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A New Family of Generalized Distributions

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Journal of Statistical Computation & Simulation Vol. 00, No. 00, August 2009, 1{17 A new family of generalized distributions Gauss M. Cordeiroa and M¶ariode Castrob¤ aUniversidade Federal Rural de Pernambuco, Departamento de Estat¶³stica e Inform¶atica, 52171-900, Recife-PE, Brasil; bUniversidade de S~aoPaulo, Instituto de Ci^encias Matem¶aticas e de Computa»c~ao,Caixa Postal 668, 13560-970, S~aoCarlos-SP, Brasil (Received December 7, 2009) Kumaraswamy [1] introduced a distribution for double bounded random processes with hy- drological applications. For the ¯rst time, based on this distribution, we describe a new family of generalized distributions (denoted with the pre¯x \Kw") to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distri- butions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments of the parent distribution. We also obtain the ordinary moments of order statistics as functions of probability weighted moments of the baseline distribution. We use the method of maximum likelihood to ¯t the distributions in the new class and illustrate the potentiality of the new model with an application to real data. Keywords: gamma distribution; Kumaraswamy distribution; moments; normal distribution; order statistics; Weibull distribution AMS Subject Classi¯cation: 62E10; 62F03; 62F05; 62F10 1. Introduction Beta distributions are very versatile and a variety of uncertainties can be usefully modeled by them. Many of the ¯nite range distributions encountered in practice can be easily transformed into the standard beta distribution. In econometrics, many times the data are modeled by ¯nite range distributions. Generalized beta distributions have been widely studied in statistics and numerous authors have developed various classes of these distributions. Eugene et al. [2] proposed a general class of distributions for a random variable de¯ned from the logit of the beta random variable by employing two parameters whose role is to introduce skewness and to vary tail weight. Following the work of Eugene et al. [2], who de¯ned the beta normal distribution, Nadarajah and Kotz [3] introduced the beta Gumbel distribution, Nadarajah and Gupta [4] proposed the beta Fr¶echet distribution and Nadarajah and Kotz [5] worked with the beta exponential distribution. However, all these works lead to some mathematical di±culties because the beta distribution is not fairly tractable and, in particular, its cumulative distribution function (cdf) involves the incomplete beta function ratio. The paper by Kumaraswamy [1] proposed a new probability distribution for dou- ble bounded random processes with hydrological applications. The Kumaraswamy's distribution appears to have received considerable interest in hydrology and related areas, see [6{9]. ¤Corresponding author. Email: [email protected] ISSN: 0094-9655 print/ISSN 1563-5163 online © 2009 Taylor & Francis DOI: 10.1080/0094965YYxxxxxxxx http://www.informaworld.com 2 G. M. Cordeiro and M. de Castro In reliability and life testing experiments, many times the data are modeled by ¯nite range distributions. See, for example, [10]. We start with the Kumaraswamy's distribution (called from now on the Kw distribution) on the interval (0; 1), having the probability density function (pdf) and the cdf with two shape parameters a > 0 and b > 0 de¯ned by f(x) = a b xa¡1(1 ¡ xa)b¡1 and F (x) = 1 ¡ (1 ¡ xa)b: (1) The density function in (1) has many of the same properties as the beta distribution but has some advantages in terms of tractability. The Kw distribution does not seem to be very familiar to statisticians and has not been investigated systematically in much detail before, nor has its relative in- terchangeability with the beta distribution has been widely appreciated. However, in a very recent paper, Jones [11] explored the background and genesis of the Kw distribution and, more importantly, made clear some similarities and di®erences between the beta and Kw distributions. For example, the Kw densities are also unimodal, uniantimodal, increasing, decreasing or constant depending in the same way as the beta distribution on the values of its parameters. He highlighted sev- eral advantages of the Kw distribution over the beta distribution: the normalizing constant is very simple; simple explicit formulae for the distribution and quantile functions which do not involve any special functions; a simple formula for random variate generation; explicit formulae for L-moments and simpler formulae for mo- ments of order statistics. Further, according to Jones [11], the beta distribution has the following advantages over the Kw