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1 This article was downloaded by: [ ] On: 14 August 2015, At: 20:34 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: 5 Howick Place, London, SW1P 1WG Click for updates American Journal of Mathematical and Management Sciences Publication details, including instructions for authors and subscription information: A new generalized gamma distribution with applications Marcelo Bourguignon a, Maria do Carmo S. Lima b, Jeremias Leão c, Abraão D. C. Nascimento b, Luis Gustavo B. Pinho b & Gauss M. Cordeiro b a Department of Statistics, Federal University of Rio Grande do Norte, Rio Grande do Norte, Brazil b Department of Statistics, Federal University of Pernambuco, Pernambuco, Brazil c Department of Statistics, Federal University of Piauí, Piauí, Brazil Published online: 14 Aug To cite this article: Marcelo Bourguignon, Maria do Carmo S. Lima, Jeremias Leão, Abraão D. C. Nascimento, Luis Gustavo B. Pinho & Gauss M. Cordeiro (2015) A new generalized gamma distribution with applications, American Journal of Mathematical and Management Sciences, 34:4, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the

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3 American Journal of Mathematical and Management Sciences, 34: , 2015 Copyright C Taylor & Francis Group, LLC ISSN: print / online DOI: / A NEW GENERALIZED GAMMA DISTRIBUTION WITH APPLICATIONS MARCELO BOURGUIGNON, 1 MARIA DO CARMO S. LIMA, 2 JEREMIAS LEÃO, 3 ABRAÃO D. C. NASCIMENTO, 2 LUIS GUSTAVO B. PINHO, 2 and GAUSS M. CORDEIRO 2 1 Department of Statistics, Federal University of Rio Grande do Norte, Rio Grande do Norte, Brazil 2 Department of Statistics, Federal University of Pernambuco, Pernambuco, Brazil 3 Department of Statistics, Federal University of Piauí, Piauí, Brazil SYNOPTIC ABSTRACT The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields. We introduce and study the gamma-nadarajah Haghighi model, which can be interpreted as a truncated generalized gamma distribution (Stacy, 1962). It can have a constant, decreasing, increasing, upside-down bathtub or bathtub-shaped hazard rate function depending on the parameter values. We demonstrate that the new density function can be expressed as a mixture of exponentiated Nadarajah Haghighi densities. Various of its structural properties are derived, including explicit expressions for the moments, quantile and generating functions, skewness, kurtosis, mean deviations, Bonferroni and Lorenz curves, probability weighted moments, and two types of entropy. We also investigate the order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived. We illustrate the flexibility of the new distribution by means of two applications to real datasets. Key Words and Phrases: generalized gamma distribution, hazard rate function, Nadarajah Haghighi distribution, maximum likelihood estimation. 1. Introduction In the last decades, several distributions have been proposed based on different modifications of the beta, gamma, and Weibull Address correspondence to Marcelo Bourguignon, Department of Statistics, Federal University of Rio Grande do Norte, Rio Grande do Norte, Brazil. m.p.bourguignon@gmail.com 309

4 310 M. Bourguignon et al. distributions, among others, to provide bathtub hazard rate functions (hrfs) as, for example, the exponentiated Weibull distribution pioneered by Mudholkar and Srivastava (1993). Cordeiro, Ortega, and Nadarajah (2010) defined the Kumaraswamy Weibull distribution, Kong, Lee, and Sepanski (2007) studied the beta gamma distribution and Cordeiro, Ortega, and Silva (2011) proposed the exponentiated generalized gamma distribution. Recently, Zografos and Balakrishnan (2009) introduced a broad class of univariate distributions called the gamma-g ( GG, for short) distributions, from any baseline cumulative density function (cdf) G(x) using an additional shape parameter a > 0, introducing skewness, and varying tail weight. The GG class is defined by the probability density function (pdf) and cdf and f (x) = g(x) Ɣ(a) F (x) = { log [1 G(x)] } a 1 (1) γ {a, log[1 G(x)]}, (2) Ɣ(a) respectively, where γ (a, z) = 1 z Ɣ(a) 0 ta 1 e t dt denotes the incomplete gamma function and Ɣ( ) is the gamma function. Each new GG distribution can be obtained from a specified G distribution. For a = 1, the G distribution is a basic exemplar with a continuous crossover toward cases with different shapes (for example, a particular combination of skewness and kurtosis). Zografos and Balakrishnan (2009) motivated the GG distribution as follows. Let X (1),...,X (n) be lower record values from a sequence of i.i.d. random variables from a population with pdf g(x). Then, the pdf of the nth lower record value is given by Equation (1) with a = n. A logarithmic transformation of the baseline distribution G transforms the random variable X, with the density function Equation (1), to a gamma distribution. In other words, if X has density Equation (1), then the random variable Z = log[1 G(X )] has a gamma density π(z; a) = Ɣ(a) 1 z a 1 e z z > 0, say Z G(a, 1). The opposite is also true, if Z G(a, 1), then the random variable X = G 1 (1 e z ) has the GG density function Equa-

