Mode Meets Mean: A New Robust Volatility

Ullah and Wang (Journal of Time Series Analysis, 46 (4), 748–773) introduced a novel nonparametric estimator for the volatility function that leverages the mode to complement traditional mean-based volatility measures, thereby highlighting unique features of the data. In this paper, we extend their framework to estimate traditional mean volatility by proposing a new approach, termed robust mode-oriented volatility. Our method treats the bandwidth parameter associate with the kernel objective function as a constant rather than a shrinkage parameter, prioritizing robustness and efficiency while maintaining its foundation in the mode-based framework. We demonstrate that under $\alpha$-mixing time series dependence, the proposed robust estimator retains the same asymptotic distribution as estimators derived under independence assumptions, while achieving the convergence rate of nonparametric mean regression. Furthermore, we theoretically establish that efficiency gains over traditional mean-based estimation can be realized by appropriately adjusting the bandwidth, particularly in the presence of outliers or heavy-tailed distributions. In contrast to Ullah and Wang (Journal of Time Series Analysis, 46 (4), 748–773), which focused on capturing volatility structures specific to the mode, our approach broadens the applicability of their framework by integrating it with traditional mean volatility estimation. The discussions on bandwidth selection and numerical examples highlight the finite sample performance and practical advantages of our robust estimation procedure.