distribution: simpler formulae for moments and moment generating function; a one-parameter sub-family of symmetric distri- butions; simpler moment estimation and more ways of generating the distribution via physical processes. Consider starting from a parent continuous distribution function G(x). A natural way of generating families of distributions on some other support from a simple starting parent distribution with pdf g(x) = dG(x)=dx is to apply the quantile function to a family of distributions on the interval (0; 1). We now combine the works of Eugene et al. [2] and Jones [11] (see also [12]) to construct a new class of Kw generalized (Kw-G) distributions. From an arbitrary parent cdf G(x), the cdf F (x) of the Kw-G distribution is de¯ned by F (x) = 1 ¡ f1 ¡ G(x)agb; (2) where a > 0 and b > 0 are two additional parameters whose role is to introduce skewness and to vary tail weights. Because of its tractable distribution function (2), the Kw-G distribution can be used quite e®ectively even if the data are censored. Correspondingly, the density function of this family of distributions has a very simple form f(x) = a b g(x) G(x)a¡1f1 ¡ G(x)agb¡1; (3) whereas the density of the beta-G distribution is given by 1 f(x) = g(x)G(x)a¡1 f1 ¡ G(x)gb¡1 ; (4) B(a; b) where B(¢; ¢) denotes the beta function. The basic di®erence (except for a scale multiplier) between (3) and (4) is the power of G(x) inside the braces. Clearly, for b = 1 both densities are identical. Journal of Statistical Computation & Simulation 3 The new density (3) has an advantage over the class of generalized beta distri- butions due to Eugene et al. [2], since it does not involve any special function. For each continuous name distribution (here name denotes the name of the parent dis- tribution), we can associate the Kw-name distribution with two extra parameters a and b from the cdf G(x) and pdf g(x) of the name distribution whose density function is de¯ned by formula (3). Special Kw generalized distributions can be generated as follows: the Kw-normal (KwN) distribution is obtained by taking G(x) in formula (3) to be the distribution function of the normal distribution. Analogously, the Kw-Weibull (KwW ), Kw- gamma (KwGa) and Kw-Gumbel (KwGu) distributions are obtained by taking G(x) to be the cdf of the Weibull, gamma and Gumbel distributions, respectively, among several others. Hence, each new Kw-G distribution can be obtained from a speci¯ed G distribution. The Kw distribution is a special case of the Kw-G distri- bution with G being the uniform distribution on [0; 1], whereas the G distribution is the distribution corresponding to a = b = 1. With a = 1, the Kw-G distribu- tion coincides with the beta-G distribution generated by the beta(1; b) distribution. Furthermore, for b = 1 and a being an integer, the Kw-G is the distribution of the maximum of a random sample of size a from G. One major bene¯t of the Kw family of generalized distributions is its ability of ¯tting skewed data that can not be properly ¯tted by existing distributions. In this article we deal with formula (3) in some generality. The mathematical properties of the Kw generalized family are usually much simpler to derive than those of the class of generalized beta distributions proposed by Eugene et al. [2]. Even if g(x) is a symmetric function around 0, then f(x) will not be a symmetric distribution even when a = b. From (1), if u is sampled from the uniform (0,1) distribution, then G¡1(f1 ¡ (1 ¡ u)1=bg1=a) is drawn from the Kw-G distribution. The paper is outlined as follows. Section 2 provides some special distributions in the Kw generalized family. In Section 3, we derive general expansions for the density of the Kw-G distribution as a function of the parent density g(x) multi- plied by power series in G(x) depending if a is integer or real non-integer. We can easily apply these expansions to several Kw-G distributions. Probability weighted moments (PWMs) are expectations of certain functions of a random variable and they can be de¯ned for any random variable whose ordinary moments exist. In Section 4, we derive two simple expansions for the moments of any Kw-G distri- bution as linear functions of PWMs of the G distribution which are valid if a is integer or real non-integer. We derive in Section 5 some expansions for the density of order statistics of the class of Kw-G distributions. In Section 6, PWMs are ob- tained for this class. Section 7 provides an alternative formula for moments of order statistics of the Kw-G distribution. The L-moments are also given in this section. Some inferential tools are discussed in Section 8. A real data set is analyzed by some distributions in the Kw-G family in Section 9.
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