The Limit Theory Of Autoregressive Time Series And Their Applications

This dissertation aims at probing into some relative discussions in autoregressivetime series. We presented the limit distribution of test statistics when dealing with the relative research on parameter and nonparametric estimation, which including the extension of the independent identically distribution (i.i.d.) assumptions on residuals to a class of mild dependence. As an application, the copy number of genome-wide single nucleotide polymorphism (SNP) array and the peak detection of the abrupt change points are studied.The first chapter focuses on a the dependent autoregressive time series. Underthe assumption of unit root, we prove that the null distribution of the unit root test statistic based on the least-square estimator can be approximated by using residual bootstrap in section 2. The third section pays attention to the autoregressive processes with a mean shift under the unit root hypothesis. We derive the asymptotic behavior of the normalized estimator and Dickey-Fuller F-statistic. In the theorems stated in this section, we see that convergence rate of the normalized estimator with a mean shift is much faster than those without a mean shift, up to O_p(T^-3/2) (n is the sample size). The consistent estimators of the prior mean and post mean are obtained in this section too. As the most popularvolatility models, GARCH models are discussed in the literature frequently. The fourth section, high order nonstationary autoregressive model with GARCH errors is discussed. Under the unit root assumption, we rewrite the DickeyFullertest statistics in the form of the self-normalized sums, and we derive the asymptotic distributions of the test statistics with or without deterministic drift term. In the literature on unit root with GARCH errors, our assumptions on the residuals are the weakest conditions for the unit root distribution to exist.The second chapter is concerning on the limit properties of nonparametricestimation in high order autoregressive models. In section two, without any restriction on the tail condition on kernel function, we obtain the asymptotic convergence rate of L_r-norms between kernel density function and it estimator in the AR(p) process by martingale difference approximation method. The third section focuses attention on quantile autoregressive (QAR) models in which the autoregressive coefficients can be depend on the quantile function. We use the self-weighted quantile regressive estimation for infnite variance QAR models. The asymptotic normality of the estimated parameters are established conditionallyon lagged values of the response. In addition, the Wald test statistics are developed for the linear restriction on the parameters.Enlightened by the study of unit root phenomena in chapter one, the third chapter applies the relative discussion into a sequence of branching process with immigration. In the second section of this chapter, we prove the mean estimation of offspring in a sequence of branching process with immigration is a function of Winner process, having the asymptotic normality with the convergence rateO_p(n^-1) in the nearly critical. In the third section, using the bootstrap method,we approximate the distribution of n(θ_n- 1) by that of m(θ_m -θ_n), with mdenoting the bootstrap sample size andθ_m indicating the bootstrap least squareestimatior.The last chapter are the studies on biomarker detection of myelodysplastic syndromes (MDS) single nucleotide polymorphism (SNP). Section two mainly focuses on the pre-processing of MDS SNP arrays. By employing a moving average algorithm and local linear regression correction of the MDS SNP arrays with different fractions, we perform the power analysis to obtain the estimated sample size. We define the risk score function and general variation level for power analysis and prognosis for the discrimination of high grade and low grade, which supported by the genetic analytic results. In the third section, we present a novel number copy inference algorithm based on conditional random pattern (CRP) model. The peak detection of kernel smoothed SNP data are presented in the fourth section. By proposing a mixture model, we can interpret the signal data more appropriately. We use a Bayesian approach to estimate parameters from the proposed mixture model, and use Markov chain Monte Carlo to perform Bayesian inference. By introducing a reversible jump method, which could automatically estimate the number of mixtures in the model.