Bivariate gamma-geometric law and its induced Levy process

In this article we introduce a three-parameter extension of the bivariate exponential-geometric (BEG) law (Kozubowski and Panorska, 2005) [4]. We refer to this new distribution as the bivariate gamma-geometric (BGG) law. A bivariate random vector (X, N) follows the BGG law if N has geometric distribution and X may be represented (in law) as a sum of N independent and identically distributed gamma variables, where these variables are independent of N. Statistical properties such as moment generation and characteristic functions, moments and a variance-covariance matrix are provided. The marginal and conditional laws are also studied. We show that BBG distribution is infinitely divisible, just as the BEG model is. Further, we provide alternative representations for the BGG distribution and show that it enjoys a geometric stability property. Maximum likelihood estimation and inference are discussed and a reparametrization is proposed in order to obtain orthogonality of the parameters. We present an application to a real data set where our model provides a better fit than the BEG model. Our bivariate distribution induces a bivariate Levy process with correlated gamma and negative binomial processes, which extends the bivariate Levy motion proposed by Kozubowski et al. (2008) [6]. The marginals of our Levy motion are a mixture of gamma and negative binomial processes and we named it BMixGNB motion. Basic properties such as stochastic self-similarity and the covariance matrix of the process are presented. The bivariate distribution at fixed time of our BMixGNB process is also studied and some results are derived, including a discussion about maximum likelihood estimation and inference. (C) 2012 Elsevier Inc. All rights reserved.

Regional and socioeconomic distribution of household food availability in Brazil, in 2008-2009

OBJECTIVE: To describe the regional and socioeconomic distribution of household food availability in Brazil. METHODS: Data from the 2008-2009 Household Budget Survey on food and beverage acquisition for household consumption, conducted by the Instituto Brasileiro de Geografia e Estatistica (Brazilian Institute of Geography and Statistics), were analyzed. The amounts of foods, recorded during seven consecutive days in the 55,970 sample households, were converted into calories and nutrients. Food quality indicators were constructed and analyzed according to the regional and socioeconomic strata of the Brazilian population. RESULTS: The amount of energy from protein was adequate in all regional and socioeconomic strata. On the other hand, an excess of free sugars and fats was observed in all regions of the country, especially in the Southern and Southeastern regions. The proportion of saturated fats was high in urban areas and consistent with the greater contribution of animal-derived products. Limited availability of fruits and vegetables was found in all regions. An increase in the fat content and reduction in carbohydrate content of the diet were observed with the increase in income. CONCLUSIONS: The negative characteristics of the Brazilian diet observed at the end of the first decade of the 21(st) century indicate the need to prioritize public policies for the promotion of healthy eating.

Levy stable distributions via associated integral transform

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g(alpha)(x), 0 <= x < infinity, 0 < alpha < 1. We demonstrate that the knowledge of one such a distribution g a ( x) suffices to obtain exactly g(alpha)p ( x), p = 2, 3, .... Similarly, from known g(alpha)(x) and g(beta)(x), 0 < alpha, beta < 1, we obtain g(alpha beta)( x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For a rational, alpha = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g(l/k)(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4709443]