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

• Autorzy: Gribkova N. , Helmers R.
• Języki publikacji: EN

Abstrakty

EN

We investigate a Bahadur–Kiefer type representation for the p_n-th empirical quantile corresponding to a sample of n i.i.d. random variables when p_n ∈ (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 k_n-th order statistic when r_n = k_n ∧ (n − k_n) → ∞, 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(r_n^-c) for arbitrary c > 0.We obtain also an ‘almost sure’ version under the additional assumption that log n/r_n → 0 as n → ∞. Finally, we establish a Bahadur–Kiefer type representation for the sum of order statistics lying between the population p_n-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.

Słowa kluczowe

EN

Bahadur–Kiefer type representation; intermediate sample quantiles; Bahadur–Kiefer processes; empirical processes; quantile processes; von Mises statistic type approximation; slightly trimmed sum

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2012
• Tom: Vol. 32, Fasc. 2
• Strony: 255--279
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Gribkova N.

• St. Petersburg State University, Mathematics and Mechanics Faculty, 198504, St. Petersburg, Stary Peterhof, Universitetsky pr. 28, Russia

autor

Helmers R.

• Centre for Mathematics and Computer Science, P.O. Box 94079, 1090 GB, Amsterdam, The Netherlands

Bibliografia

• [1] R. R. Bahadur, A note on quantiles in large samples, Ann. Math. Statist. 37 (1966), pp. 577-580.
• [2] J. Beirlant, P. Deheuvels, J. H. J. Einmahl, and D. M. Mason, Bahadur-Kiefer theorems for uniform spacings processes, Theory Probab. Appl. 36 (1991), pp. 724-743.
• [3] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge Univ. Press (Encyclopedia Math. Appl.) 27, Cambridge 1987.
• [4] K. C. Chanda, Bahadur-Kiefer representation properties of intermediate order statistics, Statist. Probab. Lett. 14 (1992), pp. 175-178.
• [5] M. Csörgő, S. Csörgő, L. Horváth, and D. M. Mason, Weighted empirical and quantile processes, Ann. Probab. 14 (1986), pp. 31-85.
• [6] P. Deheuvels, Strong approximation of quantile processes by iterated Kiefer processes, Ann. Probab. 28 (2000), pp. 909-945.
• [7] P. Deheuvels, A multivariate Bahadur-Kiefer representation for the empirical copula process, Zap. Nauchn. Sem. POMI 364 (2009), pp. 120-147. Translation in: J. Math. Sci. (N.Y.) 163 (4) (2010), pp. 382-398.
• [8] P. Deheuvels and D.M. Mason, Bahadur-Kiefer-type processes, Ann. Probab. 18 (1990), pp. 669-697.
• [9] J. H. J. Einmahl and D. M. Mason, Strong limit theorems for weighted quantile processes, Ann. Probab. 16 (1988), pp. 1623-1643.
• [10] N. V. Gribkova and R. Helmers, The empirical Edgeworth expansion for a Studentized trimmed mean, Math. Methods Statist. 15 (2006), pp. 61-87.
• [11] N. V. Gribkova and R. Helmers, On the Edgeworth expansion and the M out of N bootstrap accuracy for a Studentized trimmed mean, Math. Methods Statist. 16 (2007), pp. 142-176.
• [12] N. V. Gribkova and R. Helmers, Second order approximations for slightly trimmed sums, Theory Probab. Appl. 58 (2013) (to appear).
• [13] P. S. Griffin and W. E. Pruitt, Asymptotic normality and subsequential limits of trimmed sums, Ann. Probab. 17 (1989), pp. 1186-1219.
• [14] W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), pp. 13-30.
• [15] O. Kallenberg, Foundations of Modern Probability, Springer, New York 2002.
• [16] J. Kiefer, On Bahadur’s representation of sample quantiles, Ann. Math. Statist. 38 (1967), pp. 1323-1342.
• [17] J. Kiefer, Deviations between the sample quantile process and the sample d.f., in: Nonparametric Techniques in Statistical Inference, M. L. Puri (Ed.), Cambridge Univ. Press, London 1970, pp. 299-319.
• [18] J. Kiefer, Old and new methods for studying order statistics and sample quantiles, in: Nonparametric Techniques in Statistical Inference, M. L. Puri (Ed.), Cambridge Univ. Press, London 1970, pp. 349-357.
• [19] C. A. Leon and F. Perron, Extremal properties of sums of Bernoulli random variables, Statist. Probab. Lett. 62 (2003), pp. 345-354.
• [20] P. K. Sen, Asymptotic normality of sample quantiles for m-dependent processes, Ann. Math. Statist. 39 (1968), pp. 1724-1730.
• [21] G. R. Shorack and J. A. Wellner, Empirical Processes with Application to Statistics, Wiley, New York 1986.
• [22] M. Talagrand, The missing factor in Hoeffding’s inequalities, Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), pp. 689-702.
• [23] V. Watts, The almost sure representation of intermediate order statistics, Z. Wahrsch. Verw. Gebiete 54 (1980), pp. 281-285.
• [24] W. B. Wu, On the Bahadur representation of sample quantiles for dependent sequences, Ann. Statist. 33 (2005), pp. 1934-1963.