The Annals of Probability

The $\bm{L}_\mathbf{1}$-norm density estimator process

Evarist Giné, David M. Mason, and Andrei Yu. Zaitsev

Full-text: Open access

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 a 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 $f$.

Article information

Source
Ann. Probab. Volume 31, Number 2 (2003), 719-768.

Dates
First available in Project Euclid: 24 March 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1048516534

Digital Object Identifier
doi:10.1214/aop/1048516534

Mathematical Reviews number (MathSciNet)
MR1964947

Zentralblatt MATH identifier
1031.62026

Subjects
Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles 62G07: Density estimation

Keywords
Kernel density function estimator $L_{1}$-norm central limit theorem weak convergeance Poissonization entropy

Citation

Giné, Evarist; Mason, David M.; Zaitsev, Andrei Yu. The $\bm{L}_\mathbf{1}$-norm density estimator process. Ann. Probab. 31 (2003), no. 2, 719--768. doi:10.1214/aop/1048516534. http://projecteuclid.org/euclid.aop/1048516534.


Export citation

References

  • BARTLETT, M. S. (1938). The characteristic function of a conditional statistic. J. London Math. Soc. 13 62-67.
  • BEIRLANT, J. and MASON, D. M. (1995). On the asy mptotic normality of Lp-norms of empirical functionals. Math. Methods Statist. 4 1-19.
  • 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.
  • BERLINET, A. and DEVROy E, L. (1994). A comparison of kernel density estimates. Publ. Inst. Statist. Univ. Paris 38 3-59.
  • BICKEL, P. J. and ROSENBLATT, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071-1095.
  • BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • BORISOV, I. (2002). Moment inequalities connected with accompanying Poisson laws in Abelian groups. Preprint.
  • CSÖRG O, M. and HORVÁTH, L. (1988). Central limit theorems for Lp-norms of density estimators. Z. Wahrsch. Verw. Gebiete 80 269-291.
  • DE ACOSTA, A. (1981). Inequalities for B-valued random vectors with applications to the strong law of large numbers. Ann. Probab. 9 157-161.
  • 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.
  • DE LA PEÑA, V. and GINÉ, E. (1999). Decoupling, from Dependence to Independence. Springer, New York.
  • DEVROy E, L. (1991). Exponential inequalities in nonparametric estimation. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 31-44. Kluwer, Dordrecht.
  • DEVROy E, L. and GYÖRFI, L. (1985). Nonparametric Density Estimation: The L1 View. Wiley, New York.
  • 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.
  • DUDLEY, R. M. (1989). Real Analy sis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA.
  • DUNFORD, N. and SCHWARTZ, J. T. (1958). Linear Operators, Part I. Wiley, New York.
  • EGGERMONT, P. P. B. and LARICCIA, V. N. (2001). Maximum Penalized Likelihood Estimation 1. Density Estimation. Springer, New York.
  • EINMAHL, J. H. J. (1987). Multivariate Empirical Processes. CWI, Amsterdam.
  • EINMAHL, U. and MASON, D. M. (1997). Gaussian approximation of local empirical processes indexed by functions. Probab. Theory Related Fields 107 283-301.
  • FELLER, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • FOLLAND, G. B. (1999). Real Analy sis, 2nd ed. Wiley, New York.
  • GINÉ, E. and ZINN, J. (1984). Some limit theorems for empirical processes. Ann. Probab. 12 929- 989.
  • HOLST, L. (1979). Asy mptotic normality of sum spacings. Ann. Probab. 7 1066-1072.
  • HORVÁTH, L. (1991). On Lp-norms of multivariate density estimators. Ann. Statist. 19 1933- 1949.
  • 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.
  • LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces. Springer, Berlin.
  • MASON, D. M. and VAN ZWET, W. (1987). A refinement of the KMT inequality for the uniform empirical process. Ann. Probab. 15 871-884.
  • MONTGOMERY-SMITH, S. (1993). Comparison of sums of independent identically distributed random variables. Probab. Math. Statist. 14 281-285.
  • NABEy A, S. (1951). Absolute moments in 2-dimensional normal distributions. Ann. Inst. Statist. Math. 3 2-6.
  • 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).
  • PINELIS, I. F. (1994). On a majorization inequality for sums of independent random variables. Probab. Statist. Lett. 19 97-99.
  • 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.
  • 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).
  • SWEETING, T. J. (1977). Speeds of convergence in the multidimensional central limit theorem. Ann. Probab. 5 28-41.
  • VAN DER VAART, A. and WELLNER, J. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • 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