5 A New Generalized Gamma Distribution 311 tion (1). Nadarajah, Cordeiro, and Ortega (2013) derived some mathematical properties of Equation (1) in the most simple, explicit, and general forms for any G distribution. If G(x; α, λ) = 1 exp[ (λx) α ] is the Weibull cumulative distribution with parameters α>0 (shape parameter) and λ>0 (scale parameter), then Equation (2) yields the generalized gamma ( GeGa ) cumulative distribution (Stacy, 1962) F (x; a,α,λ) = γ (a, (λx)α ), Ɣ(a) and the corresponding pdf is given by f (x; a,α,λ) = αλαa Ɣ(a) xαa 1 exp[ (λx) α ]. The GeGa distribution having the Weibull, gamma, exponential, and Rayleigh as special models is widely used for modeling lifetime data. Ortega, Bolfarine, and Paula (2003) discussed influence diagnostics in GeGa regression models, Nadarajah and Gupta (2007) applied the GeGa distribution to drought data, Huang and Hwang (2006) presented a simple method for estimating the model parameters using its characterization and moment estimation, and Cox, Chu, Schneider, and Muñoz (2007) developed a parametric survival analysis and taxonomy for its hrf. More recently, Ortega, Paula, and Bolfarine (2008) compared three types of residuals based on the deviance component in GeGa regression models under censored observations, and Ortega, Cancho, and Paula (2009) proposed a modification of these regression models to allow the possibility that long-term survivors may might be present in the data. Nadarajah and Haghighi (2011) proposed a new generalization of the exponential distribution as an alternative to the gamma, Weibull, and exponentiated exponential (EE) distributions with cdf and pdf (for x > 0) given by G(x; α, λ) = 1 exp[1 (1 + λx) α ],

6 312 M. Bourguignon et al. and g(x; α, λ) = αλ(1 + λx) α 1 exp[1 (1 + λx) α ], respectively, where α>0 (shape parameter) and λ>0 (scale parameter). If Y follows the Nadarajah Haghighi (NH) distribution, we write Y NH(α, λ). The generalization always has its mode at zero and yet allows for increasing, decreasing, and constant hrfs (Nadarajah & Haghighi, 2011). Lemonte (2013) studied the exponentiated NH (ENH) distribution. In this study, we combined the works of Zografos and Balakrishnan (2009) and Nadarajah and Haghighi (2011) to derive some mathematical properties of a new three-parameter model named the gamma Nadarajah Haghighi (GNH) distribution, which may provide a better fit compared to the GeGa distribution, in certain practical situations. Additionally, we study some of its mathematical properties and discuss the maximum likelihood estimation (MLE) of the model parameters. This distribution is expected to have immediate application for modeling failure times due to fatigue and lifetime data in fields such as engineering, finance, economics, and insurance, among others. The rest of the article is organized as follows. In Section 2, we present the density function and hrf and provide plots of such functions for selected parameter values. In Section 3, we derive useful expansions for its cumulative and density functions. In Section 4, we provide explicit expressions for the moments, quantile and generating functions, mean deviations, Bonferroni and Lorenzcurves, and probability weightedmoments. The Rényi and Shannon entropies are derived in Section 5. The estimation of the model parameters is performed by MLE in Section 6, and two applications to real data are given in Section 7. Concluding remarks are addressed in Section The GNH Distribution Here, we introduce a new generalization of the gamma distribution. If G(x; α, λ) = 1 exp[1 (1 + λx) α ] is the NH cumulative distribution with parameters α and λ, then Equation (2) yields the

7 A New Generalized Gamma Distribution 313 GNH cumulative distribution (for x > 0): F (x; a,α,λ) = γ (a, (1 + λx)α 1), (3) Ɣ(a) where λ>0 is a scale parameter and the other positive parameters α and a are shape parameters. The corresponding density function becomes (for x > 0) f (x; a,α,λ) = αλ Ɣ(a) (1 + λx)α 1 [(1 + λx) α 1] a 1 exp{ [(1 + λx) α 1]}. (4) Hereafter, a random variable X following Equation (4) is denoted by X GNH(a,α,λ). Evidently, the density function Equation (4) does not involve any complicated function. It is a positive point of the current generalization. It can be proved that the GNH density function is log-convex if α < 1anda < 1 and log-concave if α>1anda > 1. Further, lim f (x; a,α,λ) = 0 and lim f (x; a,α,λ) = x x 0, a < 1, 0, a > 1, αλ,a = 1. The study of the GHN distribution is important because it extends some useful distributions previously studied in the literature. In fact, the NH distribution is obtained by taking a = 1. The gamma distribution corresponds to α = 1, whereas the exponential distribution arises by taking α = a = 1. For α = 1, a = η/2, and λ = 2, Equation (4) reduces to the chi-squared distribution, where η denotes the degrees of freedom. Figure 1 displays some plots of the density function Equation (4) for some parameter values. It is evident that the new distribution is much more flexible than the NH distribution. We note five motivations for the proposed distribution: 1. Ability (or the inability) of the GNH distribution to model data that have their modes fixed at zero. 2. If Y is a GeGa random variable with shape parameters α and a, and scale parameter λ, then the density Equation (4) is the

8 314 M. Bourguignon et al. FIGURE 1 Plots for the GNH density for some parameter values; λ = 1. same as that of the random variable Z = Y λ 1 truncated at zero, i.e., the GNH distribution can be interpreted as a truncated GeGa distribution. 3. As we shall see later, the GNH hrf can be constant, decreasing, increasing, upside-down bathtub, or bathtub-shaped. 4. Some distributions commonly used for parametric models in survival analysis are special cases of the GNH distribution, such as the NH, gamma, chi-squared, and exponential distributions. 5. It can be applied in some interesting situations as follows: biological and reliability studies (Cordeiro, Ortega, and Silva, 2011); failure times of fatiguing materials (see Section 7), among others. The GHN hrf is given by τ(x; a,α,λ) = αλ(1 + λx)α 1 [(1 + λx) α 1] a 1 exp{ [(1 + λx) α 1]}, Ɣ(a, (1 + λx) α 1) (5)

9 A New Generalized Gamma Distribution 315 TABLE 1 Parameter intervals with the corresponding shapes of the hazard rate α a (0, 1) (1, ) (0, 1) decreasing upside-down bathtub (1, ) bathtub-shaped increasing FIGURE 2 The GNH hrf for some parameter values; λ = 1. where Ɣ(a, z) = Ɣ(a) γ (a, z) = z t a 1 e t dt. Table 1 gives some parameter intervals for which the hrf is decreasing, increasing, upside-down bathtub, and bathtub-shaped (it is constant if α = a = 1). The parameter λ does not change the shape of the hrf because it is a scale parameter. In the Appendix A, we demonstrate the results of this table, using the idea of Qian (2012). Figure 2 displays plots of the GNH hrf for some parameter values.

