OPTIMAL SPARSE VOLATILITY MATRIX ESTIMATION FOR HIGH-DIMENSIONAL ITO PROCESSES WITH MEASUREMENT ERRORS

• 成果类型: Article
• 署名作者: Tao, Minjing; Wang, Yazhen; Zhou, Harrison H.
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; Yale University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1128
• 发表日期: 2013
• 页码: 1816-1864
• 关键词: covariance-matrix; Optimal Rates; diffusion; CONVERGENCE; time; selection; noise

摘要

Stochastic processes are often used to model complex scientific problems in fields ranging from biology and finance to engineering and physical science. This paper investigates rate-optimal estimation of the volatility matrix of a high-dimensional Ito process observed with measurement errors at discrete time points. The minimax rate of convergence is established for estimating sparse volatility matrices. By combining the multi-scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate. The minimax lower bound is derived by considering a subclass of Ito processes for which the minimax lower bound is obtained through a novel equivalent model of covariance matrix estimation for independent but nonidentically distributed observations and through a delicate construction of the least favorable parameters. In addition, a simulation study was conducted to test the finite sample performance of the optimal estimator, and the simulation results were found to support the established asymptotic theory.