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On a Bahadur-Kiefer representation of von mises statistic type for intermediate sample quantiles

Gribkova N., Helmers R.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 2

255--279

We investigate a Bahadur–Kiefer type representation for the pn-th empirical quantile corresponding to a sample of n i.i.d. random variables when pn ∈ (0; 1) is a sequence which, in particular, may tend to zero or one, 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 rn = kn ∧ (n − kn) → ∞, n → ∞, without any restrictions of the rate at which rn tends to infinity.We give a bound for the remainder term in the representation with probability 1−O(rn-c) for arbitrary c > 0.We obtain also an ‘almost sure’ version under the additional assumption that log n/rn → 0 as n → ∞. Finally, we establish a Bahadur–Kiefer type representation for the sum of order statistics lying between the population pn-quantile and the corresponding intermediate sample quantile by a von Mises type statistic approximation, especially useful in establishing second order approximations for slightly trimmed sums.