Comments on: Subsampling weakly dependent time series and application to extremes

Abstract

Professors Doukhan, Prohl, and Robert are to be congratulated for their work on extending the validity of the subsampling method to a much wider class of processes compared to the existing literature that typically requires the processes to be strongly mixing (cf. Politis et al. 1999). As described in Sects. 1 and 2, many common time series models, including the ARMA models, often fail to satisfy the strong mixing condition but they typically satisfy the ηand λ-weak dependence conditions of Doukhan and Louhichi (1999) considered in this paper. As a result, extending the validity of the subsampling method under suitable ηand λ-weak dependence conditions is an important contribution. Expectedly, the smooth version of the subsampling estimator is especially suited to the form of the ηand λ-weak dependence conditions which give covariance bounds for smooth functions of the observations. This is one reason why the validity of the smooth subsampling estimator in Theorem 1 holds under weaker conditions than those for the rough subsampling estimator in Theorem 3. However, from the applications point of view, it is worth noting that while smoothing is known to play an important role in the resampling methodology in certain inference problems (e.g., inference on quantiles), caution must be exercised in cases where the limit distribution of the (unbootstrapped) statistic has points of discontinuity. The authors also prove validity of the (smooth) subsampling estimator for the sample maximum under different sets of ηand λ-weak dependence conditions, for both the overlapping and the non-overlapping cases. This is an important problem where its natural competitor, namely, the block bootstrap method (cf. Kunsch 1989