Consistency of a hybrid block bootstrap for distribution and variance estimation for sample quantiles of weakly dependent sequences

Abstract

SummaryConsistency and optimality of block bootstrap schemes for distribution and variance estimation of smooth functionals of dependent data have been thoroughly investigated by Hall, Horowitz & Jing (), among others. However, for nonsmooth functionals, such as quantiles, much less is known. Existing results, due to Sun & Lahiri (), regarding strong consistency for distribution and variance estimation via the moving block bootstrap (MBB) require that b→∞, where b=⌊n/ℓ⌋ is the number of resampled blocks to be pasted together to form the bootstrap data series, n is the available sample size, and ℓ is the block length. Here we show that, in fact, weak consistency holds for any b such that 1≤b=O(n/ℓ). In other words we show that a hybrid between the subsampling bootstrap (b=1) and MBB is consistent. Empirical results illustrate the performance of hybrid block bootstrap estimators for varying numbers of blocks.