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\(U\)-processes indexed by Vapnik-Červonenkis classes of functions with applications to asymptotics and bootstrap of \(U\)-statistics with estimated parameters. (English) Zbl 0807.62014
New results on exponential inequalities, on the law of the iterated logarithm and on the bootstrap central limit theorem for \(U\)-processes indexed by so-called \(VC\) classes of functions [cf. R. M. Dudley, Lect. Notes Math. 1097, 1-142 (1984; Zbl 0554.60029)] are derived. These results are then applied to the asymptotics and the bootstrap of \(U\)- statistics with estimated parameters, in particular to the trimming of U- statistics. Earlier results are improved and extended.
Reviewer: L.Paditz (Dresden)

MSC:
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
62G20 Asymptotic properties of nonparametric inference
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[1] Alexander, K.S., Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. probab., 12, 1041-1067, (1984) · Zbl 0549.60024
[2] Arcones, M.A., The law of the iterated logarithm for U-processes, J. mult. anal., 47, 139-151, (1993) · Zbl 0780.60030
[3] Arcones, M.A.; Giné, E., On the bootstrap of U and V-statistics, Ann. statist., 20, 655-674, (1992) · Zbl 0760.62018
[4] Arcones, M.A.; Giné, E., Limit theorems for U-processes, Ann. probab., 21, 1494-1542, (1993), (1991) · Zbl 0789.60031
[5] Arcones, M.A.; Giné, E., On the bootstrap of M-estimators and other statistical functionals, (), 13-47 · Zbl 0842.62027
[6] Borell, C., On the integrability of Banach space valued Walsh polynomials, (), 1-3
[7] Bretagnolle, J., Lois limites du bootstrap de certaines fonctionelles, Ann. inst. Henri Poincaré, 19, 281-296, (1983) · Zbl 0535.62046
[8] R.M. Dudley, A course on empirical processes, Lecture Notes in Mathematics Vol. 1097 (Springer, New York) 1-142.
[9] Ghosh, J.K., A new proof of the bahadur representation of quantiles and an application, Ann statist., 42, 1957-1961, (1971) · Zbl 0235.62006
[10] Gijbels, I.; Janssen, P.; Veraverbeke, N., Weak and strong representation for trimmed U-statistics, Probab. th. rel. fields, 77, 179-194, (1988) · Zbl 0621.60034
[11] Giné, E.; Zinn, J., Lectures on the central limit theorem for empirical processes, (), 50-113
[12] Giné, E.; Zinn, J., Bootstrapping general empirical measures, Ann. probab., 8, 851-869, (1990) · Zbl 0706.62017
[13] Helmers, R.; Janseen, P.; Sherfling, R., Glivenko-Cantelli properties of some generalized empirical DF’s and strong convergence of generalized L-statistics, Probab. th. rel. fields, 79, 75-93, (1988) · Zbl 0631.60032
[14] Hoffmann-Jørgensen, J., Stochastic processes on Polish spaces, (1991), Matematisk Inst., Aarhus U, Various Publications Series No. 39 · Zbl 0919.60003
[15] Janssen, P., Generalized empirical distribution functions with statistical applications, ()
[16] Janssen, P.; Serfling, R.; Veraverbeke, N., Asymptotic normality of U-statistics based on trimmed samples, J. statist. planning inf., 16, 63-74, (1987) · Zbl 0617.62016
[17] Janssen, P.; Veraverbeke, N., Bootstrapping U-statistics with estimated parameters, Comm. statist. theor. method., (1991), to appear in · Zbl 0775.62108
[18] Kuelbs, J.; Dudley, R.M., Log log laws for empirical measures, Ann. probab., 8, 405-418, (1980) · Zbl 0442.60031
[19] de la Peña, V., Decoupling and Khintchine’s inequalities for U-statistics, Ann probab., 20, 1877-1892, (1992) · Zbl 0761.60014
[20] Pollard, D., Convergence of stochastic processes, (1984), Springer New York · Zbl 0544.60045
[21] Randles, R.M., On the asymptotic normality of statistics with estimated parameters, Ann. statist., 10, 462, (1982) · Zbl 0493.62022
[22] Schneemeier, W., Weak convergence and Glivenko-Cantelli results for empirical processes of U-statistic structure, Stoch. proc. appl., 33, 325-334, (1989) · Zbl 0685.60037
[23] Sukhatme, B.V., Testing the hypothesis that two populations differ only on location, Ann. math. statist., 29, 60-78, (1958) · Zbl 0085.35405
[24] de Wet, T.; Randles, R.M., On the effect of substituting parameter estimators in limiting X2 and V statistics, Ann. statist., 15, 398-412, (1987) · Zbl 0617.62017
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