DescriptionThis thesis deals with three problems. The first problem is concerned with the classical white noise test, and the second is on the estimation of autoregressive models for matrix-valued time series. These two time series problems are treated in Chapter 2 and 3. The third problem, on the construction of exact algorithm for the linear constrained LASSO, is treated in Chapter 4.
In Chapter 2, we study the asymptotic distribution of sample canonical correlations under different distributional assumptions of the time series. The joint density of the asymptotic distribution is derived explicitly for the normal and elliptical distributions. For the general non-normal case, we propose to bootstrap the canonical correlations to obtain the p-value. We carry out an extensive simulation study to illustrate the size and power performances of the proposed tests.
In Chapter 3, we propose a novel estimator of the matrix autoregressive model, based on the weighted least squares. We derive the asymptotic distributions of the estimator, and demonstrate its performance by simulations and real examples.
Chapter 4 deals with the exact algorithm of the constrained LASSO problem. We develop such an exact algorithm by exploring the geometric properties of the problem. We prove that the solution path of the problem is piece-wise linear. We also prove an exponential upper bound for the complexity of the constrained LASSO problem.