Statistical inference for dependent data

Abstract

Functional data Analysis has emerged as an important area of statistics which provides convenient and informative tool for the analysis of data objects of high dimension/high resolution. In the literature, it seems that the emphasis has been placed on independent functional data or models where the covariates and errors are assumed to be independent. However, the independence assumption is often too strong to be realistic in many application especially if the data are collected sequentially over time such as climate data and high frequency financial data. Motivated by our ongoing research on the development of high-resolution climate projections through statistical downscaling, we consider the change point problem and the two sample problem for temporally dependent functional data. Specifically, in Chapter 1, we develop a self-normalization based test to test the structural stability of temporally dependent functional observations. We propose new tests to detect the differences of the covariance operators and their associated characteristics of two functional time series in Chapter 2. The self-normalization approach introduced in the first two chapters is closely linked to the fixed-b asymptotic scheme in the econometrics literature. Motivated by recent studies on heteroskedasticity and autocorrelation consistent based robust inference, we propose a class of estimators for estimating the asymptotic covariance matrix of the generalized method of moments estimator in the stationary time series models in Chapter 3. Under mild conditions, we establish the first order asymptotic distribution for the Wald statistics when the smoothing parameter is held fixed. Furthermore, we derive higher order Edgeworth expansions for the finite sample distribution of the Wald statistics in the Gaussian location model under the fixed-smoothing paradigm. The results are used to justify the second order correctness of a new bootstrap method, the Gaussian dependent bootstrap, in the context of Gaussian location model. Finally, in Chapter 4, we describe an extension of the fixed-b approach to the empirical likelihood estimation framework.