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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.

Bounds for E׀Sn׀Q for subordinated linear processes with application to M-estimation

Furmańczyk K.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 1

129--141

Let Xj =Σ∞r=0 ArZj−r be a one-sided m-dimensional linear process, where (Zn) is a sequence of i.i.d. random vectors with zero mean and finite covariance matrix. The aim of this paper is to prove the moment inequalities of the form [formula] where G is a real function defined on Rm: The form of the constant C in (0.1) plays an important role in applications concerning the problems of M-estimation, especially the Ghosh representation.