X,θ θ 2 For the PIW distributio, the Fisher iformatio (FI that X cotais about the parameters θ (α,, γ are obtaied as follows: Usig the pdf g(x; α,, γ, lg(x;α,,γ lαγ (1 + l(αx γ(αx We have the followig partial derivatives: lg(x;α,,γ α lg(x;α,,γ α + x γα 1, 1 l(αx + γ(αx l(αx, lg(x;α,,γ γ 1 γ (αx, 2 lg(x;α,,γ α 2 α 2 α 2 (1 + γx, 2 lg(x;α,,γ 2 1 2 γ(αx l 2 (αx, 2 lg(x;α,,γ γ 2 1 γ 2, 2 lg(x;α,,γ α 1 α + x γα 1 ( l(αx + 1, 2 lg(x;α,,γ α γ x α 1,
Electroic Joural of Applied Statistical Aalysis 13 ad The, [ 2 lg(x;α,,γ E α 2 [ 2 lg(x;α,,γ E 2 [ 2 lg(x;α,,γ E γ 2 [ 2 lg(x;α,,γ E α [ 2 lg(x;α,,γ E α γ 2 lg(x;α,,γ γ (αx l(αx ( α 2 α 2 (1 + γx g(x;α,,γdx (1 + α 2 xexp( xdx α 2, 2 α 2, ( 1 2 γ(αx l 2 (αx ( 1 2 + γ(αx l 2 (αx 1 2 + 1 2 [Γ (2 2lγ Γ (2 + l 2 γ 1 + Γ (2 2Γ (2lγ + l 2 γ 2, 1 γ 2, g(x;α,,γdx αγe γ(αx (αx +1 dx 2 lg(x;α,,γ γ 2 g(x;α,,γdx 1 γ 2 αγ(αx 1 exp[ γ(αx dx 2 lg(x;α,,γ g(x;α,,γdx α [ 1 α x γα 1 ( l(αx + 1 αγe γ(αx (αx +1 dx lγ Γ (2, α αγ, 2 lg(x;α,,γ g(x;α,,γdx α γ x α 1 αγ(αx 1 exp[ γ(αx dx
14 Oluyede, Yag ad [ 2 lg(x;α,,γ E γ 1 γ Γ (2 lγ γ 2 lg(x;α,,γ γ g(x;α,,γdx (αx l(αxαγ(αx 1 exp[ γ(αx dx t[lt lγexp[ tdt Now, the Fisher Iformatio Matrix (FIM for PIW distributio is give by: [ [ E 2 logg(x;α,,γ E 2 logg(x;α,,γ α 2 α E [ [ I(α,,γ E 2 logg(x;α,,γ α E 2 logg(x;α,,γ E 2 E E E where the etries are give by [ 2 logg(x;α,,γ γ α I(1,1 2 [ 2 logg(x;α,,γ γ α 2, I(1,2 I(2,1 lγ Γ (2 α I(1,3 I(3,1 αγ, I(2,2 1 + Γ (2 2Γ (2lγ + l 2 γ 2, I(2,3 I(3,2 Γ (2 lγ, ad I(3,3 1 γ γ 2 [ 2 logg(x;α,,γ [ α γ 2 logg(x;α,,γ γ [ 2 logg(x;α,,γ γ 2 5 Estimatio of Parameters i the Proportioal Iverse Weibull Distributio I this sectio, we obtai estimates of the parameters for the PIW distributio Method of momet (MOM ad maximum lielihood (ML estimators are preseted Estimatio of the parameters of PIW distributio for complete ad right cesored data are preseted 51 Method of Momet Estimators Let X 1,X 2,,X be a idepedet sample from the PIW distributio The method of momets estimators are defied as follows:,, E(X j x j i j 1,2,
Electroic Joural of Applied Statistical Aalysis 15 We have E(X j γ j α j Γ( j, so we have the followig equatios: ( γ 1 1 α 1 Γ ( γ 2 2 α 2 Γ ( γ 3 3 α 3 Γ X, S 2 x2 i x3 i From the divisio of the first two equatios above, we get Γ 2 ( 1 Γ( 2 The we ca apply Newto-Ralphso method to obtai the solutio Let f ( Γ 2 ( 1 S2 Γ( 2 X 2, the do the iteratio to fid, say ˆ If α is ow, the X 2 S 2 +1 f ( f ( ( α ˆ X ˆγ Γ( ˆ 1 ˆ, Whe γ is ow, γ 1ˆ Γ( ˆ 1 ˆα X ˆ 52 Maximum Lielihood Estimators Let X 1,X 2,,X be a radom sample from a PIW distributio The the lielihood fuctio is give by L g(x 1,,x ;α,,γ (αγ ( The log-lielihood fuctio is give by ll l(αγ (1 + αx i 1 exp[ γ l(αx i γ (αx i (αx i