10 316 M. Bourguignon et al. The new distribution is easily simulated as follows: if V is a gamma random variable with shape parameter a > 0, then X = λ 1 { (1 + V ) 1/α 1 } has the GNH(a,α,λ) distribution. 3. Useful Expansions Expansions for Equations (3) and (4) can be derived using the concept of exponentiated distributions. Consider the ENH distribution with power parameter a > 0 defined by Y ENH(a,α,λ) with cdf and pdf given by and H a (x) = { 1 exp {1 (1 + λx) α } } a h a (x) = a αλ (1 + λx)α 1 exp {1 (1 + λx) α } [ 1 exp {1 (1 + λx)α } ] 1 a, respectively. The properties of several exponentiated distributions have been studied by some authors; for details, see Mudholkar and Srivastava (1993) and Mudholkar, Srivastava, and Friemer (1995) for exponentiated Weibull, Gupta, Gupta, and Gupta (1998) for exponentiated Pareto, Gupta and Kundu (2001) for exponentiated exponential (EE), and Nadarajah and Gupta (2007) for exponentiated gamma (EG) distributions. More recently, Cordeiro, Ortega, and Silva (2011) investigated these properties for the exponentiated generalized gamma (EGG) distribution. Based on an expansion due to Nadarajah, Cordeiro, and Ortega (2013), we can write { } a 1 ( ) k + 1 a log (1 G(x)) = (a 1) k=0 k k j=0 ( 1) j+k( k j) p j,k G(x) a+k 1, (a 1 j)

11 A New Generalized Gamma Distribution 317 where a > 0 is a real parameter and the constants p j,k (for j > 0) are determined recursively by p j,k = k 1 k m=1 ( 1) m [m( j + 1) k] (m + 1) p j,k m, for k = 1, 2,...and p j,0 = 1. Let b k = ( k+1 a ) k (a + k)ɣ(a 1) k ( 1) ( j+k k j) p j,k. (a 1 j) j=0 Then, Equation (4) can be expressed as where f (x) = b k h a+k (x) = g(x) (a + k) b k G(x) a+k 1, (6) k=0 k=0 h a+k (x) = αλ(a + k)(1+ λx) α 1 exp {1 (1 + λx) α } [ 1 exp {1 (1 + λx) α } ] a+k 1 denotes the ENH(a +k,α,λ) density function. Equation (6) is the main result of this section. It reveals that the GNH density function is a mixture of ENH densities. So, several GNH properties can be obtained by knowing those properties of the ENH distribution. The cdf corresponding to Equation (6) becomes F (x) = b k H a+k (x) = k=0 { [ b k 1 exp 1 (1 + λx) α ]} a+k, k=0 where H a+k (x) ={1 exp[1 (1 + λx) α ]} a+k denotes the ENH cdf with parameters a + k, α, andλ. (7)

12 318 M. Bourguignon et al. For z (0, 1) and any real noninteger α, wehave z α = s r (α) z r, (8) r =0 where s r (α) = ( α ( 1) r +l l l=r )( ) l. r Combining Equations (6) and (8), we obtain f (x) = g(x) d r G(x) r, (9) r =0 where d r = (r + 1) 1 k=0 b k (a + k) s r (a + k 1). 4. Mathematical Properties 4.1. Quantile Function The GNH quantile function (qf), say Q GNH (u) = F 1 (u), can be expressed in terms of the NH qf (Q NH ( )). Inverting Equation (3), it follows the qf of X (for 0 < u < 1) as Q GNH (u) = Q NH { 1 exp[ Q 1 (a, 1 u)] }, (10) where Q 1 (a, u) is the inverse function of Q(a, z) = 1 γ (a, z)/ɣ(a). Quantities of interest can be obtained from Equation (10) by substituting appropriate values for u. Further, after some algebra, the NH qf can expressed as (Nadarajah and Haghighi, 2011) { Q NH (u) = λ 1 [ ] } 1/α 1 log(1 u) 1. (11) By replacing (11) in Equation (10), we obtain Q GNH (u)

13 A New Generalized Gamma Distribution 319 = λ 1 { [ 1 log[1 (1 exp[ Q 1 (a, 1 u)])] ] 1/α 1 }. (12) The inverse function Q 1 (a, u) follows from the Wolfram website 1 as z = Q 1 (a, 1 u) = a i u i/a, (13) i=0 where a 0 = 0, a 1 = Ɣ(a + 1) 1/a, a 2 = Ɣ(a+1)2/a, a (a+1) 3 = (3a+5)Ɣ(a+1)3/a [2(a+1) 2 (a+2)], etc. We use throughout the article an equation of Gradshteyn and Ryzhik (2007, Section 0.314) for a power series raised to a positive integer j: ( ) j a i x i = i=0 c j,i x i, (14) where the coefficients c j,i (for i = 1, 2,...) are easily obtained from the recurrence equation c j,i = (i a 0 ) 1 i=0 i [m( j + 1) i] a m c j,i m (15) m=1 and c j,0 = a j 0. The coefficient c j,i can be determined from c j,0,...,c j,i 1 and then from the quantities a 0,...,a i listed above. Then, after some algebra using Equations (11), (13), and (14), we obtain Q GNH (u) = q i u i/a, (16) i=0 1 introductions/gammas/showall.html

