Rate-Optimal Estimation of the Blumenthal-Getoor Index of a Lévy Process


Journal article


Fabian Mies
Electronic Journal of Statistics, vol. 14, 2020, pp. 4165-4206


arXiv
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APA   Click to copy
Mies, F. (2020). Rate-Optimal Estimation of the Blumenthal-Getoor Index of a Lévy Process. Electronic Journal of Statistics, 14, 4165–4206. https://doi.org/10.1214/20-EJS1769


Chicago/Turabian   Click to copy
Mies, Fabian. “Rate-Optimal Estimation of the Blumenthal-Getoor Index of a Lévy Process.” Electronic Journal of Statistics 14 (2020): 4165–4206.


MLA   Click to copy
Mies, Fabian. “Rate-Optimal Estimation of the Blumenthal-Getoor Index of a Lévy Process.” Electronic Journal of Statistics, vol. 14, 2020, pp. 4165–206, doi:10.1214/20-EJS1769.


BibTeX   Click to copy

@article{mies2020a,
  title = {Rate-Optimal Estimation of the Blumenthal-Getoor Index of a Lévy Process},
  year = {2020},
  journal = {Electronic Journal of Statistics},
  pages = {4165-4206},
  volume = {14},
  doi = {10.1214/20-EJS1769},
  author = {Mies, Fabian}
}

The Blumenthal–Getoor (BG) index characterizes the jump measure of an infinitely active Lévy process. It determines sample path properties and affects the behavior of various econometric procedures. If the process contains a diffusion term, existing estimators of the BG index based on high-frequency observations achieve rates of convergence which are suboptimal by a polynomial factor. In this paper, a novel estimator for the BG index and the successive BG indices is presented, attaining the optimal rate of convergence. If an additional proportionality factor needs to be inferred, the proposed estimator is rate-optimal up to logarithmic factors. Furthermore, our method yields a new efficient volatility estimator which accounts for jumps of infinite variation. All parameters are estimated jointly by the generalized method of moments. A simulation study compares the finite sample behavior of the proposed estimators with competing methods from the financial econometrics literature. 

Keywords: Fisher information; high-frequency; jump activity; method of moments; non-diagonal rate matrix



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