16 Oluyede, Yag The ormal equatios are ll α ll ṋ α (1 + ˆ ṋ α + ˆ ˆγ ṋ l( ˆαx i + ˆγ ( ˆαx i ˆ 1 x i, [( ˆαx i ˆ l( ˆαx i, ad ll γ respectively These equatios reduces to ṋ γ ( ˆαx i ˆ, ˆγ ˆ ˆα ˆ x ˆ i l( ˆαx i ˆγ ˆα ˆ [x ˆ i l( ˆαx i From the ormal equatios, we ow that if α or γ is ow, we ca use Newto s method to solve for umerically If γ ad are ow, we solve for α to obtai Whe α ad are ow, we solve for γ to get 53 Numerical Examples ( 1 ˆα γ x i ˆγ α x i I this sectio, we provide several umerical examples to show ad illustrate the flexibility of the geeralized iverse Weibull distributio for date modelig Specifically, we cosider three data sets from Lawless (23 The first set of data represets the umber of millio revolutios before failure of each of 22 ball bearig i a life testig experimet The data are: 1788, 2892, 33, 4152, 4212, 456, 488, 5184, 5196, 5412, 5556, 678, 6864, 6864, 6888, 8412, 9312, 9864, 1512, 12792, 1284, 1734 The secod data set represets the failure times (i miutes for a sample of 15 electroic compoets i a accelerated life test ad are give by: 14, 51, 63, 18, 121, 185, 197, 222, 23, 36, 373, 463, 539, 598, 662 The third set of data are the umber of cycles of failure for 25 1-cm specimes of yar, tested at a particular strai level The data are: 15, 2, 38, 42, 61, 76, 86, 98, 121, 146, 149, 157, 175, 176, 18, 18, 198, 22, 224, 251, 264, 282, 321, 325, 653 These data sets ca be modeled by the geeralized iverse Weibull distributio The MLEs of the parameters α, ad γ are computed by maximizig the objective fuctio with the trustregio algorithm via the NLPTR subroutie i SAS The estimated values of the parameters α,
Electroic Joural of Applied Statistical Aalysis 17 ad γ, log-lielihood statistic, Kolmogorov-Smirov statistics ad the correspodig gradiet objective fuctio (ormal equatios uder the geeralized iverse Weibull distributio ad other alteratives icludig the geeralized Lidley distributio are preseted i table 2 The geeralized Lidley (GL distributio (Zaerzadeh ad Dolati (29 is give by f GL (x;α,,γ 2 (x α 1 (α + γxe x, α,,γ > (γ + Γ(α + 1 The other models cosidered are the gamma, ad logormal distributios give by f G (x;α, (Γ(α 1 α x α 1 e x, ad f LN (x;α, 1 2παx e 1 2 ( logx α 2 After usig NLPTR method, we have the followig results for the geeralized iverse Weibull parameter estimates for the first set of data: First, we set our iitial guess as α 1, 1, ad γ 1 The after 18 iteratios, we have ˆα 983, ˆ 18158 ad ˆγ 168395 The value of the log lielihood fuctio is 119885 For the secod set of data, we have ˆα 31, ˆ 8423 ad ˆγ 18676 The value of the log lielihood fuctio is 685351 For the third set of data, we have the followig: ˆα 1967, ˆ 1111 ad ˆγ 16698 The value of the log lielihood fuctio is 1585789 These values together with the values of the gradiet object fuctio for all three examples are preseted i table 2 54 Estimatio i Right Cesored Data from PIW Distributio I this sectio, the maximum lielihood estimate (MLE of the PIW distributio parameters α,, ad γ uder type I cesorig is preseted Suppose we have idepedet positive radom variables X 1,X 2,,X, where X i has a associated idicator variable δ i where δ i 1 if X i is a observed failure time ad δ i if X i is right cesored, the the lielihood fuctio is give by L(x 1,x 2,,x ;θ g δ i (x i ;α,,γ(1 G(x i ;α,,γ 1 δ i, where θ (α,,γ The log-lielihood fuctio is give by that is, l(θ ll(x 1,x 2,,x ;θ [ δi l[g(x i ;α,,γ + (1 δ i l[ḡ(x i ;α,,γ, [ δ i l(αγ δ i ( + 1l(αx i δ i γ(αx i (1 δ i l[1 exp[ γ(αx i
