Some topics on statistical theory and applications

This thesis includes four chapters, each dealing with a different topic of statistics.; In Chapter 1, after introducing the biological problem that serves as the motivation for our work, we present the detail of our model for sequence segmentation. As an extension of the well-known Hidden Markov Model (HMM), our model is a hierarchical Bayesian model which uses a continuous mixture of Gaussian distributions to model the variations within each state. Detailed and efficient algorithm and estimation procedure based on dynamic programming will be presented after the model is set up. We applied our model to the biological problem of delineating regions of human genome with different levels of sensitivity. Finally, we will present some variants and extensions of our hierarchical model.; In Chapter 2, we study the image superresolution problem using a variational approach. In our new approach, we use a formulation inspired by works in dimension reduction in the machine learning community. We approximate each pixel value by a linear combination of its neighbors, based on the provided low-resolution image, and extrapolate the pixel values in the high-resolution image. This new algorithm is demonstrated to better preserve the structure of the image by preventing over-smoothing as usually happens in other approaches.; In Chapter 3, we make some contributions to the theory of consistency and rates of convergence of the nonparametric Bayesian models. In the first part of this chapter, we study the consistency of posterior distribution in estimating step functions. This requires a slight extension of the existing theory to the case with observations that are non i.i.d. In the second part of the chapter, we give a simple proof of rates of convergence of posterior distribution under model misspecification, based on a key identity that has previously been exploited only in the well-specified case.; In Chapter 4, we extend the theory of reproducing kernel Hilbert spaces in order to deal with functional data models with functional response. This theory provides a new framework for functional regression that has clearly better performance in dealing with nonlinear models. We also show that our new estimator performs better than other simplistic ones such as the kernel regression estimate.