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On the asymptotic theory of subsampling. (English) Zbl 1006.62019

Summary: A general approach to constructing confidence intervals by subsampling was presented by D.N. Politis and J.P. Romano [Ann. Stat. 22, No. 4, 2031-2050 (1994; Zbl 0828.62044)]. The crux of the method is recomputing a statistic over subsamples of the data, and these recomputed values are used to build up an estimated sampling distribution. The method works under extremely weak conditions, it applies to independent, identically distributed (i.i.d.) observations as well as to dependent data situations, such as time series (possibly nonstationary), random fields, and marked point processes.
We present some theorems showing: a new construction for confidence intervals that removes a previous condition, a general theorem showing the validity of subsampling for data-dependent choices of the block size, and a general theorem for the construction of hypothesis tests (not necessarily derived from a confidence interval construction). The arguments apply to both the i.i.d. setting and the dependent data case.

MSC:

62F25 Parametric tolerance and confidence regions
62G20 Asymptotic properties of nonparametric inference
62G10 Nonparametric hypothesis testing
62F03 Parametric hypothesis testing
62G15 Nonparametric tolerance and confidence regions
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)

Citations:

Zbl 0828.62044
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