Joint large deviation result for empirical measures of the coloured random geometric graphs

We prove joint large deviation principle for the \emph{ empirical pair measure} and \emph{empirical locality measure} of the \emph{near intermediate} coloured random geometric graph models on $n$ points picked uniformly in a $d-$dimensional torus of a unit circumference.From this result we obtain large deviation principles for the \emph{number of edges per vertex}, the \emph{degree distribution and the proportion of isolated vertices } for the \emph{near intermediate} random geometric graph models.Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:1312.632

Asymptotics of the partition function of Ising model on inhomogeneous random graphs

For a finite random graph, we defined a simple model of statistical mechanics. We obtain an annealed asymptotic result for the random partition function for this model on finite random graphs as n; the size of the graph is very large. To obtain this result, we define the empirical bond distribution, which enumerates the number of bonds between a given couple of spins, and empirical spin distribution, which enumerates the number of sites having a given spin on the spinned random graphs. For these empirical distributions we extend the large deviation principle(LDP) to cover random graphs with continuous colour laws. Applying Varandhan Lemma and this LDP to the Hamiltonian of the Ising model defined on Erdos-Renyi graphs, expressed as a function of the empirical distributions, we obtain our annealed asymptotic result.Comment: 14 page

Large deviation principles for empirical measures of colored random graphs

For any finite colored graph we define the empirical neighborhood measure, which counts the number of vertices of a given color connected to a given number of vertices of each color, and the empirical pair measure, which counts the number of edges connecting each pair of colors. For a class of models of sparse colored random graphs, we prove large deviation principles for these empirical measures in the weak topology. The rate functions governing our large deviation principles can be expressed explicitly in terms of relative entropies. We derive a large deviation principle for the degree distribution of Erd\H{o}s--R\'{e}nyi graphs near criticality.Comment: Published in at http://dx.doi.org/10.1214/09-AAP647 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Statistical model for overdispersed count outcome with many zeros: an approach for direct marginal inference

Marginalized models are in great demand by most researchers in the life sciences particularly in clinical trials, epidemiology, health-economics, surveys and many others since they allow generalization of inference to the entire population under study. For count data, standard procedures such as the Poisson regression and negative binomial model provide population average inference for model parameters. However, occurrence of excess zero counts and lack of independence in empirical data have necessitated their extension to accommodate these phenomena. These extensions, though useful, complicates interpretations of effects. For example, the zero-inflated Poisson model accounts for the presence of excess zeros but the parameter estimates do not have a direct marginal inferential ability as its base model, the Poisson model. Marginalizations due to the presence of excess zeros are underdeveloped though demand for such is interestingly high. The aim of this paper is to develop a marginalized model for zero-inflated univariate count outcome in the presence of overdispersion. Emphasis is placed on methodological development, efficient estimation of model parameters, implementation and application to two empirical studies. A simulation study is performed to assess the performance of the model. Results from the analysis of two case studies indicated that the refined procedure performs significantly better than models which do not simultaneously correct for overdispersion and presence of excess zero counts in terms of likelihood comparisons and AIC values. The simulation studies also supported these findings. In addition, the proposed technique yielded small biases and mean square errors for model parameters. To ensure that the proposed method enjoys widespread use, it is implemented using the SAS NLMIXED procedure with minimal coding efforts.Comment: 28 page