Abstract
Given an i.i.d. sample from a probability measure P on ℝ, an estimator is constructed that efficiently estimates P in the bounded-Lipschitz metric for weak convergence of probability measures, and, at the same time, estimates the density of P — if it exists (but without assuming it does) — at the best possible rate of convergence in total variation loss (that is, in L 1-loss for densities).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. P. Bickel and Y. Ritov, “Nonparametric Estimators which can be ‘Plugged-In’,” Ann. Statist. 31, 1033–1053 (2003).
O. Bousquet, “Concentration Inequalities for Sub-Additive Functions Using the Entropy Method”, in: Progress in Probability, Vol. 56: Stochastic Inequalities and Applications, Ed. by E. Giné, C. Houdré, and D. Nualart (Birkhäuser, Boston, 2003), pp. 213–247.
A. S. Dalalyan, G. K. Golubev, and A. B. Tsybakov, “Penalized Maximum Likelihood and Semiparametric Second-Order Efficiency”, Ann. Statist. 34, 169–201 (2006).
V. de la Peña and E. Giné, Decoupling. From Dependence to Independence (Springer, New York, 1999).
L. Devroye and G. Lugosi, Combinatorial Methods in Density Estimation (Springer, New York, 2001).
D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Density Estimation by Wavelet Thresholding”, Ann. Statist. 24, 508–539 (1996).
E. Ginée and D. M. Mason, “On Local U-Statistic Processes and the Estimation of Densities of Functions of Several Sample Variables”, Ann. Statist. 35, 1105–1145 (2007).
E. Giné and R. Nickl, “Uniform Central Limit Theorems for Kernel Density Estimators”, Probab. Theory Related Fields, 141, 333–387 (2008).
E. Giné and R. Nickl, “An Exponential Inequality for the Distribution Function of the Kernel Density Estimator, with Applications to Adaptive Estimation”, Probab. Theory Related Fields, 2008 (in press).
E. Giné and J. Zinn, “Empirical Processes Indexed by Lipschitz Functions”, Ann. Probab. 14, 1329–1338 (1986).
G. K. Golubev and B. Y. Levit, “Distribution Function Estimation: Adaptive Smoothing”, Math.Methods Statist. 5, 383–403 (1996).
A. Juditsky and S. Lambert-Lacroix, “OnMinimaxDensity Estimation on ℝ”, Bernoulli 10, 187–220 (2004).
G. Kerkyacharian, D. Picard, and K. Tribouley, “L p Adaptive Density Estimation”, Bernoulli 2, 229–247 (1996).
O. V. Lepski, “Asymptotically Minimax Adaptive Estimation. I. Upper Bounds. Optimally Adaptive Estimates”, Theory Probab. Appl. 36, 682–697 (1991).
O. V. Lepski and V.G. Spokoiny, “Optimal PointwiseAdaptiveMethods inNonparametric Estimation”, Ann. Statist. 25, 2512–2546 (1997).
R. Nickl and B. M. Pötscher, “Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov-and Sobolev-Type”, J. Theoret. Probab. 20, 177–199 (2007).
M. Talagrand, “New Concentration Inequalities in Product Spaces”, Invent. Math. 126, 505–563 (1996).
A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes (Springer, New York, 1996).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Giné, E., Nickl, R. Adaptation on the space of finite signed measures. Math. Meth. Stat. 17, 113–122 (2008). https://doi.org/10.3103/S1066530708020026
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530708020026
Keywords
- kernel density estimator
- exponential inequality
- adaptive estimation
- total variation loss
- bounded Lipschitz metric
- L 1-loss