14 320 M. Bourguignon et al. where q 0 = (q 0 1)λ 1, q i = q i λ 1 (i 1), and q i and some other quantities of interest and algebraic details are reported in Appendix B. Equations (12) (16) are the main results of this section. Let Q(u) = Q GNH (u). The Galton s skewness is given by G = Q ( ( 3 4) + Q 1 ( 4) 2Q 1 ) 2 Q ( ( 3 4) Q 1 ), 4 whereas the kurtosis (Moors, 1988) is given by Q ( ( ( ) 3 8) Q 1 8) + Q 7 Q ( ) M = Q ( ( 6 8) Q 2 ) Moments The ordinary and incomplete moments of X can be immediately obtained from those moments of Y following the ENH(a,α,λ) distribution. Thus, we can write from Equation (6) μ n = E (X n ) = k=0 b k E (Y n k ). Using the moments of Y k ENH(a+k) (Lemonte, 2013), we have μ n = E (X n ) = λ n k=0 b k R n (α, a + k), (17) where R n (α, a + k) = 1 0 {[1 log(1 u)]1/α 1} n u a+k 1 du is an integral to be computed numerically. Alternatively, we can write μ n in closed-form, based on the ) (Lemonte, 2013), as quantity E (Y n k μ n = λ n (a + k) j,k=0 i=0 ( 1) n+ j i (a + k) e j+1 b k ( j + 1) i/α+1 ( ) a + k 1 j

15 A New Generalized Gamma Distribution 321 ( ) ( ) n i Ɣ i α + 1, j + 1, (18) where Ɣ(a, x) = x z a 1 e z dz. Let T n (y) denote the nth incomplete moment of X, say T n (y) = y 0 xn f (x) dx.fromequation(6),wecanwrite T n (y) = k=0 b k Tn NH (y, a + k), where T NH (y, a + k) denotes the incomplete moment of Y k. Lemonte (2013) provides two expressions for T n. The first is Tn NH (y, a + k) = (a + k)λ n 1 exp 1 (1+λy)α { [1 log(1 u)] 1/α 1 } n u a+k 1 du, which involves numerical integration. The second is given in closed-form as Tn NH (y, a + k) = λ n ( ) ( a + k 1 n j i j,k=0 i=0 ) Ɣ ( 1) n+ j i (a + k) e j+1 b k ( j + 1) i/α+1 ( i + 1, ( j + 1)(1 + λy)α α Based on the incomplete first moment, we obtain the mean deviations of X about the mean μ 1 and about the median M as ).

16 322 M. Bourguignon et al. δ 1 = 2 [ μ 1 F (μ 1 ) T 1(μ 1 )] and δ 2 = μ 1 2T 1(M), (19) respectively. For a given probability π, the Bonferroni and Lorenz curves of X are given by B(π) = T 1 (q )/(πμ 1 )andl(π) = T 1(q )/μ 1,respectively, where q = Q GNH (π) is determined from Equation (12). Next, we obtain the probability weighted moments (PWMs) of X. The primary use of these moments is to estimate the parameters of a distribution whose inverse can not be expressed explicitly. The (s, p )th PWM of X is defined as ξ s,p = E [X s F (X ) p ] = 0 x s F (x) p f (x) dx. Using Equations (6), (7), and (14), we have where ξ s,p = ω r,s,m,i = r,m,i=0 j,k,l,n=0 ( ( m ) ς p,r ω r,s,m,i Ɣ α + 1, 1, (20) i+ j+n+m+s 1 α ( 1) d l λ s k!(αi + 1) ( )( )( )( ) ) k l + r α 1 s + n, i j n m d l is defined in Section 3, v m = k=0 b k s m (a + k), s m (a + k) is given in Section 3, ς p,r = (rv 0 ) 1 rm=1 [m(p + 1) r ] v m ς p,r m (for r 1) and ς p,0 = v p 0. Equations (17), (18), and (20) are the main results of this section Generating Function A first representation for the moment generating function (mgf) of X follows from the qf M(t) = 1 0 exp [ tq GNH (u) ] du.

17 A New Generalized Gamma Distribution 323 Expanding the exponential function, using Equation (16) and after some algebra, we obtain M(t) = i,k=0 d k,i ( t k i a + 1) k!, (21) where d k,i = (i q 0 ) 1 im=1 [m(k+1) i] q m d k,i m for k 1, d k,0 = q k 0, d 0,i = 1, q i = q i λ 1 (for i 1), q 0 = (q 0 1)λ 1,andq i and the other quantities are defined in Appendix B. A second representation for M(t) is determined from the ENH generating function. We can write M(t) = k=0 b k M k (t), where b k isgiveninsection3andm k (t) is the mgf of Y k ENH(a + k) given by where η i = ζ r = k=0 ( 1) i λ k k! M k (t) = i,r =0 η i g i,r t k r/β + 1, (22) ( ) k r, g i i,r = (r ζ 0 ) 1 [n(i + 1) r ] ζ n g i,r n, r f m d m,r, d m,r = (ra 0 ) 1 [v(m + 1) r ] a v d m,r v, m=0 n=1 v=0 and f m = ( ) ( 1) j m j (α 1 ) m j /j!. j=m Here, (α 1 ) j = ( α 1) ( α 1 1 )... ( α 1 j + 1 ) is the descending factorial. Other quantities and details about Equation (22) are provided in Appendix C. Equations (21) and (22) are the main results of this section.

