Sankhya: The Indian Journal of Statistics

2006, Volume 68, Pt. 1, 130--166

Posterior

Bootstrapping the Sample Quantile of a Weakly Dependent Sequence

Shuxia Sun, Wright State University, Dayton, USA
Soumendra N. Lahiri, Iowa State University, Ames, USA

SUMMARY. In this paper, we investigate consistency properties of block bootstrap approximations  for sample quantiles of weakly dependent data. Under mild weak dependence conditions and mild smoothness conditions on the one-dimensional marginal distribution function, we  show that   the moving block bootstrap method %of K$\mathrm{\ddot{u}}$nsch (1989) and Liu and Singh (1992) provides a valid approximation to the distribution of normalized sample quantile in the almost sure sense. Strong consistency of the block bootstrap estimator of the asymptotic variance of the sample quantile is   also established under similar conditions. For the proof, we develop some exponential inequalities for block bootstrap moments and also develop some almost sure bounds on the oscillations of the empirical distribution function of strongly mixing random variables,  which may be of some independent interest.

AMS (1991) subject classification. Primary 62M10, 62G09.

Key words and phrases. Weakly dependent, moving block bootstrap, sample quantile.

Full paper (PDF)