18 Oluyede, Yag Table 2: Estimates, Log-lielihood ad Kolmogorov-Smirov Statistic Data set Model α γ LL K-S L α I (22 PIW 983 18158 168395-119885 91-111 -3387 214 GL 987 1378 9371-186875 127 32 579 327 Gamma 637 15646 - -185738 116 55-44 28 Logormal 31268 16732 - -193792 176 32 652 342 II (15 PIW 31 8423 18676-685351 91 238 518-4848 GL 654 1256 851-6411 12 44 479-433 Gamma 531 14434 - -64188 1 32 17 392 Logormal 29295 1599 - -6562 164-5367 359-43 III (25 PIW 1967 1112 16698-1585789 121-6375 -948 GL 134 1525 185-1523741 141 678 896 34 Gamma 113 17964 - -1524432 137 234-44 25 Logormal 49129 931 - -15411 156-523 -361 38 L L γ
Electroic Joural of Applied Statistical Aalysis 19 The MLEs ˆθ ( ˆα, ˆ, ˆγ are obtaied as the solutio of the followig system of equatios: [ l(θ α δ i α + δ iγ(αx i 1 x i + (1 δ iγ(αx i 1 x i h(x i, 1 h(x i [ l(θ δ i δ i l(αx i + δ i γ(αx i l(αx i + (1 δ ih(x i γ(αx i l(αx i, 1 h(x i [ l(θ δ i γ γ δ i(αx i (1 δ ih(x i (αx i, 1 h(x i where h(x h(x;α,,γ exp[ γ(αx The system does ot admit ay explicit solutio, therefore the ML estimates ˆθ ( ˆα, ˆ, ˆγ ca be obtaied oly by meas of umerical procedures Uder the usual regularity coditios, the well-ow asymptotic properties of the maximum lielihood method esure that ( ˆθ θ d N(,Σ θ, where Σ θ [I(θ 1 is the asymptotic variace-covariace matrix ad I(θ is the Fisher Iformatio Matrix, whose etries were calculated earlier 6 Geeralizatio via Kumaraswamy Distributio I this sectio, we preset results o the geeralized iverse Weibull Distributio via the Kumaraswamy distributio I particular, we derive the probability desity fuctio (pdf, cumulative distributio fuctio (cdf, momets, ad some additioal properties 61 Cumulative Distributio ad Probability Desity Fuctios The cdf ad pdf of the Kumaraswamy proportioal iverse Weibull (Kum-PIW distributio are give by ad g (x;α,,γ,λ,ϕ g (x G (x;α,,γ,λ,ϕ G (x 1 [1 G λ (x ϕ 1 {1 exp[ γλ(αx } ϕ, αγλϕ(αx 1 exp[ γλ(αx {1 exp[ γλ(αx }, for α >, >, γ >, λ >, ad ϕ > respectively 62 Mode of Kum-PIW Distributio To obtai the mode of the Kum-PIW distributio, we solve the equatio lg (x;α,,γ,λ,ϕ x for x Note that lg (x l(αγλϕ (1 + l(αx γλ(αx + (ϕ 1l{1 exp[ γλ(αx }
11 Oluyede, Yag The derivative of lg (x;α,,γ,λ,ϕ with respect to x is give by lg (x x 1 + x + αγλ(αx 1 + (ϕ 1{ exp[ γλ(αx αγλ(αx 1 } 1 exp[ γλ(αx Now lg (x;α,,γ,λ,ϕ x implies (1 + {1 exp[ γλ(αx } + γλ(αx {1 ϕ exp[ γλ(αx } Whe ϕ 1, the above equatio reduces to: ad we obtai x {1 exp[ γλ(αx }[γλ(αx 1, ( α γλ 1+ 1, which the same result as the mode of PIW distributio whe λ 1 Whe ϕ 1, we ca use umerical method to solve for the mode 63 Hazard Fuctio The hazard fuctio of the Kum-PIW distributio is give by λ G (x;α,,γ,λ,ϕ g (x;α,,γ,λ,ϕ Ḡ (x;α,,γ,λ,ϕ αγλϕ(αx 1 exp[ γλ(αx 1 exp[ γλ(αx, for α >, >, λ >, γ >, ad ϕ > We ca apply Glaser s Theorem to see that the hazard fuctio of Kum-PIWD has a upside dow bathtub shape (UBT 64 Reverse Hazard Fuctio The reverse hazard fuctio of the Kum-PIW distributio is give by τ G (x;α,,γ,λ,ϕ g (x;α,,γ,λ,ϕ G (x;α,,γ,λ,ϕ for α >, >, λ >, γ >, ad ϕ > 65 Momets αγλϕ(αx 1 e [ γλ(αx {1 e [ γλ(αx } 1 {1 e [ γλ(αx } ϕ, The c th o-cetral momet of the Kum-PIW distributio is give by E(X c x c g (x;α,,γ,λ,ϕdx x c αγλϕ(αx 1 exp[ γλ(αx {1 exp[ γλ(αx } dx
Electroic Joural of Applied Statistical Aalysis 111 Note that whe ϕ 1, this turs out to be the momets of the PIW distributio with parameters α α, ad γ γλ, that is ( E(X c (γλ c c α c Γ, for > c Whe ϕ > 1 ad a iteger, ote that ad we have E(X c {1 exp[ γλ(αx } ( ϕ 1 x c αγλϕ(αx 1 exp[ γλ(αx ( 1 exp[ γλ(αx, ( ϕ 1 ( 1 exp[ γλ(αx dx ( ϕ 1 x c αγλϕ(αx 1 exp[ γλ(αx ( ϕ 1 {αγλϕ( 1 x c (αx 1 exp[ γλ(1 + (αx dx} Let γλ(1 + (αx t, the we have E(X c 66 Shao Etropy (γλ c ϕ α c Γ ( ϕ( 1 ( α c ( c (1 + c 1 (γλ c Shao etropy for Kum-PIW distributio is give by ( 1 exp[ γλ(αx dx t c 1 exp[ tdt ( ( ϕ 1 ( 1 (1 + c 1 H(g E[ lg (X lg (x g (xdx (A + B +C + D, where A l(αγλϕ g (xdx l(α γλ ϕ, B (1 + l(αxαγλϕ(αx 1 exp[ γλ(αx {1 exp[ γλ(αx } dx,
112 Oluyede, Yag C γλ(αx g (xdx, ad D (ϕ 1l{1 exp[ γλ(αx } g (xdx We simplify the quatities A, B, C, ad D Note that B [ ( ϕ 1 (1 + αγλϕ ( 1 l(αx (αx 1 exp[ γλ( + 1(αx dx Let γλ( + 1(αx t, the the quatity B ca be rewritte as B (1 + ϕ ( ( 1 (1 + (1 + ϕ ( (1 + [lt l(γλ(1 + exp[ tdt ( 1 [Γ (1 l(γλ(1 + Also, for the quatity C, we have C αγ 2 λ 2 ϕ αγ 2 λ 2 ϕ γλ(αx g (xdx (αx 2 1 exp[ γλ(αx {1 exp[ γλ(αx } dx [( ϕ 1 ( 1 Now, let γλ( + 1(αx t, the C reduces to C ϕ ϕ [( ( 1 (1 + 2 [( (αx 2 1 exp[ γλ(1 + (αx dx ( 1 (1 + 2 t exp[ tdt Note that D (ϕ 1l{1 exp[ γλ(αx } g (xdx ( ϕ 1 (ϕ 1αγλϕ ( 1 (αx 1 l{1 exp[ γλ(αx } exp[ γλ(1 + (αx dx
Electroic Joural of Applied Statistical Aalysis 113 To simplify D, we let γλ( + 1(αx t, the D (ϕ 1ϕ [( ( 1 ( [ l 1 exp 1 + We ow that Taylor expasio for l(1 x is: l(1 x 1 x, for x < 1, t exp[ t dt 1 + so that ad ( [ l 1 exp t exp[ t dt 1 + D (ϕ 1ϕ (ϕ 1ϕ exp[ t 1+ 1 exp[ (1++t 1 1 + 1 (1 + + +1 1 [( ( 1 1 + [( ( 1 1 + 1, +1 ( 1 +1 1 1 1 1+ exp[ t dt dt Fially, we obtai Shao etropy: that is, H(g (x l(αγλϕ ( ( ϕ [ ( 1 1 + H(g (x [A + B +C + D, +1 1 1 + + (ϕ 1 1 1 1 + [Γ (1 l[γλ(1 + 67 Estimatio i Right Cesored Data from Kum-PIW Distributio I this sectio, the maximum lielihood estimatio (MLE of the Kum-PIW distributio parameters α,, γ, λ ad ϕ uder type I cesorig is preseted Suppose we have idepedet positive radom variables X 1,X 2,,X where X i has a associated idicator variable δ i where
