On non parametric statistical inference for densities under long-range dependence

ABSTRACT

Statistical inference for kernel estimators of the marginal density is considered for stationary processes with long-range dependence. The asymptotic behavior is known to differ sharply between small and large bandwidths. The statistical implications of this dichotomy have not been fully explored in the literature. The optimal rate and a functional limit theorem are obtained for large bandwidths, if the long-memory parameter exceeds a certain threshold. The threshold can be lowered arbitrarily close to the lower bound of the long-memory range. This result is extended to processes with infinite variance, and the construction of simultaneous finite-sample confidence bands is considered.

KEYWORDS:

• Infinite variance
• kernel density estimation
• long-range dependence
• simultaneous confidence band
• smoothing dichotomy

MATHEMATICS SUBJECT CLASSIFICATION:

• 62Gxx
• 62Mxx
• 62G07
• 62G20
• 60K99

Acknowledgments

We would like to thank the referees for very useful remarks that helped to improve the presentation of the results.

Funding

This research was supported in part by the DFG-Research [grant number BE 2123/11-1]. The river discharge series were obtained from the River Discharge Database of The Center of Sustainability and Global Environment, Gaylord Nelsen Institute for Environmental Studies, University of Wisconsin-Madison.