On the Bahadur - Kiefer Representation for Intermediate Sample Quantiles

TL;DR

This paper extends the Bahadur-Kiefer representation to intermediate sample quantiles, providing probabilistic and almost sure versions under mild conditions, and analyzes the sum of order statistics between population and empirical quantiles.

Contribution

It introduces new probabilistic and almost sure Bahadur-Kiefer representations for intermediate sample quantiles, broadening understanding of their asymptotic behavior.

Findings

Established in probability Bahadur-Kiefer representation for intermediate quantiles.

Proved almost sure version under additional regularity conditions.

Derived representation for sum of order statistics between population and empirical quantiles.

Abstract

We investigate a Bahadur-Kiefer type representation for the p-th empirical quantile corresponding to a sample of n i.i.d. random variables, when 0<p<1 is a sequence which, in particular, may tend to 0 or 1, i.e. we consider the case of intermediate sample quantiles. We obtain an 'in probability' version of the Bahadur -- Kiefer type representation for a $\kn$-th order statistic when $r_n=\kn\wedge (n-k_n)\to \infty$ under some mild regularity conditions, and an 'almost sure' version under additional assumption that $\log n/r_n\to 0$, $\nty$. A representation for the sum of order statistics laying between the population p-quantile and the corresponding empirical quantile is also established.