Nonparametric estimation for Lévy processes from low-frequency observations

2007/09/13 by Neumann, Michael H., Reiss, Markus · 3 citations
#62G15 #62M15 #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.0709.2007

Abstract

We suppose that a Lévy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the Lévy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific C²-criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line.

Cited by

• Learning Lévy density via adaptive RKHS regression with bi-level optimization
• Bayesian Non-Parametric Inference for Lévy Measures in State-Space Models
• Statistical Inference for Quasi-Infinitely Divisible Distributions via Fourier Methods

Related

• Adaptive minimax estimation for discretely observed Lévy processes
• Nonparametric estimation for irregularly sampled Lévy processes
• A Lepskiĭ-type stopping rule for the covariance estimation of multi-dimensional Lévy processes
• Optimal estimation of some random quantities of a Lévy process
• Nonparametric density estimation for the small jumps of Lévy processes