Nonparametric Gaussian Inference for Stable Processes

Journal article

Fabian Mies, Ansgar Steland
Statistical Inference for Stochastic Processes, vol. 22, 2019, pp. 525-555

Cite

APA Click to copy
Mies, F., & Steland, A. (2019). Nonparametric Gaussian Inference for Stable Processes. Statistical Inference for Stochastic Processes, 22, 525–555. https://doi.org/10.1007/s11203-018-9193-9

Chicago/Turabian Click to copy
Mies, Fabian, and Ansgar Steland. “Nonparametric Gaussian Inference for Stable Processes.” Statistical Inference for Stochastic Processes 22 (2019): 525–555.

MLA Click to copy
Mies, Fabian, and Ansgar Steland. “Nonparametric Gaussian Inference for Stable Processes.” Statistical Inference for Stochastic Processes, vol. 22, 2019, pp. 525–55, doi:10.1007/s11203-018-9193-9.

BibTeX Click to copy

@article{mies2019a,
  title = {Nonparametric Gaussian Inference for Stable Processes},
  year = {2019},
  journal = {Statistical Inference for Stochastic Processes},
  pages = {525-555},
  volume = {22},
  doi = {10.1007/s11203-018-9193-9},
  author = {Mies, Fabian and Steland, Ansgar}
}

Jump processes driven by α-stable Lévy processes impose inferential difficulties as their increments are heavy-tailed and the intensity of jumps is infinite. This paper considers the estimation of the functional drift and diffusion coefficients from high-frequency observations of a stochastic differential equation. By transforming the increments suitably prior to a regression, the variance of the emerging quantities may be bounded while allowing for identification of drift and diffusion in a single framework. These findings are applied to obtain a novel nonparametric kernel estimator, for which asymptotic normality and consistency of subsampling approximations are derived, and to a parametric volatility estimator for the Ornstein–Uhlenbeck process. The proposed approach also suggests a semiparametric estimator for the index of stability α. Finite sample properties of the proposed estimators, in terms of mean (integrated) absolute error, are investigated by a simulation study and compared to their non-tempered counterparts.

Keywords: high-frequency data; infinitesimal generator; jumps; kernel regression; stable process; subsampling