Minimax rates for the covariance estimation of multi-dimensional Lévy processes with high-frequency data

2019/03/15 by Papagiannouli, Katerina
#60G51 #60J75 #62C20 #62G05 #62G10 #FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1903.06585

Abstract

This article studies nonparametric methods to estimate the co-integrated volatility for multi-dimensional Lévy processes with high frequency data. We construct a spectral estimator for the co-integrated volatility and prove minimax rates for an appropriate bounded nonparametric class of semimartingales. Given n observations of increments over intervals of length 1/n, the rates of convergence are 1 / √(n) if r ≤ 1 and (nlog n)^(r-2)/2 if r>1 , which are optimal in a minimax sense. We bound the co-jump index activity from below with the harmonic mean. Finally, we assess the efficiency of our estimator by comparing it with estimators in the existing literature.

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