The Extended Weibull Geometric Family

Transcription

1 The Exteded Weibull Geometric Family Giovaa Oliveira Silva 1 Gauss M. Cordeiro 2 Edwi M. M. Ortega 3 1 Itroductio The literature o Weibull models is vast, disjoited, ad scattered across may differet jourals. Whe modelig mootoe hazard rates, the Weibull distributio may be a iitial choice because of its egatively ad positively skewed desity shapes ad it provides simple ad elegat forms to may problems i reliability aalysis. However, it does ot provide a reasoable parametric fit for modelig pheomeo with o-mootoe failure rates such as the bathtub shaped ad the uimodal failure rates which are commo i reliability ad biological studies. Over the last two decades several ew models have bee proposed that are i some way related to the Weibull distributio. They provide a richess that makes them appropriate to model complex data sets. From the statistical modelig poit of view, we itroduce a geeralized class of distributios havig two importat aspects: i oe had, the proposed class of models ivolves oe additioal shape parameter to the parameters of the baselie class of distributios, ad, o the other had, the ew parameter has a clear physical iterpretatio. I the last few years, ew classes of distributios were proposed by extedig the Weibull distributio to cope with bathtub shaped failure rates. A good review of some of these models is doe by Pham ad Lai (2007). Amog these models, we poit out the expoetiated Weibull, additive Weibull (AW) modified Weibull, beta expoetial, BLZ, geeralized modified Weibull ad beta modified Weibull (Silva, Ortega ad Cordeiro, 2010) distributios. This ote itroduces a ew family of lifetime models called the exteded Weibull geometric (EWG) family by compoudig the class of exteded Weibull (EW) (Nadarajah ad Kotz, 2005) ad the geometric distributios. We hope that the ew family will attract wider applicatios i reliability ad biology ad i other areas of research. Several ew distributios are obtaied as special cases of this family icludig the exteded expoetial geometric (EEG), modified Weibull geometric (MWG), expoetial power geometric (EPG), log-weibull geometric (LWG), geeralized power Weibull geometric (GPWG), amog several others. Besides these distributios, the EWG family cotais other promisig ew distributios as, for example, 1 Departameto de Estatística - UFBA giovaa@ufba.br 2 Departameto de Estatística - UFPE. 3 Esalq/USP 1

2 the additive Weibull geometric (AWG) ad XTG geometric (XTGG) distributios. Due to its flexibility i accommodatig differet forms of the risk fuctio, the ew family is a importat tool to be used i a variety of problems i modelig survival data. Various EWG distributios are ot oly coveiet for modelig comfortable bathtub-shaped failure rates but they are also suitable for testig goodess-of-fit of special models as sub-hypotheses. Alteratively, various works had itroduced distributios which are more flexible i modelig mootoe or uimodal failure rates but it is ot useful for modelig the bathtub shaped failure rates. Adamidis ad Loukas (1998) defied the expoetial geometric (EG) distributio to model lifetime data with decreasig failure rate fuctio. Gupta ad Kudu (1999, 2001a,b) proposed the geeralized expoetial (GE) (also called the expoetiated expoetial) distributio, ad ivestigated some of its mathematical properties. This distributio has oly icreasig or decreasig failure rate fuctio. Followig the same idea of the GE distributio, Silva, Barreto-Souza ad Cordeiro (2010) proposed the geeralized expoetial geometric (GEG) model ad demostrated that its failure rate fuctio ca be icreasig, decreasig or uimodal. Aother geeralizatio of the Weibull for modelig mootoe or uimodal failure rates, refereed to as the Weibull geometric (WG) distributio, was ivestigated by Barreto-Souza et al. (2010). 2 Methods 2.1 Exteded Weibull geometric family The cumulative distributio fuctio (cdf) of the class of EW distributios is give by G α,τ(y) = 1 exp{ αh(y)}, y > 0, (1) where α > 0 is a scale parameter, H(y) is a mootoically icreasig fuctio of y with the oly limitatio H(y) 0 ad τ deotes the vector of ukow parameters i H(y). If H(y) is a power law fuctio, the equatio (1) reduces to the Weibull distributio. Its pdf is give by g α,τ(y) = αh(y) exp{ αh(y)}, y > 0, (2) where h(y) = H(y)/ y. We deote by Y α EW(α,τ) a radom variable Y α havig desity fuctio (2). The modified Weibull (MW) distributio is a special case of (2) whe H(y) = y γ exp(λy), where γ > 0 ad λ 0. Clearly, the Weibull distributio is obtaied as a basic exemplar whe λ = 0. developmets i this paper. It is give by The EW quatile fuctio plays a importat role i the algebraic Q α,τ(u) = H 1 { 1 α log(1 u) }. (3) 2

