Optimal Sparse Volatility Matrix Estimation for High-Dimensional Ito Processes with Measurement Errors

• Activity Number: 347
• Type: Topic Contributed
• Date/Time: Tuesday, August 6, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Business and Economic Statistics Section
• Author(s): Minjing Tao*+ and Yazhen Wang and Harrison Zhou
• Companies: Department of Statistics, University of Wisconsin-Madison and University of Wisconsin-Madison and Yale University
• Keywords: large matrix estimation ; measurement error ; minimax lower bound ; optimal convergence rate ; sparsity ; volatility matrix estimator

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 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 non-identically 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.