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The Exponentiated Generalized Gumbel Distribution Revista Colombiana de Estadística January 2015, Volume 38, Issue 1, pp. 123 to 143 DOI: http://dx.doi.org/10.15446/rce.v38n1.48806 The Exponentiated Generalized Gumbel Distribution Distribución Gumbel exponencializada generalizada Thiago Andrade1;a, Heloisa Rodrigues1;b, Marcelo Bourguignon2;c, Gauss Cordeiro1;d 1Departamento de Estatística, Centro de Ciências Exatas e da Natureza, Universidade Federal de Pernambuco, Recife, Brasil 2Departamento de Estatística, Centro de Ciências da Natureza, Universidade Federal do Piauí, Teresina, Brasil Abstract A class of univariate distributions called the exponentiated generalized class was recently proposed in the literature. A four-parameter model within this class named the exponentiated generalized Gumbel distribution is de- fined. We discuss the shapes of its density function and obtain explicit ex- pressions for the ordinary moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. The density function of the order statistic is derived. The method of maximum likelihood is used to estimate model parameters. We determine the observed informa- tion matrix. We provide a Monte Carlo simulation study to evaluate the maximum likelihood estimates of model parameters and two applications to real data to illustrate the importance of the new model. Key words: Gumbel Distribution, Maximum Likelihood, Moment, Rényi Entropy. Resumen Recientemente fue propuesta una clase de distribuciones univariadas cono- cida como la clase exponencializada generalizada. Dentro de esta clase se de- fine un modelo con cuatro parámetros conocido como distribución Gumbel exponencializada generalizada. En este artículo estudiamos las formas de la función de densidad de este modelo, obtenemos expresiones explicitas para los momentos ordinarios, las funciones generadora de momentos y cuantílica, aGraduate student. E-mail: [email protected] bGraduate student. E-mail: [email protected] cAssistent Professor. E-mail: [email protected] dProfessor. E-mail: [email protected] 123 124 Thiago Andrade, Heloisa Rodrigues, Marcelo Bourguignon & Gauss Cordeiro para los desvíos medios, las curvas de Bonferroni y Lorenz, y, para la en- tropía de Rényi. Derivamos la función de densidad de la estadística de orden. Usamos el método de máxima verosimilitud para estimar los parámetros del modelo. Determinamos la matriz de información observada. Presentamos una simulación de Monte Carlo que evalúa las estimativas de máxima ve- rosimilitud de los parámetros del modelo y presentamos dos aplicaciones a datos reales que ilustran la importancia del modelo nuevo. Palabras clave: distribución Gumbel, entropía de Rényi, máxima verosi- militud, momentos. 1. Introduction The Gumbel distribution is a very popular statistical model due to its wide applicability. An extensive list of the Gumbel model applications can be obtained in Kotz & Nadarajah (2000). In the area of climate modeling, for example, some applications of the Gumbel model include: global warming problems, offshore modeling, rainfall and wind speed modeling (Nadarajah 2006). We can find ap- plications of this model in various areas of engineering such as flood frequency analysis, network, nuclear, risk-based, space, software reliability, structural and wind engineering (Cordeiro, Nadarajah & Ortega 2012). Due to its wide appli- cability, several works aimed at extending the Gumbel model become important. Some examples are mentioned in: Nadarajah & Kotz (2004), Nadarajah (2006) and Cordeiro et al. (2012). The cumulative distribution function (cdf) G(x) and probability density func- tion (pdf) g(x) of the Gumbel (Gu) distribution are given by x − µ G(x; µ, σ) = exp −exp − (1) σ and 1 x − µ x − µ g(x; µ, σ) = exp − + exp − ; (2) σ σ σ respectively, for x 2 R, µ 2 R and σ > 0. In recent years, some different generalizations of continuous distributions have received great attention in the literature. Here, we refer to the papers: Marshall & Olkin (1997) for the Marshall-Olkin class, Eugene, Lee & Famoye (2002) for the Beta class, Zografos & Balakrishnan (2009) and Ristic & Balakrishnan (2011) for the Gamma class and Cordeiro & de Castro (2011) for the Kumaraswamy class of distributions. In a similar manner, for any baseline cdf G(x), and x 2 R, Cordeiro, Ortega & Cunha (2013) defined the exponentiated generalized (EG) class of distributions with two extra parameters α > 0 and β > 0 and cdf F (x) and pdf f(x) given by F (x) = f1 − [1 − G(x)]αgβ (3) and f(x) = αβ[1 − G(x)]α−1f1 − [1 − G(x)]αgβ−1g(x); (4) Revista Colombiana de