18 324 M. Bourguignon et al Order Statistics Suppose X 1,...,X n is a random sample from the standard GNH distribution and let X 1:n < < X i:n denote the corresponding order statistics. Using Equations (6) and (7), the pdf of X i:n can be expressed as f i:n (z) = K = K n i ( ( 1) j n i j j=0 n i ( ( 1) j n i j, j=0 ) i+ j 1 f (z) F (z) ) [ g(z) [ ] i+ j 1 b k G(z) a+k, k=0 ] (a + r ) b r G(z) a+r 1 where K = n!/[(i 1)!(n i)!] and the coefficients b k saregiven in Section 3. Based on Equations (14) and (15), we obtain [ ] i+ j 1 b k G(z) a+k = k=0 r =0 η i+ j 1,k G(z) a+k, k=0 i+ j 1 where η i+ j 1,0 = κ0 and η i+ j 1,k = (kb 0 ) 1 k k] b m η i+ j 1,k m. Thus, the pdf of X i:n reduces to m=1 [m(i + j) f i:n (z) = g(z) m j,k,r G(z) 2a+k+r 1, (23) k,r =0 where m j,k,r = n i j=0 ( 1) j n! b r η i+ j 1,k (i 1)! (n i j)! j!.

19 A New Generalized Gamma Distribution 325 Equation (23) can be expressed as f i:n (z) = n i j=0 k,r =0 f j,k,r h 2a+k+r (x), (24) where f j,k,r = m j,k,r 2a + k + r. Equation (24) is the main result of this section. It reveals that the pdf of the GNH order statistics is a double linear combination of ENH densities with parameters 2a + k + r, α, andλ. So, several mathematical properties associated with the GNH order statistics, such as ordinary and incomplete moments, mgf, and mean deviations, can be obtained from those of the ENH distribution. 5. Entropies Entropy can be understood as a measure of variation or uncertainty of a random variable X. The Rényi and Shannon entropies are the two more common measures (Shannon, 1948; Rényi, 1961). The Rényi entropy of a random variable with pdf f (x; θ)is defined by H R,c (θ) = 1 1 c log { E [ f c 1 (X ; θ) ]} = 1 1 c log ( 0 ) f c (x; θ) dx, (25) for c > 0andc 1. The Shannon entropy of a random variable X is defined by H S (θ) = E { log [ f (X ; θ)] }. It is the special case of the Rényi entropy when c 1. Direct calculation gives the Shannon entropy of the random variable X as H S (θ) = {log(α) + log(λ) log[ɣ(a)] } (α 1) E log(1 + λ X ) 1 + E (1 + λ X ) α

20 326 M. Bourguignon et al. (a 1) E log[(1 + λx ) α 1]. After some algebraic manipulations, we obtain E [log(1 + λ X )] = 1 [ ψ(a) α E [log(1 + Z)] α ; a ], α where Z Ɣ(a, 1), E [(1 + λ X ) α ] = a + 1, E { log[(1 + λx ) α 1] }= ψ(a) andψ( ) is the digamma function. Next, we obtain an expansion for the GNH Rényi entropy. From a result by Lemonte and Cordeiro (2011) where x λ = f j = f j (λ) = k= j j=0 and using Equation (9), we obtain f j x j, ( ) k ( 1) k j (λ)k j k!, ( c f c (x) = g c (x) d r G (x)) r = g c (x) r =0 By applying Equation (14), we have j=0 ( j f j d r G (x)) r. r =0 f c (x) = g c (x) j,r =0 f j c j,r }{{} w j,r G r (x) = j,r =0 w j,r g c (x) G r (x). Finally, using this result in Equation (25), the Rényi entropy reduces to H R,c (θ) = 1 1 c log j,r =0 w j,r g c (x) G r (x)dx 0

21 A New Generalized Gamma Distribution 327 = 1 1 c log j,r =0 [ w j,r E Y g c 1 (Y ) G r (Y ) ], where E Y denotes the expected value based on the ENH random variable Y defined at the beginning of Section Maximum Likelihood Estimation Here, we propose a method for obtaining the maximum likelihood estimates (MLEs) of the GNH model parameters. Let x 1,...,x n be a random sample of size n from X GHN(a,α,λ). The log-likelihood function for the vector of parameters θ = (a,α,λ) can be expressed as l(θ) = n { log(α) + log(λ) log[ɣ(a)] } +(α 1) log(1 + λ x i ) + n + (a 1) log[(1 + λx i ) α 1]. The components of the score vector are given by U θ = ( U a, U α, U λ ) = (1 + λ x i ) α ( ) dl(θ) da, dl(θ) dα, dl(θ), dλ U a = n ψ (0) (a) + log[ (1 + λ x i ) α 1], U α = n α + + (a 1) log(1 + λ x i ) (1 + λ x i ) α log(1 + λ x i ) (1 + λ x i ) α log(1 + λ x i ) (1 + λ x i ) α 1

22 328 M. Bourguignon et al. and U λ = n λ + (α 1) + α (a 1) ( ) x i α 1 + λ x i [ xi (1 + λ x i ) α 1 (1 + λ x i ) α 1 ], x i (1 + λ x i ) α 1 where ψ( ) is the digamma function. Setting these equations to zero, U (θ) = 0, and solving them simultaneously yields the MLE θ of θ. These equations cannot be solved analytically and statistical software computes them numerically using iterative techniques such as the Newton Raphson algorithm. For interval estimation of the model parameters, we can adopt the 3 3 total observed information matrix J (θ) given by whose elements are J (θ) = J aa = n ψ (1) (a), J aα = J aλ = α J aa J aα J aλ J aα J αα J αλ, J aλ J αλ J λλ x i (1 + λ x i ) α 1 (1 + λ x i ) α 1, (1 + λ x i ) α log(1 + λ x i ), (1 + λ x i ) α 1 J αα = n α 2 { (1 + λ x i ) α log 2 (1 + λ x i ) + (a 1) (1 + λ x i ) α log 2 (1 + λ x i ) [(1 + λ x i ) α 1] (1 + λ x i ) 2α log 2 (1 + λ x i ) [(1 + λ x i ) α 1] 2 ( ) x i J αλ = α x i (1 + λ x i ) α 1 log(1 + λ x i ) x i (1 + λ x i ) α λ x i α x i (1 + λ x i ) α 1 log(1 + λ x i ) + x i (1 + λ x i ) α 1 + (a 1) (1 + λ x i ) α 1 α x i (1 + λ x i ) 2α 1 log(1+λx i ) [(1 + λ x i ) α 1] 2 },

