## 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
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### 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
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