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.

LAGOA SANTA REVISITED: AN OVERVIEW OF THE CHRONOLOGY, SUBSISTENCE, AND MATERIAL CULTURE OF PALEOINDIAN SITES IN EASTERN CENTRAL BRAZIL

Lagoa Santa, a karstic area in eastern Central Brazil, has been subject to research on human paleontology and archaeology for 175 years. Almost 300 Paleoindian human skeletons have been found since Danish naturalist Peter Lund's pioneering work. Even so, some critical issues such as the role of rockshelters in settlement systems, and the possible paleoclimatic implications of the peopling of the region have yet to be addressed. We present some results obtained from recent excavations at four rockshelters and two open-air sites, new dates for human Paleoindian skeletons, and a model to explain the cultural patterns observed so far. It is also argued that the Paleoindian subsistence system at Lagoa Santa was similar to other locations in South America: generalized small-game hunting complemented by fruits, seed, and root gathering.