23 A New Generalized Gamma Distribution 329 and J λλ = n ( ) λ 2 (α 1) x 2 i α (α 1) xi 2 (1 + λ x i ) α λ x i { (α 1) xi (1 + λ x i ) α 2 [(1 + λ x i ) α 1] α x i (1 + λ x i ) 2α 2 } + α (a 1) x i [(1 + λ x i ) α 1] 2. The multivariate normal N 3 (0, J ( θ) 1 ) distribution can be used to construct approximate confidence intervals for the parameters under standard asymptotic theory (Cox & Hinkley, 1974). 7. Applications to Real Data We conduct two applications of the GNH distribution to real data for illustrative purposes. We estimate the unknown parameters of the fitted distributions by the MLE method, as discussed in Section 6. All computations are performed using the Ox matrix programming language (Doornik, 2006). We compare the GNH model with other three-parameter models such as the GeGa (Stacy, 1962), ENH (Lemonte, 2013), and gamma exponentiated exponential (GEE; Ristić and Balakrishnan 2012) distributions. Their densities (for x > 0) are given by f GeGa (x) = αλαa Ɣ(a) xαa 1 exp[ (λx) α ], f ENH (x) = a αλ (1 + λx)α 1 exp {1 (1 + λx) α } [ 1 exp {1 (1 + λx)α } ] 1 a, f GEE (x) = λαa Ɣ(a) e λx (1 e λx ) α 1 [ log(1 e λx )] a 1, respectively, where λ > 0,α > 0, and a > 0 are unknown parameters Example 1: Stress Carbon Fiber Data The first example is a dataset from Nichols and Padgett (2006) consisting of 100 observations on breaking stress of carbon fibers

24 330 M. Bourguignon et al. FIGURE 3 TTT-plot for the stress carbon fibers data. (in Gba). Table 2 gives some descriptive measures for these data, which include central tendency statistics, standard deviation (SD), skewness (CS), and kurtosis (CK). One important device that can help in selecting a particular model is the total time on test (TTT) plot (Aarset, 1987). Hence, Figure 3 provides a TTT plot for the fibers data, thus indicating an increasing hrf. Table 3 provides the MLEs and corresponding standard errors (SEs), the values of the Cramér-von Mises (W ) and Anderson-Darling (A ) statistics described by Chen and Balakrishnan (1995), and the Kolmogorov Smirnov (KS) statistic and p-values for the stress carbon fiber data. In general, the smaller the values of these statistics are the better the fit to the data. TABLE 2 Descriptive statistics n Min. Median Mean Max. Mode SD CS CK

25 A New Generalized Gamma Distribution 331 TABLE 3 Parameter estimates, goodness-of-fit tests for the stress carbon fibers data Estimate (SE) Goodness-of-Fit Test Statistic (p-value) α λ â KS W A GNH (2.2944) (0.2677) (0.7864) (0.7998) (0.2720) (0.3317) GeGa (0.1335) (0.7722) (0.7690) (0.5548) (0.0286) (0.0550) ENH (0.0632) (0.0365) (0.9971) (0.4514) (0.0801) (0.1397) GEE (1.1726) (0.0243) (1.0632) (<0.001) (0.0165) (0.0318) We note that α >1andâ > 1 for the GNH and ENH models, which implies that the hrf for these distributions are increasing in accordance with Figure 3. From the figures of Table 3 and according to these formal tests, the GNH model fits the current data better than other models, i.e., these values indicate that the null hypothesis is strongly not rejected for the GNH distribution. These results illustrate the potentiality of the GNH distribution and the importance of the additional shape parameter. In Figure 4(a), we present the quantile against quantile (QQ) plot with envelope, which allows us to compare the empirical distribution of the data with the GNH distribution. This graphical goodness-of-fit method supports the result indicated by the KS, W,andA statistics. The plot of the estimated GNH density of the model fitted to these data is displayed in Figure 4(b). TABLE 4 Descriptive statistics n Min. Median Mean Max. Mode SD CS CK

26 332 M. Bourguignon et al. FIGURE 4 (a) QQ plot with envelope for the GNH distribution and (b) fitted to the GNH distribution for stress carbon fibers data Example 2: Number of Successive Failures Data For the second example, we consider the dataset consisting of the number of successive failures for the air conditioning system of each member in a fleet of 13 Boeing 720 jet airplanes (Proschan, 1963). Table 4 gives some descriptive statistics. Figure 5 displays an upside-down bathtub hrf for the number of successive failures. Therefore, these plots indicate the appropriateness of the GNH distribution to fit these data, since the new model can present both forms of the hrf. Table 5 gives the MLEs and correponding SEs, the values of the W,A, and KS statistics, and the p-values for number of successive failures data. In this example, we have α <1andâ > 1 for the GNH and ENH models, thus implying that the hrf is an upside-down bathtub shape, in accordance with Figure 5. In the same way of the first application, we can note from Table 5 that for these formal tests, the GNH model fits the current data better than the other models. Figure 6(a) displays the QQ plot with envelope. The plot of the estimated GNH density to these data is given in Figure 6(b).

