Various Statistical Inferences for High-dimensional Time Series: Bootstrap, Homogeneity Pursuit and Autocovariance Test

Abstract

This thesis aims to study various statistical inferences for high-dimensional data, especially high-dimensional time series, including sieve bootstrap, homogeneity pursuit, and an equivalence test for spiked eigenvalues of autocovariance matrix. The primary techniques used in this thesis are novel dimension-reduction methods developed from factor models and principal component analysis (PCA).

Chapter 2 proposes a novel sieve bootstrap method for high-dimensional time series and applies it to sparse functional time series where the actual observations are not dense, and pre-smoothing is misleading. Chapter 3 introduces an iterative complement-clustering principal component analysis (CPCA) to study high-dimensional data with group structures, where both homogeneity and sub-homogeneity (group-specific information) can be identified and estimated. Lastly, Chapter 4 proposes a novel test statistic named the autocovariance test to compare the spiked eigenvalues of the autocovariance matrices for two high-dimensional time series. In all chapters, dimension-reduction methods are applied for novel statistical inferences. In particular, Chapters 2 and 4 focus on the spiked eigenstructure of autocovariance matrix and use factors to capture the temporal dependence of the high-dimensional time series. Meanwhile, Chapter 3 aims to simultaneously estimate homogeneity and sub-homogeneity, which form a more complicated spiked eigenstructure of the covariance matrix, despite that the group-specific information is relatively weak compared with the homogeneity and traditional PCA fails to capture it.

The theoretical and asymptotic results of all three statistical inferences are provided in each chapter, respectively, where the numerical evidence on the finite-sample performance for each method is also discussed. Finally, these three statistical inferences are applied on particulate matter concentration data, stock return data, and age-specific mortality data for multiple countries, respectively, to provide valid statistical inferences.