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August 2013 Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors
Minjing Tao, Yazhen Wang, Harrison H. Zhou
Ann. Statist. 41(4): 1816-1864 (August 2013). DOI: 10.1214/13-AOS1128

Abstract

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 Itô 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 Itô 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.

Citation

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Minjing Tao. Yazhen Wang. Harrison H. Zhou. "Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors." Ann. Statist. 41 (4) 1816 - 1864, August 2013. https://doi.org/10.1214/13-AOS1128

Information

Published: August 2013
First available in Project Euclid: 5 September 2013

zbMATH: 1281.62178
MathSciNet: MR3127850
Digital Object Identifier: 10.1214/13-AOS1128

Subjects:
Primary: 62G05 , 62H12
Secondary: 62M05

Keywords: Large matrix estimation , measurement error , minimax lower bound , multi-scale , optimal convergence rate , Sparsity , subGaussian tail , threshold , volatility matrix estimator

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 4 • August 2013
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