Rate-optimal estimation of the Blumenthal-Getoor index of a L\'evy process
Pith reviewed 2026-05-25 20:13 UTC · model grok-4.3
The pith
A new GMM estimator for the Blumenthal-Getoor index and successive indices of Lévy processes attains the optimal rate of convergence, with a simulation study comparing it to prior methods.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
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.
Load-bearing premise
The paper assumes a Lévy process observed at high frequency with the standard semimartingale structure that allows separation of diffusion and jump components; this modeling choice underpins the rate claims and is invoked when contrasting with existing suboptimal estimators.
read the original abstract
The Blumenthal-Getoor (BG) index characterizes the jump measure of an infinitely active L\'evy 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 only 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a novel GMM-based estimator for the Blumenthal-Getoor index (and successive indices) of an infinitely active Lévy process observed at high frequency. It claims this estimator attains the optimal rate of convergence; when a proportionality factor must also be estimated, the rate remains optimal up to logarithmic factors. The approach simultaneously yields an efficient volatility estimator that accounts for jumps of infinite variation. All parameters are estimated jointly, and the finite-sample performance is assessed via simulation against existing methods in the financial econometrics literature.
Significance. If the rate-optimality result holds under the stated semimartingale Lévy assumptions, the contribution would be significant: it removes the polynomial-factor suboptimality that has characterized prior high-frequency BG-index estimators in the presence of a diffusion component. The joint GMM construction and the accompanying volatility estimator are additional strengths. The simulation study supplies practical evidence, though its design details would need to be verified for full reproducibility.
minor comments (3)
- [Abstract and main theorem] The abstract states that the estimator is 'rate-optimal up to logarithmic factors' when a proportionality factor is inferred, but the precise logarithmic term and the conditions under which it appears should be stated explicitly in the main theorem (likely §4 or §5).
- [Section 2 (model and notation)] Notation for the successive BG indices and the truncation levels used in the high-frequency scheme should be introduced once and used consistently; current usage appears to switch between β and β_k without a single defining display.
- [Section 6 (simulation)] The simulation study reports finite-sample behavior but does not specify the exact data-generating processes, the range of sampling frequencies, or the number of Monte Carlo replications; these details are needed for reproducibility.
Simulated Author's Rebuttal
We thank the referee for the positive and constructive report. We are pleased that the significance of the rate-optimality result under semimartingale Lévy assumptions, the joint GMM estimation, and the accompanying volatility estimator are recognized. The recommendation for minor revision is appreciated. No specific major comments were listed in the report, so we have no points requiring detailed rebuttal or revision at this stage.
Circularity Check
No significant circularity
full rationale
The paper introduces a novel GMM estimator for the Blumenthal-Getoor index (and successive indices) plus volatility in a standard high-frequency Lévy semimartingale model. The abstract and reader's summary present the optimality claims as arising from asymptotic analysis of the proposed estimator relative to the usual separation of diffusion and infinite-variation jumps; no equations, fitted parameters, or self-citations are shown that reduce the target rates or indices to the inputs by construction. The modeling assumptions are the conventional ones for this literature and are invoked explicitly rather than smuggled in. The derivation chain is therefore self-contained against external benchmarks.
discussion (0)
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