Annals of Statistics

Estimating a concave distribution function from data corrupted with additive noise

Geurt Jongbloed and Frank H. van der Meulen

Full-text: Open access


We consider two nonparametric procedures for estimating a concave distribution function based on data corrupted with additive noise generated by a bounded decreasing density on (0, ∞). For the maximum likelihood (ML) estimator and least squares (LS) estimator, we state qualitative properties, prove consistency and propose a computational algorithm. For the LS estimator and its derivative, we also derive the pointwise asymptotic distribution. Moreover, the rate n−2/5 achieved by the LS estimator is shown to be minimax for estimating the distribution function at a fixed point.

Article information

Ann. Statist., Volume 37, Number 2 (2009), 782-815.

First available in Project Euclid: 10 March 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62G05: Estimation

Asymptotic distribution deconvolution decreasing density least squares maximum likelihood minimax risk


Jongbloed, Geurt; van der Meulen, Frank H. Estimating a concave distribution function from data corrupted with additive noise. Ann. Statist. 37 (2009), no. 2, 782--815. doi:10.1214/07-AOS579.

Export citation


  • Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
  • Chernoff, H. (1964). Estimation of the mode. Ann. Statist. Math. 16 31–41.
  • Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249–267.
  • Delaigle, A. and Hall, P. (2006). On optimal kernel choice for deconvolution. Statist. Probab. Lett. 76 1594–1602.
  • Dümbgen, L. and Rufibach, K. (2004). Maximum likelihood estimation of a log-concave density: Basic properties and uniform consistency. Preprint.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
  • Gripenberg, G., Londen, S.-O. and Staffans, O. (1990). Volterra Integral and Functional Equations. Cambridge Univ. Press, Cambridge.
  • Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Boston.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). A canonical process for estimation of convex functions: “The invelope” of integrated Brownian motion + t4. Ann. Statist. 29 1620–1652.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653–1698.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2008). The support reduction algorithm for computing nonparametric function estimates in mixture models. Scand. J. Statist. 35 385–399.
  • Jongbloed, G. (2000). Minimax lower bounds and moduli of continuity. Statist. Probab. Lett. 50 279–284.
  • Jongbloed, G., van der Meulen, F. H. and van der Vaart, A. W. (2005). Nonparametric inference for Lévy-driven Ornstein–Uhlenbeck processes. Bernoulli 11 759–791.
  • Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219.
  • Mammen, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741–759.
  • Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033–2053.
  • Pipkin, A. C. (1991). A Course on Integral Equations. Springer, New York.
  • Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York.
  • Stefanski, L. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 2 169–184.
  • van de Geer, S. A. (2000). Empirical Processes in M-Estimation. Cambridge Univ. Press, Cambridge.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • Van Es, B. J., Jongbloed, G. and van Zuijlen, M. C. A. (1998). Isotonic inverse estimators for nonparametric deconvolution. Ann. Statist. 26 2395–2406.