A Grand Journey of Statistical Hierarchical Modelling

Abstract

This thesis presents three research reports composed by the candidate and his collaborators on different perspectives and applications of statistical hierarchical modelling, which seeks to connect the observed quantities via the assumed and unobserved variables. The topics range from random graph modelling, Shrinkage estimation, to studying tick-by-tick financial data. In Chapter 1, we discuss the modeling of unlabeled dense network, whose probabilistic property can be uniquely characterized by a sequence of uniformly distributed latent variables that mean the inter-connectivity of each node, and a bivariate function of interests, graphon, that will map each pair of these latent variables into the generative probability of each edge on the graph. A consistent estimation methodology for graphon is proposed. For the case of simultaneously estimation, studied in Chapter 2, the introduction of an unknown prior distribution over all the parameters that we want to estimate naturally lead to the so-called shrinkage estimation, originally introduced in the frequentist sense. We then take an in-depth look into the theoretical property of these shrinkage estimators under the mean square error in the scenario where independent variables (or covariates) are available to use. Finally, when studying the dynamics of high-frequency financial data in Chapter 3, assuming an non-observable Poisson random fields whose realizations are thought as pulses or innovations that indeed drive the seen movements of trade price processes, will naturally lead to a new family of theoretically coherent models for discrete-valued stochastic processes that could address different empirical features for market microstructure. In this internet age, hierarchical modelling has become an effective methodology that could help to organize the tremendous amount of data in a logical manner, by pooling information together, reducing idiosyncratic noises, and finally achieving a better estimation (or prediction) that can still generalize to future data. In addition to the viewpoint from the statistical stability, correctly identifying the pieces that we cannot see and, at the same time, explicitly considering these quantities into our modelling framework, will often lead to a more descriptive statistical model that can better represents different characteristics of the observed data. Even though these research topics were introduced by the candidates and his collaborators from varied original considerations, it turns out that these seemly disconnected studies could all be seen as different representation and realization of the statistical hierarchical modeling. The candidate is eager to present this grand journey, during which he feels amazed, blessed, and breathtaking.