27 A New Generalized Gamma Distribution 333 FIGURE 5 TTT-plot for the number of sucessive failures air conditioning system data. 8. Concluding Remarks In this article, we propose a new generalized gamma model called the gamma-nadarajah Haghighi (GNH) distribution. We demon- TABLE 5 Parameter estimates, goodness-of-fit tests for the number of successive failures data Estimate (SE) Goodness-of-Fit Test Statistic (p-value) α λ â KS W A GNH (0.0955) (0.0795) (0.4855) (0.8554) (0.8316) (0.7735) GeGa (5.7261) (0.1747) (4.5431) (0.8543) (0.5714) (0.5231) ENH (0.0080) (0.0262) (0.1291) (0.8175) (0.3039) (0.2744) GEE (0.3448) (10 6 ) (1.7474) (<0.001) (0.7783) (0.7204)

28 334 M. Bourguignon et al. FIGURE 6 (a) QQ plot with envelope for the GNH distribution and (b) fitted to the GNH distribution for number of successive failures for the air conditioning system. strate that the hazard rate function of the GNH distribution can be increasing, decreasing, bathtub-shaped, and upside-down bathtub shaped. A detailed study on some mathematical properties of the new distribution is presented. The model parameters are estimated by MLE and the observed information matrix is determined. The flexibility of the new model is demonstrated by means of two real datasets. In fact, the GNH distribution model fits the two datasets well. We hope that the proposed model might attract wider applications in statistics. Acknowledgments The authors would like to thank the editors and the reviewers who helped to substantially improve this article. Funding We gratefully acknowledge the grants from CAPES and CNPq. References Aarset, M. V. (1987). How to identify bathtub hazard rate. IEEE Transactions on Reliability, 36, Chen, G., & Balakrishnan, N. (1995). A general purpose approximate goodnessof-fit test. Journal of Quality Technology, 27,

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30 336 M. Bourguignon et al. Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. M. (2013). The gamma-g family of distributions. Mathematical properties and applications. To appear in Communications Statistics Theory and Methods. Nichols, M. D., & Padgett, W. J. (2006). A bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International, 22, Ortega, E. M. M., Bolfarine, H., & Paula, G. A. (2003). Influence diagnostics in generalized log-gamma regression models. Computational Statistics and Data Analysis, 42, Ortega, E. M. M., Paula, G. A., & Bolfarine, H. (2008). Deviance residuals in generalized log-gamma regression models with censored observations. Journal of Statistical Computation and Simulation, 78, Ortega, E. M. M., Cancho, V. G., & Paula, G. A. (2009). Generalized log-gamma regression models with cure fraction. Lifetime Data Analysis, 15, Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics, 5, Qian, L. (2012). The Fisher information matrix for a three-parameter exponentiated Weibull distribution under type II censoring. Statistical Methodology, 9, Rényi, A. (1961). On measures of information and entropy In Proceedings of the 4th Berkeley Symposium on Mathematics. InStatistics and Probability 1960, Berkeley, CA: University of California Press. Ristić, M. M., & Balakrishnan, N. (2012). The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82, Shannon, C. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics, 33, Zografos, K., & Balakrishnan, N. (2009). On families of beta and generalized gamma-generated distributions and associated inference. Statistical Methodology, 6, Appendix A: Shapes of the Hazard Rate Function We demonstrate for which parameter intervals the hrf of the GHN distribution is decreasing, increasing, bathtub-shaped, and upside-down bathtub-shaped. For doing this, we follow the idea of Qian (2012) and we will show graphically that the regions {(a,α):a < 1andα<1}, {(a,α):a > 1andα>1}, {(a,α):a < 1andα>1}, and{(a,α):a > 1andα<1} correspond to the decreasing, increasing, bathtub-shaped, and upsidedown bathtub-shaped hrfs, respectively. Let z = (1 + λx) α, which implies z > 1forx > 0. We have x = (z 1/α 1)/λ. Rewriting the hrf as a function in z and denoting

31 it by r (z), we have A New Generalized Gamma Distribution 337 r (z) = τ((z 1/α 1)/λ) = αλz(α 1)/α exp[ (z 1)](z 1) a 1. Ɣ(a, z 1) Thus, taking the logarithm on both sides, we obtain log[r (z)] = log(αλ) + α 1 log(z) + 1 z + (a 1) log(z 1) α log[ɣ(a, z 1)]. By differentiating, we have which implies r (z) r (z) = α a 1 α z z 1 Ɣ (a, z 1) Ɣ(a, z 1), [ α 1 r (z) = r (z) 1 + a 1 ] α z z 1 Ɣ (a, z 1), z > 1. Ɣ(a, z 1) However, Ɣ (a, x) = x a 1 /e x. Thus, [ α 1 r (z) = r (z) 1 + a 1 ] α z z 1 + (z 1)a 1 e z 1 Ɣ(a, z 1) = r (z) s(z), z > 1. Hence, the sign of r (z) is the same as the sign of s(z) since r (z) > 0. Figure 7 displays the plots of s(z) for the parameters in the four regions. From the left panel, one observes that s(z)takes purely positive and purely negative values in region I and region II, respectively, which implies that the hrf is monotone-increasing in region I and monotone-decreasing in region II. From the right panel, one observes that s(z) takes positive values first, then drops to negative values in region III, which indicates the upside-down bathtub-shaped hrf. In region IV, it is shown that s(z) takesnegative values first, then positive values, indicating a bathtub shape for the hrf.