Estadística 38 (2015) 123–143 The EGGu Distribution 125 respectively. In this paper, we study the so-called exponentiated generalized Gum- bel (“EGGu” for short) distribution by inserting (1) in equation (3). The rest of the paper is organized as follows. In Section 2, we define the EGGu distribution. Shapes of the density function are discussed in Section 3. Explicit expressions for cumulative and density functions, quantile function, ordinary mo- ments, mean deviations, Bonferroni and Lorenz curves, generating function, Rényi entropy and order statistics are derived in Section 4. We discuss maximum like- lihood estimation and present a Monte Carlo simulation experiment to evaluate the maximum likelihood estimates (MLEs) of the model parameters in Section 5. Two applications in Section 6 illustrate the usefulness of the new distribution for data modeling. Lastly, concluding remarks are given in Section 7. 2. The EGGu distribution The EGGu distribution was proposed by Cordeiro et al. (2013), but they did not study its mathematical properties. The cdf and pdf of the EGGu distribution are given by x − µαβ F (x) = F (x; α; β; µ, σ) = 1 − 1 − exp − exp − (5) σ and αβ x − µ x − µ f(x) = f(x; α; β; µ, σ) = exp − + exp − σ σ σ x − µα−1 1 − exp − exp − (6) σ x − µαβ−1 1 − 1 − exp − exp − ; σ respectively. Henceforth, a random variable X having density function (6) is denoted by X ∼ EGGu(α; β; µ, σ). We write F (x) = F (x; α; β; µ, σ) in order to eliminate the dependence on the model parameters. In this model, µ 2 R and σ > 0 are the location and scale parameters, respectively, whereas α > 0 and β > 0 are the shape parameters. The Gumbel distribution is clearly a special case of (5) when α = β = 1. Setting β = 1 we obtain the exponentiated Gumbel distribution defined by Nadarajah (2006). 3. Shape The main features of the density shape can be perceived through the study of its first and second derivative. Regarding the EGGu distribution, the first derivative of logff(x)g is Revista Colombiana de Estadística 38 (2015) 123–143 126 Thiago Andrade, Heloisa Rodrigues, Marcelo Bourguignon & Gauss Cordeiro d logff(x)g 1 (α − 1) z α (β − 1) z (1 − z)α−1 = −1 − ln(z) 1 − + ; dx σ (1 − z) [1 − (1 − z)α] x−µ where z = exp − exp − σ . Here, 0 < z < 1. The critical values of f(x) are the roots of the equation: (α − 1) z α (β − 1) z (1 − z)α−1 ln(z) + 1 − = : (7) (1 − z) [1 − (1 − z)α] ln(z) If the point x = x0 is a root of (7), then we can classify it as local maximum, local minimum or inflection point when we have, respectively, λ(x0) < 0, λ(x0) > 0 2 2 and λ(x0) = 0, where λ(x) = d logff(x)g=dx . Plots of the EGGu density function for selected parameter values are displayed in Figure 1. β = 1 β = 5 1.0 α = 2 1.0 α = 2 α = 3 α = 3 α = 4 α = 4 α = 5 0.8 0.8 α = 5 0.6 0.6 Density Density 0.4 0.4 0.2 0.2 0.0 0.0 −2 −1 0 1 234 −2 −1 0 1 234 x x α = 2 α = 9 1.5 β = 3 1.5 β = 3 β = 5 β = 5 β = 7 β = 7 β = 9 β = 9 1.0 1.0 Density Density 0.5 0.5 0.0 0.0 −2 −1 0 1 234 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 x x Figure 1: Plots of the EGGu pdf for µ = 0, σ = 1 and some values of α and β. Revista Colombiana de Estadística 38 (2015) 123–143 The EGGu Distribution 127 4. Properties In this section, we study some structural properties of the EGGu distribution. 4.1. A Useful Representation Several properties of the new distribution can be derived using the concept of exponentiated distributions. The class of exponentiated distributions has been studied by many authors in recent years. Here, we refer to a few of these pa- pers: Gupta & Kundu (2001) for exponentiated exponential, Nadarajah & Kotz (2003) for exponentiated Fréchet, Nadarajah (2006) for exponentiated Gumbel, Shirke & Kakde (2006) for exponentiated lognormal, Nadarajah & Gupta (2007) for exponentiated Gamma, Carrasco, Ortega & Cordeiro (2008) for exponentiated modified Weibull and Cordeiro, Ortega & Silva (2011) for exponentiated general- ized Gamma. For an arbitrary baseline cdf G(x), a random variable is said to have the exponentiated-G (“exp-G” for short) distribution with power parameter a > 0, say a a−1 Y ∼exp-G(a), if its cdf and pdf are Ha(x) = G(x) and ha(x) = a g(x)G(x) , respectively. We consider the generalized binomial expansion 1 X b (1 − z)b = (−1)k zk; (8) k k=0 which holds for any real non-integer b and jzj < 1. Using expansion (8) twice in equation (5), we can express the EGGu cdf as 1 X F (x) = wj+1 Hj+1(x); (9) j=0 P1 j+m+1 β m α j+1 where wj+1 = m=1(−1) m j+1 and Hj+1(x) = G(x) is the expo- nentiated Gumbel (exp-Gu) cdf with power parameter j + 1.
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