114 Oluyede, Yag δ i 1 if X i is a observed failure time ad δ i if X i is right cesored, the the lielihood fuctio is give by L The log-lielihood fuctio is, that is ll l(θ ll g δ i (x i;α,,γ,λ,ϕ[1 G (x i ;α,,γ,λ,ϕ 1 δ i {δ i l[g (x i ;α,,γ,λ,ϕ + (1 δ i l[1 G (x i ;α,,γ,λ,ϕ}, [δ i l(αγλϕ δ i ( + 1l(αx i + δ i lh(x i + (ϕ δ i l[1 h(x i, where h(x exp[ γλ(αx The derivative of h(x i with respect to the parameters are: h(x i α h(x i h(x i γ h(x i λ The ormal equatios are give by: h(x i [γλ(αx i 1 x i, h(x i [γλ(αx i l(αx i, h(x i [ λ(αx i, h(x i [ γ(αx i l(θ α l(θ l(θ γ l(θ λ l(θ ϕ { δ i α + B(x i,δ i γλ(αx i 1 x i }, { δ i δ i l(αx i + B(x i,δ i γλ(αx i l(αx i }, { δ i γ + B(x i,δ i [ λ(αx i }, { δ i λ + B(x i,δ i [ γ(αx i }, { δ i ϕ + l[1 h(x i}, where B(x i,δ i δ i ϕh(x i 1 h(x i These equatios do ot have a closed form solutio We ca use umerical methods to solve this problem
Electroic Joural of Applied Statistical Aalysis 115 7 Cocludig Remars Results o the geeralized iverse Weibull ad Kumaraswamy geeralized iverse Weibull distributios are preseted This class of distributios cotais a fairly large umber of distributios with potetial applicatios to a wide area of probability ad statistics Properties of the geeralized iverse Weibull ad Kumaraswamy geeralized iverse Weibull distributios icludig the pdfs, cdfs, momets, hazard fuctios, reverse hazard fuctios, coefficiets of variatio, sewess ad urtosis, Fisher iformatio, Shao etropy ad -etropy are preseted Estimatio of the parameters of the models for complete ad right cesored data are also preseted Acowledgmet The authors wish to express their gratitude to the referees ad editor for their valuable ad isightful commets Refereces Bloc, HW, Savits, TH (1998 The Reverse hazard Fuctio, Probability i the Egieerig ad Iformatioal Scieces, 12, 69-9 Calabria, R, Pulcii, G (1989 Cofidece Limits for Reliability ad Tolerace Limits i the Iverse Weibull Distributio, Egieerig ad System Safety, 24, 77-85 Calabria, R, Pulcii, G (199 O the Maximum Lielihood ad Lease Squares Estimatio i Iverse Weibull Distributio, Statistica Applicata, 2, 53-66 Calabria, R, Pulcii, G (1994 Bayes 2-Sample Predictio for the Iverse Weibull Distributio, Commuicatios i Statistics-Theory Meth, 23(6, 1811-1824 Chadra, NK, Roy, D (21 Some results o Reverse Hazard Rate, Probability i the Egieerig ad Iformatio Scieces, 15, 95-12 Cordeiro, GM, Ortega, EM, Nadarajah, S (21 The Kumaraswamy Weibull distributio with applicatio to failure data, Joural of Frali Istitute, 347, 1399-1429 Glaser, RE (198 Bathtub ad Related Failure Rate Characterizatios, Joural of America Statistical Associatio, 75, 667-672 Johso, NL, Kotz, S, Balarisha, N (1984 Cotiuous Uivariate Distributios-1, Secod Editio, Joh Wiley ad Sos Joes, MC (29 Kumaraswamy s distributio: a beta-type distributio with some tractability advatages, Statistical Methodology, 6, 7-91 Keilso, J, Sumita, U (1982 Uiform Stochastic Orderig ad Related Iequalities, Caadia Joural of Statistics, 1, 181-198 Keller, AZ, Gibli, MT, Farworth, NR (1985 Reliability Aalysis of Commercial Vehicle Egies, Reliability Egieerig, 1, 15-25, 89-12 Kha, MS, Pasha, GR, Pasha, AH, (28 Theoretical Aalysis of Iverse Weibull Distributio, Issue 2, Vol 7, 3-38
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