The ew class of Kummer beta geeralized distributios Rodrigo Rossetto Pescim 12 Clarice Garcia Borges Demétrio 1 Gauss Moutiho Cordeiro 3 Saralees Nadarajah 4 Edwi Moisés Marcos Ortega 1 1 Itroductio Geeralized beta distributios have bee widely studied i statistics ad umerous authors have developed various classes of these distributios. I the last decade, [2] proposed a geeral class of distributios for a radom variable defied from the logit of the beta radom variable by employig two parameters whose role is to itroduce skewess ad to vary tail weight. However, the beta geeralized distributios do ot offer more flexibility to extremes (right ad left) of the curves of the desity fuctios ad therefore they are ot suitable for aalyzig data sets with high degree of asymmetry. The Kummer beta (KB) distributio o the uit iterval (,1) was proposed by [4] with cumulative distributio fuctio (cdf) ad probability desity fuctio (pdf) give by ad x F KB (x) = K t a 1 (1 t) b 1 exp( ct)dt, f KB (x) = K x a 1 (1 x) b 1 exp( cx), < x < 1, (1) where a >, b > ad < c <. Here, ad 1F 1 (a;a + b; c) = 1 LCE - ESALQ/USP. e-mail: rrpescim@usp.br 2 Agradecimeto ao CNPq pelo apoio fiaceiro. 3 DE - UFPE. 4 Uiversity of Machester - UK K 1 = Γ(a)Γ(b) Γ(a + b) 1 F 1 (a;a + b; c) (2) Γ(a + b) 1 t a 1 (1 t) b 1 (a) k ( c) exp( ct)dt = Γ(a)Γ(b) k k= (a + b) k k! 1
is the cofluet hypergeometric fuctio, Γ( ) is the gamma fuctio ad (d) k = d(d +1)...(d + k 1) deotes the ascedig factorial. This distributio is a extesio of the beta distributio, ad for a < 1 (ad certai values of the parameter c) yields bimodal distributios o fiite rage. Cosider startig from a paret cotiuous distributio fuctio G(x). A atural way of geeratig families of distributios o some other support from a simple startig paret distributio with desity fuctio g(x) = dg(x)/dx is to apply the quatile fuctio to a family of distributios o the iterval (,1). We ow use the same methodology of [2] ad [1] to costruct a ew class of Kummer beta geeralized (KBG) distributios. From a arbitrary paret cumulative distributio G(x), the cdf F(x) of the KBG family of distributios is defied by G(x) F(x) = K t a 1 (1 t) b 1 exp( ct)dt, (3) where a > ad b > are shape parameters which itroduce skewess, ad thereby promote weight variatio of the tails, whereas the parameter < c < squeezes the desity fuctio to the left or right, i.e., it leads the tail weights of the desity to extreme values. The desity fuctio correspodig to (3) ca be writte as f (x) = K g(x)g(x) a 1 {1 G(x)} b 1 exp{ c G(x)}, (4) where K is defied i (2). For each cotiuous G distributio (here ad heceforth G deotes the baselie distributio), we ca associate the KBG-G distributio with three extra parameters a, b ad c defied by the desity fuctio (4). 2 Special KBG Geeralized Distributios The KBG desity fuctio (4) allows for greater flexibility of its tails ad promotes the variatio of the tail weights to the extremes of the distributio. It ca be widely applied i may areas of egieerig ad biological scieces. The desity fuctio (4) will be most tractable whe the cdf G(x) ad the pdf g(x) have simple aalytic expressios. We ow defie some of the may distributios which ca arise as special sub-models withi the KBG class of distributios. 2.1 KBG-Normal The KBGN desity fuctio is obtaied from (4) by takig G( ) ad g( ) to be the cdf ad pdf of the ormal distributio, N(µ, 2 ), so that f (x) = K ( ){ x µ ϕ Φ )} a 1 { 1 Φ )} b 1 exp{ c Φ )}, 2
where x R, µ R is a locatio parameter, > is a scale parameter, a ad b are positive shape parameters, c R, ad ϕ( ) ad Φ( ) are the pdf ad cdf of the stadard ormal distributio, respectively. A radom variable with the above desity fuctio is deoted by X KBGN(a,b,c,µ, 2 ). For µ = ad = 1, we have the stadard KBGN distributio. 