Asymptotics for statistical functionals of long-memory sequences. (English) Zbl 1250.62023
This paper establishes two general results that can be used to obtain the asymptotic behaviour for statistical functionals based on long memory sequences. The idea of quasi-Hadamard differentiability was introduced by E. Beutner and H. Zähle [J. Multivariate Anal. 101, No. 10, 2452–2463 (2010; Zbl 1213.62055)] to enable a modified functional delta method to be applied to a class of functionals. In general, to apply this technique a weak convergence result is required for the underlying empirical process with respect to a weighted-sup norm. The first result of this paper establishes such a result for the empirical process of a linear long memory sequence. This is then applied to obtain properties of certain L- and V-statistics. The second result enables the asymptotics for degenerate V-statistics to be developed when the kernel is only of locally bounded variation.
Reviewer: Neville Weber (Sydney)
MSC:
| 62G20 | Asymptotic properties of nonparametric inference |
| 60F05 | Central limit and other weak theorems |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 60F17 | Functional limit theorems; invariance principles |
Keywords:
noncentral limit theorem; linear long memory sequence; quasi-Hadamard differentiability; L-statistic; U-statistics; V-statistics; weighted empirical process; modified functional delta methodCitations:
Zbl 1213.62055Software:
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