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Dattner, I.; Goldenshluger, A.; Juditsky, A.

On deconvolution of distribution functions. (English) Zbl 1232.62056

Ann. Stat. 39, No. 5, 2477-2501 (2011).
Summary: The subject of this paper is the problem of nonparametric estimation of a continuous distribution function from observations with measurement errors. We study the minimax complexity of this problem when the unknown distribution has a density belonging to the Sobolev class, and the error density is ordinary smooth. We develop rate optimal estimators based on direct inversion of the empirical characteristic function. We also derive minimax affine estimators of the distribution function which are given by an explicit convex optimization problem. Adaptive versions of these estimators are proposed, and some numerical results demonstrating good practical behavior of the developed procedures are presented.

Cited in 1 Review
Cited in 23 Documents

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62C20 Minimax procedures in statistical decision theory
62G05 Nonparametric estimation

Keywords:

adaptive estimator; minimax risk; rates of convergence

Software:

Mosek
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\textit{I. Dattner} et al., Ann. Stat. 39, No. 5, 2477--2501 (2011; Zbl 1232.62056)
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References:

[1] Andersen, E. D. and Andersen, K. D. The MOSEK optimization tools manual. Version 5.0. Available at . · Zbl 1185.76898
[2] Ben-Tal, A. and Nemirovski, A. (2001). Lectures on Modern Convex Optimization : Analysis, Algorithms, and Engineering Applications . SIAM, Philadelphia, PA. · Zbl 0986.90032
[3] Butucea, C. and Comte, F. (2009). Adaptive estimation of linear functionals in the convolution model and applications. Bernoulli 15 69-98. · Zbl 1200.62022
[4] Butucea, C. and Tsybakov, A. B. (2008). Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl. 52 24-39. · Zbl 1141.62021
[5] Butucea, C. and Tsybakov, A. B. (2008). Sharp optimality in density deconvolution with dominating bias. II. Theory Probab. Appl. 52 237-249. · Zbl 1142.62017
[6] Cai, T. T. and Low, M. G. (2003). A note on nonparametric estimation of linear functionals. Ann. Statist. 31 1140-1153. · Zbl 1105.62325
[7] Cai, T. T. and Low, M. G. (2004). Minimax estimation of linear functionals over nonconvex parameter spaces. Ann. Statist. 32 552-576. · Zbl 1048.62054
[8] Dattner, I., Goldenshluger, A. and Juditsky, A. (2011). Supplement to “On deconvolution of distribution functions.” . · Zbl 1232.62056
[9] Donoho, D. and Liu, R. (1987). Geometrizing rates of convergence. I. Technical Report 137a, Dept. Statistics, Univ. California, Berkeley. · Zbl 0754.62028
[10] Donoho, D. and Liu, R. (1991). Geometrizing rates of convergence. II. Ann. Statist. 19 633-667. · Zbl 0754.62028
[11] Donoho, D. and Liu, R. (1991). Geometrizing rates of convergence. III. Ann. Statist. 19 668-701. · Zbl 0754.62029
[12] Donoho, D. L. (1994). Statistical estimation and optimal recovery. Ann. Statist. 22 238-270. · Zbl 0805.62014
[13] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257-1272. · Zbl 0729.62033
[14] Gaffey, W. R. (1959). A consistent estimator of a component of a convolution. Ann. Math. Statist. 30 198-205. · Zbl 0092.36602
[15] Gil-Pelaez, J. (1951). Note on the inversion theorem. Biometrika 38 481-482. · Zbl 0045.07204
[16] Gurland, J. (1948). Inversion formulae for the distribution of ratios. Ann. Math. Statist. 19 228-237. · Zbl 0032.03403
[17] Hall, P. and Lahiri, S. N. (2008). Estimation of distributions, moments and quantiles in deconvolution problems. Ann. Statist. 36 2110-2134. · Zbl 1148.62028
[18] Johannes, J. (2009). Deconvolution with unknown error distribution. Ann. Statist. 37 2301-2323. · Zbl 1173.62018
[19] Juditsky, A. B. and Nemirovski, A. S. (2009). Nonparametric estimation by convex programming. Ann. Statist. 37 2278-2300. · Zbl 1173.62024
[20] Kendall, M., Stuart, A. and Ord, J. K. (1987). Kendall’s Advanced Theory of Statistics. Vol. 1, Distribution Theory , 5th ed. Oxford Univ. Press, New York. · Zbl 0621.62001
[21] Le Cam, L. (1973). Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 38-53. · Zbl 0255.62006
[22] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory . Springer, New York. · Zbl 0605.62002
[23] Lepskiĭ, O. V. (1990). A problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454-466. · Zbl 0725.62075
[24] Meister, A. (2009). Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics 193 . Springer, Berlin. · Zbl 1178.62028
[25] Petrov, V. V. (1995). Limit Theorems of Probability Theory : Sequences of Independent Random Variables. Oxford Studies in Probability 4 . Oxford Univ. Press, New York. · Zbl 0826.60001
[26] Spokoiny, V. G. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477-2498. · Zbl 0898.62056
[27] Stefanski, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229-235. · Zbl 0686.62026
[28] Zhang, C.-H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806-831. · Zbl 0778.62037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.