On the Bahadur - Kiefer Representation for Intermediate Sample Quantiles

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 $\text{kn}$-th order statistic when $r_n=\text{kn}\wedge (n-k_n)\to \mathrm{\infty }$ under some mild regularity conditions, and an ‘almost sure’ version under additional assumption that $\log n\mathrm{/}{r}_n\to 0$, $\text{nty}$. A representation for the sum of order statistics laying between the population p-quantile and the corresponding empirical quantile is also established.