We propose several statistics to test the Markov hypothesis for -mixing stationary processes sampled at discrete time intervals. Our tests are based on the ChapmanKolmogorov equation. We establish the asymptotic null distributions of the proposed test statistics, showing that Wilkss phenomenon holds. We compute the power of the test and provide simulations to investigate the finite sample performance of the test statistics when the null model is a diffusion process, with alternatives consisting of models with a stochastic mean reversion level, stochastic volatility and jumps.

Generalized linear models and quasi-likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response. Nonparametric methods need no explicit parametric specification and the resulting model is completely determined by the data themselves. However nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in Fan, Heckman and Wand (1995). In this work, we propose two parametrically guided nonparametric estimation schemes by incorporating prior shape information on the link transformation of the response variables conditional mean in terms of the predictor variable. Asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart.

Measuring conditional dependence is an important topic in statistics with broad applications including graphical models. Under a factor model setting, a new conditional dependence measure based on projection is proposed. The corresponding conditional independence test is developed with the asymptotic null distribution unveiled where the number of factors could be high-dimensional. It is also shown that the new test has control over the asymptotic significance level and can be calculated efficiently. A generic method for building dependency graphs without Gaussian assumption using the new test is elaborated. Numerical results and real data analysis show the superiority of the new method.

Functional linear regression analysis aims to model regression relations which include a functional predictor. The analog of the regression parameter vector or matrix in conventional multivariate or multiple-response linear regression models is a regression parameter function in one or two arguments. If, in addition, one has scalar predictors, as is often the case in applications to longitudinal studies, the question arises how to incorporate these into a functional regression model. We study a varying-coefficient approach where the scalar covariates are modeled as additional arguments of the regression parameter function. This extension of the functional linear regression model is analogous to the extension of conventional linear regression models to varying-coefficient models and shares its advantages, such as increased flexibility; however, the details of this extension are more challenging in the functional case. Our

This paper introduces an efficient approach to integrating non-local statistics into the higher-order Markov Random Fields (MRFs) framework. Motivated by the observation that many non-local statistics (eg, shape priors, color distributions) can usually be represented by a small number of parameters, we reformulate the higher-order MRF model by introducing additional latent variables to represent the intrinsic dimensions of the higher-order cliques. The resulting new model, called NC-MRF, not only provides the flexibility in representing the configurations of higher-order cliques, but also automatically decomposes the energy function into less coupled terms, allowing us to design an efficient algorithmic framework for maximum a posteriori (MAP) inference. Based on this novel modeling/inference framework, we achieve state-of-the-art solutions to the challenging problems of class-specific image segmentation and template-based 3D facial expression tracking, which demonstrate the potential of our approach.