Sieve-based confidence intervals and bands for Lévy densities

Abstract

Abstract: A Lévy process combines a Brownian motion and a pure-jump homogeneous process, such as a compound Poisson process. The estimation of the Lévy density, the infinite-dimensional parameter controlling the jump dynamics of the process, is considered here under a discrete-sampling scheme. In that case, the jumps are latent variables which statistical properties can be assessed when the frequency and time horizon of observations increase to infinity at suitable rates. Nonparametric estimators for the Lévy density based on Grenander’s method of sieves had been proposed in [11]. In this paper, central limit theorems for these sieve estimators, both pointwise and uniform on an interval away from the origin, are obtained, leading to point-wise confidence intervals and bands for the Lévy density. In the point-wise case, we find feasible estimators which converge to s at a rate that is arbitrarily close to the rate of the minimax risk of estimation for smooth Lévy densities. We determine how frequently one needs to sample to attain the desired rate. In the case of uniform bands and discrete regular sampling, our results are consistent with the case of density estimation, achieving a rate of order arbitrarily close to log −1/2 (n) · n −1/3, where n is the number of observations. The rate is valid provided that s is smooth enough, and that the time horizon Tn and the dimension of the sieve are appropriately chosen in terms of n.

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