×

Estimation of distributions, moments and quantiles in deconvolution problems. (English) Zbl 1148.62028

Summary: When using the bootstrap in the presence of measurement error, we must first estimate the target distribution function; we cannot directly resample, since we do not have a sample from the target. These and other considerations motivate the development of estimators of distributions, and of related quantities such as moments and quantiles, in errors-in-variables settings. We show that such estimators have curious and unexpected properties. For example, if the distributions of the variable of interest, \(W\), say, and of the observation error are both centered at zero, then the rate of convergence of an estimator of the distribution function of \(W\) can be slower at the origin than away from the origin. This is an intrinsic characteristic of the problem, not a quirk of particular estimators; the property holds true for optimal estimators.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G09 Nonparametric statistical resampling methods
62G07 Density estimation
65C60 Computational problems in statistics (MSC2010)
62C20 Minimax procedures in statistical decision theory
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Booth, J. G. and Hall, P. (1993). Bootstrap confidence regions for functional relationships in errors-in-variables models. Ann. Statist. 21 1780-1791. · Zbl 0789.62035
[2] Butucea, C. (2004). Deconvolution of supersmooth densities with smooth noise. Canad. J. Statist. 32 181-192. JSTOR: · Zbl 1056.62047
[3] Butucea, C. and Tsybakov, A. B. (2008). Sharp optimality for density deconvolution with dominating bias. Theory Probab. Appl. · Zbl 1141.62021
[4] Carroll, R.J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184-1186. JSTOR: · Zbl 0673.62033
[5] Cordy, C. and Thomas, D. R. (1997). Deconvolution of a distribution function. J. Amer. Statist. Assoc. 92 1459-1465. JSTOR: · Zbl 0912.62030
[6] Cui, H. (2005). Asymptotics of mean transformation estimators with errors in variables model. J. Syst. Sci. Complex. 18 446-455. · Zbl 1076.62031
[7] Delaigle, A. and Gijbels, I. (2002). Estimation of integrated squared density derivatives from a contaminated sample. J. Roy. Statist. Soc. Ser. B 64 869-886. JSTOR: · Zbl 1067.62034
[8] Delaigle, A. and Gijbels, I. (2004a). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249-267. · Zbl 1429.62125
[9] Delaigle, A. and Gijbels, I. (2004b). Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Ann. Inst. Statist. Math. 56 19-47. · Zbl 1050.62038
[10] Delaigle, A. and Hall, P. (2006). On optimal kernel choice for deconvolution. Statist. Probab. Lett. 76 1594-1602. · Zbl 1099.62035
[11] Devroye, L. (1989). Consistent deconvolution in density estimation. Canad. J. Statist. 17 235-239. JSTOR: · Zbl 0679.62029
[12] Diggle, P. J. and Hall, P. (1993). A Fourier approach to nonparametric deconvolution of a density estimate. J. Roy. Statist. Soc. Ser. B 55 523-531. JSTOR: · Zbl 0783.62030
[13] Fan, J. (1991a). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257-1272. · Zbl 0729.62033
[14] Fan, J. (1991b). Global behavior of deconvolution kernel estimates. Statist. Sinica 1 541-551. · Zbl 0823.62032
[15] Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600-610. · Zbl 0785.62038
[16] Fan, J. and Koo, J.-Y. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734-747. · Zbl 1071.94511
[17] Groeneboom, P. and Jongbloed, G. (2003). Density estimation in the uniform deconvolution model. Statist. Neerlandica 57 136-157. · Zbl 1090.62527
[18] Groeneboom, P. and Wellner, J. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation . Birkhäuser, Basel. · Zbl 0757.62017
[19] Hesse, C. H. (1995). Distribution function estimation from noisy observations. Publ. Inst. Stat. Paris Sud 39 21-35. · Zbl 0837.62034
[20] Hesse, C. H. (1999). Data-driven deconvolution. J. Nonparametr. Statist. 10 343-373. · Zbl 0936.62038
[21] Hesse, C. H. and Meister, A. (2004). Optimal iterative density deconvolution. J. Nonparametr. Statist. 16 879-900. · Zbl 1059.62033
[22] Ioannides, D. A. and Papanastassiou, D. P. (2001). Estimating the distribution function of a stationary process involving measurement errors. Statist. Inference Stoch. Process. 4 181-198. · Zbl 0984.62061
[23] Jongbloed, G. (1998). Exponential deconvolution: Two asymptotically equivalent estimators. Statist. Neerlandica 52 6-17. · Zbl 0937.62035
[24] Koo, J.-A. (1999). Logspline deconvolution in Besov space. Scand. J. Statist. 26 73-86. · Zbl 0924.65150
[25] Neumann, M. H. (1997). On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Statist. 7 307-330. · Zbl 1003.62514
[26] Pensky, M. (2002). Density deconvolution based on wavelets with bounded supports. Statist. Probab. Lett. 56 261-269. · Zbl 0994.62027
[27] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033-2053. · Zbl 0962.62030
[28] Qin, H.-Z. and Feng, S.-Y. (2003). Deconvolution kernel estimator for mean transformation with ordinary smooth error. Statist. Probab. Lett. 61 337-346. · Zbl 1038.62039
[29] Stefanski, L. A. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169-184. · Zbl 0697.62035
[30] Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040-1053. · Zbl 0511.62048
[31] van de Geer, S. (1995). Asymptotic normality in mixture models. ESAIM Probab. Statist. 1 17-33. · Zbl 0867.62026
[32] van Es, B., Spreij, P. and van Zanten, H. (2003). Nonparametric volatility density estimation. Bernoulli 9 451-465. · Zbl 1044.62037
[33] Zhang, C. H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806-830. · 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.