32 338 M. Bourguignon et al. FIGURE 7 The plots of s(z) for the four defined regions. Region I ={(a,α): a > 1andα>1}, region II ={(a,α):a < 1andα<1}, region III ={(a,α): a > 1andα<1} and region IV ={(a,α):a < 1andα>1}, respectively. Appendix B: Quantile Function We derive a power series for Q GNH (u) in the following way. First, we use a known power series for Q 1 (a, 1 u). Second, we obtain a power series for the argument 1 exp[ Q 1 (a, 1 u)]. Third, we consider the NH qf given in Nadarajah and Haghighi (2011). We introduce the following quantities defined by Cordeiro and Lemonte (2011). Let Q 1 (a, z) be the inverse function of Q(a, z) = 1 γ (a, z) Ɣ(a) = Ɣ(a, z) Ɣ(a) = u. The inverse quantile function Q 1 (a, 1 u) is determined from the Wolfram website 2 as Q 1 (a, 1 u) = w + w 2 3 (3a + 5)w + a + 1 2(a + 1) 2 (a + 2) 4 [a(8a + 33) + 31]w + 3(a + 1) 3 (a + 2)(a + 3) { } a(a[a(125a ) ] ) w O(w 6 ), 24(a + 1) 4 (a + 2) 2 (a + 3)(a + 4) /03/

33 A New Generalized Gamma Distribution 339 where w = [uɣ(a + 1)] 1/a. We can write the last equation as in Equation (12), where the δ i s are given by δ i = b i Ɣ(a+1) i/a. Here, b 0 = 0, b 1 = 1 and any coefficient b i+1 (for i 1) can be obtained from the cubic recurrence equation b i+1 = 1 i(a + i) { i r =1 i s+1 s=1 b r b s b i r s+2 s (i r s + 2) } i b r b i r +2 r [r a (1 a)(i + 2 r )]. r =2 The first coefficients are b 2 = 1/(a + 1), b 3 = (3a + 5)/[2(a + 1) 2 (a + 2)],... Now, we present some algebraic details for the GNH qf, say Q GNH (u). The cdf of X isgivenbyequation(3).by inverting F (x) = u, we obtain Equation (12). The NH qf is given by Equation (11). So, using Equation (13), we have 1 + Q 1 (a, 1 u) = r i u i/a, where r 0 = 1andr i = a i (i 1). Now, replacing the last result in Equation (12), we obtain i=0 ( ) 1/α Q GNH (u) = λ 1 r i u i/a 1. i=0 By expanding ( i=0 r i u i/a) 1/α and using Equations (14) and (15), we have ( ) 1/α r i u i/a = i=0 j=0 ( ) j f i (α 1 ) r i u i/a = i=0 i, j=0 f i ɛ j,i u i/a,

34 340 M. Bourguignon et al. where f j (α 1 ) = ( ) k )(α k= j ( 1)k j j, 1 ) k /k!, ɛ j,i = (ir 0 ) 1 i m=1 [m( j + 1) i] r m ɛ j,i m (for i 1) and ɛ j,i = r j 0. Using the last result, we obtain Q GNH (u) = q i u i/a, i=0 where q 0 = (q 0 1)λ 1, q i = q i λ 1 (i 1) and q i = f i (α 1 ) j=0 ɛ j,i. 0 Appendix C: Generating function Here, we present the algebraic details of the second representation for M(t) based on the quantile power series of X.FromEquation (16), we can write 1 M(t) = exp [ tq GNH (u) ] [ ( 1 )] du = exp t q i u i/a du, q 0 = (q 0 1)λ 1, q i = q i λ 1 (i 1), q i = f i (α 1 ) j=0 ɛ j,i, f j = ( ) k )(β)k k= j ( 1)k j /k!, ɛ j, j,i = (ir 0 ) 1 i m=1 [m( j + 1) i] r m ɛ j,i m and β = 1/α. Expanding the exponential function, we have M(t) = 1 0 k=0 0 t k ( i=0 q i u i/a) k k! du = i,k=0 i=0 d k,i ( t k i a + 1) k!, where d k,i = (i q 0 ) 1 im=1 [m(k + 1) i] q m d k,i m (for i 1), d k,0 = q i 0, d 0,0 = 1. Next, we obtain the ENH generating function using the ENH qf as follows: M ENH (t) = 1 0 exp [ tq ENH (u) ] du

35 A New Generalized Gamma Distribution 341 = 1 0 exp[tλ 1 [1 log(1 u 1/β )] 1/α 1] du. Expanding the exponential function, we obtain M ENH (t) = k=0 t k λ k k! 1 0 { [1 log(1 u 1/β )] 1/α 1 } k du, (26) where 1 log(1 u 1/β ) can be expressed as 1 log(1 u 1/β ) = v r u r/β, with v r = k=0 b k s r (a + k). Using Equations (14) and (15), we have ( ) 1/α ( ) m v r u r/β = f m v r u r/β r =0 = m=0 m=0 f m r =0 r =0 r =0 ν m,r u r/β = m,r =0 f m ν m,r u r/β, (27) where ν m,r = (rv 0 ) 1 r n=1 [n(m + 1) r ] v nν m,r n. Further, using Equation (27) and the binomial expansion, we can write { [1 log(1 u 1/β )] 1/α 1 } k = ( = = m,r =0 f m ν m,r u r/β 1 ) k ( ) ( ) ( 1) i k i γ j, r u r/β i=0 i,r =0 r =0 ( ) ( 1) i k g j, i,r u r/β, (28)

36 342 M. Bourguignon et al. where γ r = m=0 f m ν m,r and g i,r = (r γ 0 ) 1 r s=1 [s(i + 1) r ] γ s g i,r s (for r 1) and g i,0 = γ0 i. Thus, replacing (28) in Equation (26), we obtain M(t) = = k=0 i,r =0 t k λ k k! ( ) ( 1) i k 1 g i, i,r u r/β du i,r =0 η i g i,r t k r/β