Exact rates in density support estimation

  • Authors:

  • Gérard Biau;Benoít Cadre;Bruno Pelletier

  • Affiliations:

  • LSTA & LPMA, Universitéé Pierre et Marie Curie-Paris VI, Boíte 158, 175 rue du Chevaleret, 75013 Paris, France;IRMAR, ENS Cachan Bretagne, CNRS, UEB, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France;Institut de Mathématiques et de Modélisation de Montpellier, UMR CNRS 5149, Equipe de Probabilités et Statistique, Université Montpellier II, CC 051, Place Eugène Bataillo ...

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2008

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Abstract

Let f be an unknown multivariate probability density with compact support S"f. Given n independent observations X"1,...,X"n drawn from f, this paper is devoted to the study of the estimator S@?"n of S"f defined as unions of balls centered at the X"i and of common radius r"n. To measure the proximity between S@?"n and S"f, we employ a general criterion d"g, based on some function g, which encompasses many statistical situations of interest. Under mild assumptions on the sequence (r"n) and some analytic conditions on f and g, the exact rates of convergence of d"g(S@?"n,S"f) are obtained using tools from Riemannian geometry. The conditions on the radius sequence are found to be sharp and consequences of the results are discussed from a statistical perspective.