Improving the Lag Window Estimators of the Spectrum and Memory for Long-Memory Stationary Gaussian Processes

Abstract

Gaussian process (GP) is a stochastic process that has been successfully applied in finance, black-box modeling of biosystems, machine learning, geostatistics, multitask learning or robotics and reinforcement learning. Effectively estimating the spectral density function (SDF) and degree of memory (DOM) of a long-memory stationary GP (LMSGP) is needed to get accurate information about the process. The practice demonstrated that the periodogram estimator (PE) and lag window estimator (LWE) that are the extremely used estimators of the SDF and DOM have some drawbacks, especially for LMSGPs. The behaviors of the PEs and LWEs are soundly investigated numerically; however, the theoretical justifications are limited and thus the challenge to improve them is still daunting. This paper gives a closer look at the theoretical justifications of the efficiency of the LWEs that provides new sufficient conditions (NSCs) for improving the LWEs of the SDF and DOM for LMSGPs. The precision, the convergence rate of the bias and variance, and the asymptotic distributions of the LWEs under the NSCs are studied. A comparison study among the LWEs under the NSCs, the LWEs without the NSCs and the PEs is given to investigate the significance of the NSCs. The main theoretical and simulation results show that: the LWEs under the NSCs are asymptotically unbiased and consistent and better than the LWEs without the NSCs and the PEs, and the asymptotic distributions of the LWEs under the NSCs are chi-square for SDF and normal for DOM.

Keywords

Gaussian process / Spectral density / Degree of memory / Lag window / Periodogram / Local Whittle method / Regression method

Funding

uic(R201810)

uic(R202010)