2.2 KBG-Weibull The cdf of the Weibull distributio with parameters β > ad α > is G(x) = 1 exp{ (βx) α } for x >. Correspodigly, the KBG-Weibull desity fuctio, say KBGW(a, b, c, α, β), reduces to f (x) = K α β α x α 1 [1 exp{ (βx) α }] a 1 exp{ c[1 exp{ (βx) α }] b(βx) α }, where x,a,b,β > ad c R. For α = 1, we obtai the KBG-expoetial (KBGE) distributio. The KBGW(1, b,, 1, β) distributio correspods to the expoetial distributio with parameter β = bβ. 2.3 KBG-Gamma Let Y be a gamma radom variable with cdf G(y) = Γ βy (α)/γ(α) for y,α,β >, where Γ( ) is the gamma fuctio ad Γ z (α) = z t α 1 e t dt is the icomplete gamma fuctio. The desity fuctio of a radom variable X havig the KBGGa distributio, say X KBGGa(a,b,c,β,α), ca be expressed as f (x) = K βα x α 1 e βx Γ(α) a+b 1 { exp c Γ } βx(α) Γ Γ(α) βx (α) a 1 { Γ(α) Γ βx (α) } b 1. For α = 1 ad c =, we obtai the KBGE distributio. The KBGGa(1,b,,β,1) distributio reduces to the expoetial distributio with parameter β = bβ. 3 Iferece Let γ be the p-dimesioal parameter vector of the baselie distributio i equatios (3) ad (4). We cosider idepedet radom variables X 1,...,X, each X i followig a KBG-G distributio with parameter vector θ = (a, b, c, γ). The log-likelihood fuctio l = l(θ) for the model parameters obtaied from (4) is l(θ) = log(k) + + (a 1) logg(x i ;γ) c G(x i ;γ) log{g(x i ;γ)} + (b 1) log{1 G(x i ;γ)}. 3
We ca ote that the elemets of score vector deped o the specified baselie distributio. Numerical maximizatio of the log-likelihood above is accomplished by usig the RS method which is available i the gamlss package i statistical software R. 4 Applicatio - Voltage data Here, we compare the results of the fits of the KBGW, beta Weibul (BW), expoetiated Weibull (EW) ad Weibull distributios to the data set studied by [3]. The data represet the times of failure ad ruig times for a sample of devices from a field-trackig study of a larger system. At a certai poit i time, 3 uits were istalled i ormal service coditios. Two causes of failure were observed for each uit that failed: the failure caused by a accumulatio of radomly occurrig damage from power-lie voltages pikes durig electric storms ad failure caused by ormal product wear. Table 1 lists the MLEs of the model parameters ad the values of the followig statistics for some models: Akaike Iformatio Criterio (AIC) ad Bayesia Iformatio Criterio (BIC). The computatios were doe usig statistical software R. These Tabela 1: MLEs ad iformatio criteria for voltage data. Model d β a b c AIC BIC KBGW 3.483.27.1648 17.6979-24.951 354.74 361.74 BW 5.6997.44.1522.2634 357.12 362.73 EW 4.6369.31.1972 1 362.92 367.12 Weibull 1.267.53 1 1 372.63 375.43 results idicate that the KBGW model has the smallest values for the AIC ad BIC statistics amog the fitted models, ad therefore it could be chose as the best model. More iformatio ca be provided by a visual compariso of the histograms of the data with the fitted desities. I order to assess if the model is appropriate, we provide i Figure 1a the histogram of the data ad the fitted KBGW, BW, EW ad Weibull desity fuctios. Further, i Figure 1b, we plot the empirical ad estimated survival fuctios of the KBGW, BW, EW ad Weibull distributios. We coclude that the KBGW distributio yields a good fit for these data. 5 Coclusios Followig the idea of the class of beta geeralized distributios ad the distributio by [4], we defie a ew family of Kummer beta geeralized (KBG) distributios to exted several widely kow distributios such as the ormal, Weibull ad Gumbel distributios. For each cotiuous G distributio, we ca defie the correspodig KBG-G distributio usig simple formulae. We discuss maximum likelihood estimatio ad iferece o the parameters. A 4
(a) (b) f(x)..2.4.6.8 KBGW BW EW Weibull S(x)..2.4.6.8 1. Kapla Meier KBGW BW EW Weibull 5 1 15 2 25 3 5 1 15 2 25 3 x x Figura 1: (a) Estimated KBGW, BW, EW ad Weibull desity fuctios for voltage data. (b) Estimated survival fuctios ad the empirical survival for voltage data. applicatio of the ew family to real data set is give to show the feasibility of the proposed class of models. We hope this geeralizatio may attract wider applicatios i statistics. Referêcias [1] CORDEIRO, G.M.; de CASTRO, M. A ew family of geeralized distributios. Joural of Statistical Computatio ad Simulatio. Elselvier. v. 81, p. 883-893, 211. [2] EUGENE, N.; LEE, C.; FAMOYE, F. Beta-ormal distributio ad its applicatios. Commuicatio i Statistics - Theory ad Methods. Elselvier. v. 31, p. 497-512, 22. [3] MEEKER, W.Q.; ESCOBAR, L.A. Statistical Methods for Reliability Data. Joh Wiley, New York. 1998. 346p. [4] NG, K.W.; KOTZ, S. Kummer-Gamma ad Kummer-Beta uivariate ad multivariate distributios. Research Report. v. 84, Departmet of Statistics, The Uiversity of Hog Kog, Hog Kog, 1995. 5