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The \(L_1\)-norm density estimator process. (English) Zbl 1031.62026

Authors’ abstract: The notion of an \(L_1\)-norm density estimator process indexed by a class of kernels is introduced. Then a functional central limit theorem and Glivenko-Cantelli theorem are established for this process. While assembling the necessary machinery to prove these results, a body of Poissonization techniques and restricted chaining methods is developed, which is useful for studying weak convergence of general processes indexed by a class of functions.
None of the theorems imposes any condition at all on the underlying Lebesgue density \(f\). Also, somewhat unexpectedly, the distribution of the limiting Gaussian process does not depend on \(f\).

MSC:

62G07 Density estimation
60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
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[1] BARTLETT, M. S. (1938). The characteristic function of a conditional statistic. J. London Math. Soc. 13 62-67. · Zbl 0018.22503
[2] BEIRLANT, J. and MASON, D. M. (1995). On the asy mptotic normality of Lp-norms of empirical functionals. Math. Methods Statist. 4 1-19. · Zbl 0831.62019
[3] BEIRLANT, J., GYÖRFI, L. and LUGOSI, G. (1994). On the asy mptotic normality of L1and L2-errors in histogram density estimation. Canad. J. Statist. 22 309-318. JSTOR: · Zbl 0816.62037
[4] BERLINET, A. and DEVROy E, L. (1994). A comparison of kernel density estimates. Publ. Inst. Statist. Univ. Paris 38 3-59. · Zbl 0804.62039
[5] BICKEL, P. J. and ROSENBLATT, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071-1095. · Zbl 0275.62033
[6] BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[7] BORISOV, I. (2002). Moment inequalities connected with accompanying Poisson laws in Abelian groups. · Zbl 1031.60005
[8] CSÖRG O, M. and HORVÁTH, L. (1988). Central limit theorems for Lp-norms of density estimators. Z. Wahrsch. Verw. Gebiete 80 269-291. · Zbl 0657.60026
[9] DE ACOSTA, A. (1981). Inequalities for B-valued random vectors with applications to the strong law of large numbers. Ann. Probab. 9 157-161. · Zbl 0449.60002
[10] DEHEUVELS, P. and MASON, D. M. (1992). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20 1248-1287. · Zbl 0767.60028
[11] DE LA PEÑA, V. and GINÉ, E. (1999). Decoupling, from Dependence to Independence. Springer, New York. · Zbl 0918.60021
[12] DEVROy E, L. (1991). Exponential inequalities in nonparametric estimation. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 31-44. Kluwer, Dordrecht. · Zbl 0739.62025
[13] DEVROy E, L. and GYÖRFI, L. (1985). Nonparametric Density Estimation: The L1 View. Wiley, New York. · Zbl 0546.62015
[14] DUDLEY, R. M. (1984). A course on empirical processes. École d’été de Probabilités de Saint-Flour XII. Lecture Notes in Math. 1097 1-142. Springer, Berlin. · Zbl 0554.60029
[15] DUDLEY, R. M. (1989). Real Analy sis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA.
[16] DUNFORD, N. and SCHWARTZ, J. T. (1958). Linear Operators, Part I. Wiley, New York. · Zbl 0635.47001
[17] EGGERMONT, P. P. B. and LARICCIA, V. N. (2001). Maximum Penalized Likelihood Estimation 1. Density Estimation. Springer, New York. · Zbl 0984.62026
[18] EINMAHL, J. H. J. (1987). Multivariate Empirical Processes. CWI, Amsterdam. · Zbl 0619.60031
[19] EINMAHL, U. and MASON, D. M. (1997). Gaussian approximation of local empirical processes indexed by functions. Probab. Theory Related Fields 107 283-301. · Zbl 0878.60025
[20] FELLER, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York. · Zbl 0138.10207
[21] FOLLAND, G. B. (1999). Real Analy sis, 2nd ed. Wiley, New York.
[22] GINÉ, E. and ZINN, J. (1984). Some limit theorems for empirical processes. Ann. Probab. 12 929- 989. · Zbl 0553.60037
[23] HOLST, L. (1979). Asy mptotic normality of sum spacings. Ann. Probab. 7 1066-1072. · Zbl 0421.60017
[24] HORVÁTH, L. (1991). On Lp-norms of multivariate density estimators. Ann. Statist. 19 1933- 1949. · Zbl 0765.62045
[25] JOHNSON, W. B., SCHECHTMAN, G. and ZINN, J. (1985). Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13 234-253. · Zbl 0564.60020
[26] LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces. Springer, Berlin. · Zbl 0748.60004
[27] MASON, D. M. and VAN ZWET, W. (1987). A refinement of the KMT inequality for the uniform empirical process. Ann. Probab. 15 871-884. · Zbl 0638.60040
[28] MONTGOMERY-SMITH, S. (1993). Comparison of sums of independent identically distributed random variables. Probab. Math. Statist. 14 281-285. · Zbl 0827.60005
[29] NABEy A, S. (1951). Absolute moments in 2-dimensional normal distributions. Ann. Inst. Statist. Math. 3 2-6. · Zbl 0045.07005
[30] PINELIS, I. F. (1990). Inequalities for sums of independent random vectors and their application to estimating a density. Theory Probab. Appl. 35 605-607 (translated from Russian). · Zbl 0733.62043
[31] PINELIS, I. F. (1994). On a majorization inequality for sums of independent random variables. Probab. Statist. Lett. 19 97-99. · Zbl 0801.60010
[32] Py KE, R. and SHORACK, G. R. (1968). Weak convergence of a two-sample empirical process and a new approach to Chernoff-Savage theorems. Ann. Math. Statist. 39 755-771. · Zbl 0159.48004
[33] SHERGIN, V. V. (1979). On the convergence rate in the central limit theorem for m-dependent random variables. Theory Probab. Appl. 24 782-796 (translated from Russian). · Zbl 0437.60018
[34] SWEETING, T. J. (1977). Speeds of convergence in the multidimensional central limit theorem. Ann. Probab. 5 28-41. · Zbl 0362.60041
[35] VAN DER VAART, A. and WELLNER, J. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[36] STORRS, CONNECTICUT 06269-3009 E-MAIL: gine@uconnvm.uconn.edu D. M. MASON DEPARTMENT OF FOOD AND RESOURCE ECONOMICS 206 TOWNSEND HALL UNIVERSITY OF DELAWARE NEWARK, DELAWARE 19717 E-MAIL: davidm@udel.edu A. YU. ZAITSEV LABORATORY OF STATISTICAL METHODS ST. PETERSBURG BRANCH OF THE STEKLOV MATHEMATICAL INSTITUTE 27 FONTANKA ST. PETERSBURG 191011 RUSSIA E-MAIL: zaitsev@pdmi.ras.ru
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