Dependent Data Modeling and Inference in the Age of AI

The central limit theorem (CLT) and its extensions, such as the Berry-Esseen theorem, are among the most useful results in classic statistics. In high dimensions, the analogous results to the CLT are Gaussian approximation (GA) schemes on various collections of subsets of the multi-dimensional Euclidean space. In this talk, by considering two important collections of subsets: the convex sets and the Borel sets, we establish general GA theory for a very general class of high-dimensional non-stationary (HDNS) time series. Our approximation rates are nearly optimal with respect to both dimension and time series length. A block multiplier bootstrap procedure is theoretically verified for the implementation of our GA theory. We demonstrate by applications the use of the GA and bootstrap theory as a unified tool for a wide range of statistical inference problems of HDNS time series. 

Conditional heteroskedastic processes are commonly described by the GARCH model. GARCH models have been widely studied in the uni- and multivariate real-valued case. More recently first steps were taken to introduce GARCH models in function spaces, which can be relevant for the description of intra-day volatility. This talk extends the concept of functional GARCH models which so far have been defined on function spaces in a pointwise sense. In contrast, the new functional GARCH model discussed in this talk is defined in general, separable Hilbert spaces, replacing pointwise definitions with general operator-valued definitions. The talk will provide sufficient conditions for the unique strictly stationary solutions, moment properties, necessary and sufficient conditions for weak stationarity, and Yule-Walker estimators for the parameters to analyze asymptotic upper bounds as well as the asymptotic distribution of the estimation errors. The talk is based on joint work with Sebastian Kühnert (Bochum), Gregory Rice and Jeremy Vanderdoes (both Waterloo).  

In this talk, I will introduce a new spectral clustering-based method for transfer learning in community detection of network data. Our goal is to improve the clustering performance of the target network using auxiliary source networks, which are locally stored across various sources, privacy-preserved, and heterogeneous. We allow the source networks to have distinct privacy and heterogeneity levels that often happen in practice. To better utilize the information from the heterogeneous and privacy-preserved source networks, we propose a novel adaptive weighting method to first aggregate the eigenspaces of the source networks multiplied by different weights chosen to incorporate the effects of privacy and heterogeneity. Then we propose a regularization method that combines the weighted average eigenspace of the source networks with the eigenspace of the target network to automatically achieve an optimal balance between them. Theoretically, we show that the adaptive weighting method enjoys the oracle property, where the bound of estimated eigenspace only depends on informative source networks, and the adaptive weighting strategy leads to an order-wise smaller bound compared to the equal weighting strategy. We also demonstrate that the bound of the estimated eigenspace is tighter than either the weighted average of eigenspace of source networks or the eigenspace of the target network alone. 

In this talk we study the problem of detecting abrupt dense changes in the mean of a high-dimensional time series. We shall focus on the dense change in the sense that a large proportion of the elements in the mean vectors can change, although our method can also handle sparse change if the jump size is large. Specifically we developed a nonparametric methodology to identify the change-point location for a time series with both temporal and cross-sectional dependence. We construct a MOSUM statistic which can also be considered as a trimmed U-statistic for the l_2 norm of the mean change at each location, and the local maximizer of the MOSUM statistics serves as a natural estimator for the true change point location. The limiting distribution of the proposed estimator can be derived, which is a two-sided Brownian motion with a drift. We further compare the constructed MOSOM statistics with some threshold to determine the number of changes points in the data so that the same method can work under both single change point model or multiple change point model. Simulation results show that the method can accurately estimate both the number and the location of change points, and the method is not sensitive to the tuning parameters.