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.

WATER HOMEOSTASIS, FRAILTY AND COGNITIVE FUNCTION IN THE NURSING HOME

Objective: - To develop and test a practical clinical method to assess frailty in nursing homes; - To investigate the relationship between cognitive status of the elderly and the balance between water compartments of their body composition. Design and subjects: Cross-sectional study, conducted at two nursing homes in Boston-MA. Methods: Body mass and height (Ht) were evaluated to calculate BMI (body mass index, in Kg/m(2)). The cognitive decline was evaluated based on the scores obtained from the Mini-Mental State Examination (MMSE); The extracellular to total body water ratio (ECW/TBW) was calculated after the analysis of TBW from deuterium and tritium dilution and ECW from bromide dilution. Single-frequency BIA analysis data were investigated for resistance (R) and reactance (Xc), plotted in an R/Ht Xc/Ht graph (vectorial analysis-BIVA). The BIVA results of nursing home residents were compared against the data obtained from the NHANES Ill study. TBW and ECW values were compared with a group of free-living elderly volunteers. Results: The ECW/TBW was significantly higher in nursing home residents than in the free-living individuals. BIVA analysis showed significantly higher Xc/Ht values in the reference subjects. The MMSE did not present a significant correlation with ECW/TBW for either gender. Conclusion: We proposed the ECW/TBW ratio and BIVA as surrogate methods for the clinical assessment of frailty. We tested successfully both approaches with nursing home patients and free-living volunteers and compared them to a national data base. The advent of new, portable instruments will enable field tests to further validate our proposed "Frailty Factor" in future studies. We found no correlation between frailty and cognitive decline in the nursing home.