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Consistency of a hybrid block bootstrap for distribution and variance estimation for sample quantiles of weakly dependent sequences
Author(s) -
Kuffner Todd A.,
Lee Stephen M. S.,
Young G. A.
Publication year - 2018
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/anzs.12206
Subject(s) - mathematics , quantile , consistency (knowledge bases) , estimator , variance (accounting) , block (permutation group theory) , statistics , distribution (mathematics) , strong consistency , series (stratigraphy) , sample variance , sample (material) , sample size determination , combinatorics , discrete mathematics , mathematical analysis , paleontology , chemistry , accounting , chromatography , business , biology
Summary Consistency 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 ([Hall, P., 1995]), among others. However, for nonsmooth functionals, such as quantiles, much less is known. Existing results, due to Sun & Lahiri ([Sun, S., 2006]), 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.