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Rate-optimal estimation of the Blumenthal-Getoor index of a Lévy process. (English) Zbl 1455.62163

Summary: 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.

MSC:

62M05 Markov processes: estimation; hidden Markov models
60G51 Processes with independent increments; Lévy processes
62P05 Applications of statistics to actuarial sciences and financial mathematics
62B10 Statistical aspects of information-theoretic topics

Software:

NLopt; GitHub
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Full Text: DOI arXiv Euclid

References:

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