On the asymptotic theory of subsampling

A general approach to constructing confidence intervals by subsampling was presented in Politis and Romano (1994). The crux of the method is based on 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 (LLd.) observations as well as to dependent data situations, such as time series (possible non stationary) , random fields, and marked point processes. In this article, we present some new theorems showing: a new construction for confidence intervals that removes a previous condition, a general theorem showing the validity of subsampling for datadependent choices of the block size, and a general theorem for the construction of hypothesis tests (which is not necessarily derived from a confidence interval construction). The arguments apply to both the Li.d. setting as well as the dependent data case