3 We oly require the iverse fuctio of H(y) to obtai the EW quatile fuctio (3). Let Z be a geometric radom variable with probability mass fuctio give by P(z; p) = (1 p) p z 1 for z N ad p (0,1). Suppose that {Y i } Z are idepedet ad idetically distributed (i.i.d.) radom variables havig the same desity fuctio (2), where the ukow umber of radom variables is modeled by the geometric distributio of Z. We assume that the radom variables Z ad Y are idepedet ad defie X = mi({y i } Z ). It the follows the coditioal desity fuctio of X give Z = z f (x z;α,τ) = zαh(x) exp{ αh(x)}[1 exp{ αh(x)}], ad the the EWG desity fuctio reduces to f (x; p,α,τ) = αh(x)exp{ αh(x)}(1 p)[1 pexp{ αh(x)}] 2, x > 0. (4) The EWG cdf is give by F(x; p,α,τ) = 1 exp{ αh(x)}, x > 0. (5) 1 pexp{ αh(x)} Hereafter, a radom variable X havig desity fuctio (4) is deoted by X EWG(p,α,τ). The failure rate fuctio correspodig to (4) becomes h(x; p,α,τ) = αh(x), x > 0. (6) 1 pexp{ αh(x)} The MWG distributio is obtaied from (4) whe H(x) = x γ exp(λx), where γ > 0 ad λ 0. Further, the WG distributio is also obtaied as a special case whe λ = 0. The EW distributio follows as the limitig distributio (the limit is defied i terms of the covergece i distributio) of the EWG distributio whe p 0 +. O the other had, if p 1, we obtai the distributio of a radom variable Y such that P(Y = 0) = 1. Hece, the parameter p ca be iterpreted as a degeeratio parameter, because the EWG distributio coverges to a distributio degeerated i zero, whe p varies from zero to oe. The EWG family of distributios cotais as special models some well-kow distributios. 2.2 Maximum Likelihood Estimatio We determie the maximum likelihood estimates (MLEs) of the parameters of the EWG family from complete samples oly. Let x 1,...,x be a radom sample of size from the EWG(p,α,τ) distributio. The log-likelihood fuctio for the vector of parameters θ = (p,α,τ) T becomes l(θ) = [log(α) + log(1 p)] + 3 log[h(x i )] α H(x i ) (7)

4 2 log[1 p exp( αh(x i ))]. The log-likelihood ca be maximized either directly by usig the MaxBFGS routie i the matrix programmig laguage Ox (see, Doorik, 2007). For iterval estimatio ad hypothesis tests o the model parameters, we require the observed iformatio matrix J = J(θ), whose elemets ca be computed umerically. Uder coditios that are fulfilled for parameters i the iterior of the parameter space but ot o the boudary, the asymptotic distributio of ( θ θ) is Np+2 (0,I(θ) 1 ), where I(θ) is the expected iformatio matrix. We ca substitute I(θ) by J( θ), i.e., the observed iformatio matrix evaluated at θ. The multivariate ormal N p+2 (0,J( θ) 1 ) distributio ca be used to costruct approximate cofidece itervals for the idividual parameters ad for the hazard rate ad survival fuctios. 3 Results ad Discussio We cosider a ucesored data set from Nichols ad Padgett [9] cosistig of 100 observatios o breakig stress of carbo fibres (i Gba). We fit some distributios to these data to illustrate the superiority of some ew distributios over their sub-models. Cosiderig the reparametrizatios, for MWG α = α γ 1 ad for XTGG α = α β 1. Table 1 lists the MLEs of the model parameters (the stadard errors are give i paretheses). Sice the value of the AIC is smaller for the BLZG distributio compared to those values of other models, this distributio yields the best fit to these data. It ca easily checked usig AIC that the ew distributios provide superior fits tha the correspodig EW special models to these data. 4 Cocludig remarks I this paper, we itroduce ad study a ew family of distributios called the exteded Weibull geometric (EWG) family that geeralizes the Weibull geometric ad modified Weibull geometric distributios proposed by Barreto-Souza et al. (2009) ad Cordeiro et al. (2010), respectively, amog other distributios. The applicatio to real lifetime data idicate that the EWG family could provide a better fit tha other well-kow lifetime models. We hope that the ew model may attract wider applicatios i statistics. 4

5 Tabela 1: MLEs of the model parameters for the breakig stress of carbo fibres data, the correspodig SEs (give i paretheses) ad the AIC statistics. Model p α 1 β AIC BLZG ( ) ( (116.19) BLZ ( ) ) Model p α 1 λ γ AIC MWG ( ) ( ) ( ) MW e (2.5402) ( ) ( ) p α 1 λ β AIC XTGG e ( ) (2.0880e-016) ( ) ( ) XTG e ( ) (0.0000) Ackowledgmets The fiacial support from CNPq is gratefully ackowledged. Refereces Barreto-Souza, W., de Morais, A.L. ad Cordeiro, G. M., The Weibull-geometric distributio. Joural of Statistical Computatio ad Simulatio, DOI: / Doorik, J. A., A Object-Orieted Matrix Laguage Ox 5. Timberlake Cosultats Press: Lodo. Nichols, M.D. ad Padgett, W.J. (2006). A bootstrap cotrol chart for Weibull percetiles. Quality ad Reliability Egieerig Iteratioal, 22, Pham, H. ad Lai, C. D. (2007). O recet geeralizatios of the Weibull distributio. IEEE Trasactios o Reliability, 56, Silva, G.O.; Ortega, E.M.M. ad Cordeiro, G.M. (2010). The Beta Modified Weibull Distributio. Lifetime Data Aalysis, 16, Silva, R.B.; Barreto-Souza, W. ad Cordeiro, G.M. (2010). A ew distributio with decreasig, icreasig ad upside-dow bathtub failure rate. Computatioal Statistics & Data Aalysis, 54,