Electronic Journal of Statistics

Adaptive density estimation in deconvolution problems with unknown error distribution

Johanna Kappus and Gwennaëlle Mabon

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We investigate the data driven choice of the cutoff parameter in density deconvolution problems with unknown error distribution. To make the target density identifiable, one has to assume that some additional information on the noise is available. We consider two different models: the framework where some additional sample of the pure noise is available, as well as the model of repeated measurements, where the contaminated random variables of interest can be observed repeatedly, with independent errors. We introduce spectral cutoff estimators and present upper risk bounds. The focus of this work lies on the optimal choice of the bandwidth by penalization strategies, leading to non-asymptotic oracle bounds.

Article information

Electron. J. Statist., Volume 8, Number 2 (2014), 2879-2904.

First available in Project Euclid: 8 January 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G99: None of the above, but in this section

Adaptive estimation deconvolution density estimation mean squared risk nonparametric methods replicate observations


Kappus, Johanna; Mabon, Gwennaëlle. Adaptive density estimation in deconvolution problems with unknown error distribution. Electron. J. Statist. 8 (2014), no. 2, 2879--2904. doi:10.1214/14-EJS976. https://projecteuclid.org/euclid.ejs/1420726194

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  • [1] Baudry, J., Maugis, C., and Michel, B. (2012). Slope heuristics: Overview and implementation., Statistics and Computing, 22(2):455–470.
  • [2] Birgé, L. (1999). An alternative point of view on Lepski’s method. In, State of the Art in Probability and Statistics: Festschrift for Willem R. van Zwet, pages 113–133. IMS Lecture Notes-Monograph Series.
  • [3] Birgé, L. and Massart, P. (1997). From model selection to adaptive estimation. In, Festschrift for Lucien Le Cam, pages 55–87. Springer, New York.
  • [4] Bonhomme, S. and Robin, J.-M. (2010). Generalized non-parametric deconvolution with an application to earnings dynamics., Review of Economic Studies, 77(2):491–533.
  • [5] Butucea, C. (2004). Deconvolution of supersmooth densities with smooth noise., The Canadian Journal of Statistics, 32(2):181–192.
  • [6] Butucea, C. and Comte, F. (2009). Adaptive estimation of linear functionals in the convolution model and applications., Bernoulli, 15(1):69–98.
  • [7] Butucea, C. and Tsybakov, A. (2008a). Sharp optimality in density deconvolution with dominating bias I., Theory Proba. Appl., 52(1):24–39.
  • [8] Butucea, C. and Tsybakov, A. (2008b). Sharp optimality in density deconvolution with dominating bias II., Theory Proba. Appl., 52(2):237–249.
  • [9] Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density., Journal of the American Statistical Association, 83(404):1184–1186.
  • [10] Comte, F. and Kappus, J. (2014). Density deconvolution from repeated measurements without symmetry assumption on the errors. Preprint, hal-01010409.
  • [11] Comte, F. and Lacour, C. (2011). Data-driven density estimation in the presence of additive noise with unknown distribution., Journal of the Royal Statistical Society: Series B, 73:601–627.
  • [12] Comte, F. and Lacour, C. (2013). Anisotropic adaptive kernel deconvolution., Ann. Inst. H. Poincaré Probab. Statist., 49(2):569–609.
  • [13] Comte, F., Rozenholc, Y., and Taupin, M.-L. (2006). Penalized contrast estimator for adaptive density deconvolution., Canadian Journal of Statistics, 3(34):431–452.
  • [14] Comte, F., Rozenholc, Y., and Taupin, M.-L. (2007). Finite sample penalization in adaptive density deconvolution., Journal of Statistical Computation and Simulation, 77(11):977–1000.
  • [15] Comte, F. and Samson, A. (2012). Nonparametric estimation of random-effects densities in linear mixed-effects model., Journal of Nonparametric Statistics, 24(4):951–975.
  • [16] Comte, F., Samson, A., and Stirnemann, J. (2012). Deconvolution estimation of onset of pregnancy with replicate observations., Scandinavian Journal of Statistics, 41(2):325–345.
  • [17] Dattner, I., Reiß, M., and Trabs, M. (2013). Adaptive quantile estimation in deconvolution with unknown error distribution., arXiv:1303.1698.
  • [18] Delaigle, A., Hall, P., and Meister, A. (2008). On deconvolution with repeated measurements., The Annals of Statistics, 36(2):665–685.
  • [19] Dion, C. (2013). New strategies for nonparametric estimation in a linear mixed model., Journal of Statistical Planning and Inference, 150:30–48.
  • [20] Efromovich, S. (1997). Density estimation for the case of supersmooth measurement errors., Journal of the American Statistical Association, 92:526–535.
  • [21] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems., The Annals of Statistics, 19(3):1257–1272.
  • [22] Johannes, J. (2009). Deconvolution with unknown error distribution., The Annals of Statistics, 37(5a):2301–2323.
  • [23] Johannes, J. and Schwarz, M. (2013). Adaptive circular deconvolution by model selection under unknown error distribution., Bernoulli, 19(5A):1576–1611.
  • [24] Kappus, J. (2014). Adaptive nonparametric estimation for Lévy processes observed at low frequency., Stochastic Processes and Their Applications, 124:730–758.
  • [25] Li, T. and Vuong, Q. (1998). Nonparametric estimation of the measurement error model using multiple indicators., Journal of Multivariate Analysis, 65:139–165.
  • [26] Massart, P. (2003)., Concentration Inequalities and Model Selection. Number 1896 in Lecture Notes in Mathematics. Springer.
  • [27] Meister, A. (2009)., Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics. Springer.
  • [28] Neumann, M. (2007). Deconvolution from panel data with unknown error distribution., Journal of Multivariate Analysis, 98(10):1955–1968.
  • [29] Neumann, M. and Reiß, M. (2009). Nonparametric estimation for Lévy processes from low frequency observations., Bernoulli, 15(1):223–248.
  • [30] Neumann, M. H. (1997). On the effect of estimating the error density in nonparametric deconvolution., Journal of Nonparametric Statistics, 7(4):307–330.
  • [31] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution., The Annals of Statistics, 27(6):2033–2053.
  • [32] Stefanski, L. (1990). Rates of convergence of some estimators in a class of deconvolution problems., Statistics and Probability Letters, 9(3):229–235.
  • [33] Stefanski, S. and Carroll, R. (1990). Deconvoluting kernel density estimators., Statistics, 21(2):169–184.