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.

Structural, Spectroscopic (NMR, IR, and Raman), and DFT Investigation of the Self-Assembled Nanostructure of Pravastatin-LDH (Layered Double Hydroxides) Systems

Layered double hydroxide (LDH) nanocontainers, suitable as carriers for anionic drugs, were intercalated with Pravastatin drug using magnesium-aluminum and zinc-aluminum in a M-II/Al molar ratio equal 2 and different Al3+/Pravastatin molar ratios. Postsynthesis treatments were used in order to increase the materials crystallinity. Hybrid materials were characterized by a set of physical chemical techniques: chemical elemental analysis, X-ray diffraction (XRD), mass coupled thermal analyses, vibrational infrared and Raman spectroscopies, and solid-state C-13 nuclear magnetic resonance (NMR). Results were interpreted in light of computational density functional theory (DFT) calculations performed for Sodium Pravastatin in order to assign the data obtained for the LDH intercalated materials. XRD peaks of LDH-Pravastatin material and the one-dimensional (1D) electron density map pointed out to a bilayer arrangement of Pravastatin in the interlayer region, where its associated carboxylate and vicinal hydroxyl groups are close to the positive LDH. The structural organization observed for the stacked assembly containing the unsymmetrical and bulky monoanion Pravastatin and LDH seems to be promoted by a self-assembling process, in which local interactions are maximized and chloride ion cointercalation is required. It is observed a high similarity among vibrational and C-13 NMR spectra of Na-Pravastatin and LDH-Pravastatin materials. Those features indicate that the intercalation preserves the drug structural integrity. Spectroscopic techniques corroborate the nature of the guest species and their arrangement between the inorganic layers. Changes related to carboxylate, alcohol, and olefinic moieties are observed in both vibrational Raman and C-13 NMR spectra after the drug intercalation. Thus, Pravastatin ions are forced to be arranged as head to tail through intermolecular hydrogen bonding between adjacent organic species. The thermal decomposition profile of the hybrid samples is distinct of that one observed for Na-Pravastatin salt, however, with no visible increase in the thermal behavior when the organic anion is sequestrated within LDH gap.