Block-based inference for Gaussian subordinated long-range dependent processes
Date
2022-08
Authors
Zhang, Qihao
Major Professor
Advisor
Nordman, Daniel
Yu, Cindy
Arka, Ghosh
Dutta, Somak
Wu, Huaiqing
Committee Member
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Abstract
With rapid growth in data collection capacity, more and more complicated structures for time series data are available; some are long-memory/long-range dependent (LRD), such as in economics, finance, geology, and traffic network. However, LRD is notorious for causing failure in some commonly used statistical techniques intended for short-range dependence time series. Therefore, my purpose is to build new methodological theory for a broad class of stationary LRD series and propose new corresponding statistical tools for inference. This dissertation addresses two inference problems about LRD data, through developing new theoretical understandings as well as novel methods driven by modern statistics.
Chapter 2 focuses on my work supervised by Prof. Daniel J. Nordman on optimal block resampling under Gaussian-subordinated LRD processes. In this project, we consider block resampling techniques for LRD time series, which are useful for allowing statistical inference without stringent assumptions on the data. However, block resampling requires data blocks to be created from an observed time sequence, and there has been little understanding about how strong dependence affects the choice of data blocks. In contrast, much more theoretical investigation and practical study exists only for weakly dependent data. To establish guideposts for block size selection under long-memory, we consider the fundamental problem of choosing a block size for good performance in variance estimation of sample means through block-based resampling; block selections can then be extended to other statistics and evaluated in other inference contexts, such as distribution estimation of statistics through block resampling.
We derive the explicit large-sample bias and variance properties of the moving block resampling technique in a long-run variance estimation problem, which helps us to determine the theoretically optimal block size (say $\ell_{opt}$) that achieves the best estimation rate possible in terms of mean squared error (MSE). Based on the form of optimal block size, we propose a novel consistent estimator $\widehat{\ell}_n/\ell_{opt} \overset{p}{\to} 1$ and generalize our results beyond the sample mean by considering wider statistical functionals, such as smoothing function estimation, M-estimation, and L-estimation. Our theoretical development is based on graph-theoretic moment expansions (cf. Taqqu (1977); Malyshev (1980)) to derive covariance rates and higher-order cumulants. We also conduct several numerical studies to show that the block selection rule applies to several inference problems under LRD where data blocking and block choices are required in practice, such as variance estimation by block resampling, block bootstrap distributional estimation, and block-based tests of Hermite rank. For the latter, we also propose a simple bootstrap method to test whether Hermite rank equals one. The numerical experiments confirm and complement our theoretical results in finite sample cases.
Chapter 3 discusses my second work with Prof. Daniel J. Nordman and Prof. Soumendra N. Lahiri on a Hermite rank estimation problem. We consider a common model for LRD data based on a Gaussian subordinated process $\{X_t = G(Z_t)\}$, where $G(\cdot)$ represents an unobserved function and $Z_t$ is a stationary long-memory Gaussian process (with lag $k$ covariance satisfying a slow decay $\gamma_Z(k) \sim k^{-\alpha}$ depending on a long-memory exponent $\alpha < \frac{1}{m}$). The Hermite rank $m$ of the transformation $G(\cdot)$ is the smallest non-zero term in the Hermite expansions of $G(\cdot) = \sum_{k=m}^{\infty} J_k/k! \cdot H_k(\cdot)$ with respect to so-called Hermite polynomials $\{H_k(\cdot)\}_{k=1}^{\infty}$. The Hermite rank $m$ is known to have a critical impact on inference. For example, with LRD Gaussian subordinated processes, the distributions of statistics can change with $m$. Despite its importance, no method has existed for estimating the Hermite rank $m$ from data $X_1,\ldots,X_n$.
We propose a block-based log-regression estimator of the Hermite rank $m$, and establish the consistency of this estimation approach. The second project is motivated by covariance properties discovered in the first project regarding squared block averages from data. In particular, the fourth-order cumulants of a transformed process $X_t=G(Z_t)$ play a role in determining a decay rate of covariances between block means through the memory-exponent $\alpha$. Based on this understanding, we use the large-sample properties of covariance estimators from block sample means to formulate an estimator of the long-memory exponent $\alpha$ and Hermite rank $m$, sequentially. In this project, a Monte-Carlo simulation study of finite-sample performance is included, along with two data examples. The first example validates the assumption of a Hermite rank $m=1$ model for log returns from the Dow Jones Index and the Standard & Poor's 500 Index. The second example illustrates the nature of LRD and Hermite rank estimation for tree ring width series.
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dissertation