Abstract
Both locally stationary processes and irregular models have had a long story of success in statistics and time series analysis. We combine both concepts and consider a nonparametric, first-order autoregressive model with irregular, positive innovations, where we assume that the coefficient function is Hölder continuous and positive. To estimate this function, we use a quasi-maximum likelihood based approach. A precise control of this method demands a delicate analysis of extremes of certain weakly dependent processes, our main result being a concentration inequality for such quantities. Based on our analysis, upper and matching minimax lower bounds are derived, showing the optimality of our estimators. Unlike the regular case, the information theoretic complexity depends both on the smoothness and an additional shape parameter, characterizing the irregularity of the underlying distribution. The results and ideas for the proofs are very different from classical and more recent methods in connection with locally stationary processes.
Acknowledgments
We would like to thank Thomas Mikosch for his valuable comments regarding literature in extreme value theory. Moreover, we are indebted to the Editors and the Reviewer for the many thoughtful comments and suggestions that greatly enhanced the quality.
Citation
H. Gruber. M. Jirak. "Irregular nonparametric autoregression." Bernoulli 31 (1) 731 - 758, February 2025. https://doi.org/10.3150/24-BEJ1748
Information