## Electronic Journal of Statistics

### Improved rates for Wasserstein deconvolution with ordinary smooth error in dimension one

#### Abstract

This paper deals with the estimation of a probability measure on the real line from data observed with an additive noise. We are interested in rates of convergence for the Wasserstein metric of order $p\geq1$. The distribution of the errors is assumed to be known and to belong to a class of supersmooth or ordinary smooth distributions. We obtain in the univariate situation an improved upper bound in the ordinary smooth case and less restrictive conditions for the existing bound in the supersmooth one. In the ordinary smooth case, a lower bound is also provided, and numerical experiments illustrating the rates of convergence are presented.

#### Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 234-265.

Dates
First available in Project Euclid: 17 February 2015

https://projecteuclid.org/euclid.ejs/1424187776

Digital Object Identifier
doi:10.1214/15-EJS997

Mathematical Reviews number (MathSciNet)
MR3314482

Zentralblatt MATH identifier
1307.62092

Subjects
Primary: 62G05: Estimation 62C20: Minimax procedures

#### Citation

Dedecker, Jérôme; Fischer, Aurélie; Michel, Bertrand. Improved rates for Wasserstein deconvolution with ordinary smooth error in dimension one. Electron. J. Statist. 9 (2015), no. 1, 234--265. doi:10.1214/15-EJS997. https://projecteuclid.org/euclid.ejs/1424187776

#### References

• [1] Bobkov, S. and Ledoux, M., One-dimensional empirical measures, order statistics and Kantorovich transport distances., Preprint, 2014.
• [2] Butucea, C. and Tsybakov, B., Sharp optimality in density deconvolution with dominating bias. I., Theory Probab. Appl., 52:24–39, 2008a.
• [3] Butucea, C. and Tsybakov, B., Sharp optimality in density deconvolution with dominating bias. II., Theory Probab. Appl., 52:237–249, 2008b.
• [4] Caillerie, C., Chazal, F., Dedecker, J., and Michel, B., Deconvolution for the Wasserstein metric and geometric inference., Electron. J. Stat., 5 :1394–1423, 2011.
• [5] Carlsson, G., Topology and data., Bull. Amer. Math. Soc., 46:255–308, 2009.
• [6] Carroll, R.J. and Hall, P., Optimal rates of convergence for deconvolving a density., J. Amer. Statist. Assoc., 83 :1184–1186, 1988.
• [7] Chazal, F., Cohen-Steiner, D., and Mérigot, Q., Geometric inference for probability measures., Found. Comput. Math., 11:733–751, 2011.
• [8] Chazal, F., Fasy, B.T., Lecci, F., Michel, B., Rinaldo, A., and Wasserman, L., Subsampling methods for persistent homology., arXiv :1406.1901, 2014.
• [9] Dattner, I., Goldenshluger, A., and Juditsky, A., On deconvolution of distribution functions., Ann. Statist., 39 :2477–2501, 2011.
• [10] Dedecker, J. and Michel, B., Minimax rates of convergence for Wasserstein deconvolution with supersmooth errors in any dimension., J. Multivar. Anal., 122:278–291, 2013.
• [11] del Barrio, E., Giné, E., and Matrán, C., The central limit theorem for the Wasserstein distance between the empirical and the true distributions., Ann. Probab., 27 :1009–1971, 1999.
• [12] del Barrio, E., Giné, E., and Utzet, F., Asymptotics for $\mathbbL_2$ functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances., Bernoulli, 11:131–189, 2005.
• [13] Delaigle, A. and Gijbels, I., Bootstrap bandwidth selection in kernel density estimation from a contaminated sample., Ann. I. Stat. Math., 56(1):19–47, 2004.
• [14] Dereich, S., Scheutzow, M., and Schottstedt, R., Constructive quantization: Approximation by empirical measures., Ann. Inst. H. Poincaré Probab. Statist., 49 :1183–1203, 2013.
• [15] Èbralidze, Š.S., Inequalities for the probabilities of large deviations in terms of pseudomoments., Teor. Verojatnost. i Primenen., 16:760–765, 1971.
• [16] Fan, J., Global behavior of deconvolution kernel estimates., Statist. Sinica, 2:541–551, 1991a.
• [17] Fan, J., On the optimal rates of convergence for nonparametric deconvolution problems., Ann. Stat., 19 :1257–1272, 1991b.
• [18] Fan, J., Adaptively local one-dimensional subproblems with application to a deconvolution problem., Ann. Stat., 21:600–610, 1993.
• [19] Fournier, N. and Guillin, A., On the rate of convergence in Wasserstein distance of the empirical measure., To appear in Probability Theory and Related Fields, 2014.
• [20] Guibas, L., Morozov, D., and Mérigot, Q., Witnessed k-distance., Discrete Comput. Geom., 49:22–45, 2013.
• [21] Hall, P. and Lahiri, S.N., Estimation of distributions, moments and quantiles in deconvolution problems., Ann. Statist, 36 :2110–2134, 2008.
• [22] Mair, P., Hornik, K., and de Leeuw, J., Isotone optimization in R: pool-adjacent-violators algorithm (PAVA) and active set methods., J. Stat. Softw., 32(5):1–24, 2009.
• [23] Meister, A., Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics. Springer, 2009.
• [24] Rachev, S.T. and Rüschendorf, L., Mass Transportation Problems, volume II of Probability and Its Applications. Springer-Verlag, 1998.
• [25] van der Vaart, A.W. and Wellner, J.A., Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, 1996.
• [26] Villani, C., Optimal Transport: Old and New. Grundlehren Der Mathematischen Wissenschaften. Springer-Verlag, 2008.