Estimation of distributions, moments and quantiles in deconvolution problems



The Annals of Statistics

Estimation of distributions, moments and quantiles in deconvolution problems

Peter Hall and Soumendra N. Lahiri

Source: Ann. Statist. Volume 36, Number 5 (2008), 2110-2134.

Abstract

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.

Primary Subjects: 62G20
Secondary Subjects: 62C20
Keywords: Bandwidth; errors in variables; ill-posed problem; kernel methods; measurement error; minimax; optimal convergence rate; smoothing; regularization

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Permanent link to this document: http://projecteuclid.org/euclid.aos/1223908086
Digital Object Identifier: doi:10.1214/07-AOS534

References

Booth, J. G. and Hall, P. (1993). Bootstrap confidence regions for functional relationships in errors-in-variables models. Ann. Statist. 21 1780–1791.
Mathematical Reviews (MathSciNet): MR1245768
Digital Object Identifier: doi:10.1214/aos/1176349397
Project Euclid: euclid.aos/1176349397
Butucea, C. (2004). Deconvolution of supersmooth densities with smooth noise. Canad. J. Statist. 32 181–192.
Mathematical Reviews (MathSciNet): MR2064400
Digital Object Identifier: doi:10.2307/3315941
Butucea, C. and Tsybakov, A. B. (2008). Sharp optimality for density deconvolution with dominating bias. Theory Probab. Appl. To appear.
Mathematical Reviews (MathSciNet): MR2354572
Carroll, R.J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
Mathematical Reviews (MathSciNet): MR997599
Digital Object Identifier: doi:10.2307/2290153
Cordy, C. and Thomas, D. R. (1997). Deconvolution of a distribution function. J. Amer. Statist. Assoc. 92 1459–1465.
Mathematical Reviews (MathSciNet): MR1615256
Digital Object Identifier: doi:10.2307/2965416
Cui, H. (2005). Asymptotics of mean transformation estimators with errors in variables model. J. Syst. Sci. Complex. 18 446–455.
Mathematical Reviews (MathSciNet): MR2172109
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.
Mathematical Reviews (MathSciNet): MR1979392
Digital Object Identifier: doi:10.1111/1467-9868.00366
Delaigle, A. and Gijbels, I. (2004a). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249–267.
Mathematical Reviews (MathSciNet): MR2045631
Delaigle, A. and Gijbels, I. (2004b). Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Ann. Inst. Statist. Math. 56 19–47.
Mathematical Reviews (MathSciNet): MR2053727
Digital Object Identifier: doi:10.1007/BF02530523
Delaigle, A. and Hall, P. (2006). On optimal kernel choice for deconvolution. Statist. Probab. Lett. 76 1594–1602.
Mathematical Reviews (MathSciNet): MR2248846
Devroye, L. (1989). Consistent deconvolution in density estimation. Canad. J. Statist. 17 235-239.
Mathematical Reviews (MathSciNet): MR1033106
Digital Object Identifier: doi:10.2307/3314852
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.
Mathematical Reviews (MathSciNet): MR1224414
Fan, J. (1991a). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
Mathematical Reviews (MathSciNet): MR1126324
Digital Object Identifier: doi:10.1214/aos/1176348248
Project Euclid: euclid.aos/1176348248
Fan, J. (1991b). Global behavior of deconvolution kernel estimates. Statist. Sinica 1 541–551.
Mathematical Reviews (MathSciNet): MR1130132
Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600–610.
Mathematical Reviews (MathSciNet): MR1232507
Digital Object Identifier: doi:10.1214/aos/1176349139
Project Euclid: euclid.aos/1176349139
Fan, J. and Koo, J.-Y. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734–747.
Mathematical Reviews (MathSciNet): MR1889978
Digital Object Identifier: doi:10.1109/18.986021
Groeneboom, P. and Jongbloed, G. (2003). Density estimation in the uniform deconvolution model. Statist. Neerlandica 57 136–157.
Mathematical Reviews (MathSciNet): MR2035863
Digital Object Identifier: doi:10.1111/1467-9574.00225
Groeneboom, P. and Wellner, J. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel.
Mathematical Reviews (MathSciNet): MR1180321
Zentralblatt MATH: 0757.62017
Hesse, C. H. (1995). Distribution function estimation from noisy observations. Publ. Inst. Stat. Paris Sud 39 21–35.
Hesse, C. H. (1999). Data-driven deconvolution. J. Nonparametr. Statist. 10 343–373.
Mathematical Reviews (MathSciNet): MR1717098
Digital Object Identifier: doi:10.1080/10485259908832766
Hesse, C. H. and Meister, A. (2004). Optimal iterative density deconvolution. J. Nonparametr. Statist. 16 879–900.
Mathematical Reviews (MathSciNet): MR2094745
Digital Object Identifier: doi:10.1080/10485250410001690086
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.
Mathematical Reviews (MathSciNet): MR1856173
Digital Object Identifier: doi:10.1023/A:1017996326631
Jongbloed, G. (1998). Exponential deconvolution: Two asymptotically equivalent estimators. Statist. Neerlandica 52 6–17.
Mathematical Reviews (MathSciNet): MR1615570
Digital Object Identifier: doi:10.1111/1467-9574.00065
Koo, J.-A. (1999). Logspline deconvolution in Besov space. Scand. J. Statist. 26 73–86.
Mathematical Reviews (MathSciNet): MR1685303
Digital Object Identifier: doi:10.1111/1467-9469.00138
Neumann, M. H. (1997). On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Statist. 7 307–330.
Mathematical Reviews (MathSciNet): MR1460203
Digital Object Identifier: doi:10.1080/10485259708832708
Pensky, M. (2002). Density deconvolution based on wavelets with bounded supports. Statist. Probab. Lett. 56 261–269.
Mathematical Reviews (MathSciNet): MR1892987
Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033–2053.
Mathematical Reviews (MathSciNet): MR1765627
Digital Object Identifier: doi:10.1214/aos/1017939249
Project Euclid: euclid.aos/1017939249
Qin, H.-Z. and Feng, S.-Y. (2003). Deconvolution kernel estimator for mean transformation with ordinary smooth error. Statist. Probab. Lett. 61 337–346.
Mathematical Reviews (MathSciNet): MR1959070
Stefanski, L. A. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169–184.
Mathematical Reviews (MathSciNet): MR1054861
Digital Object Identifier: doi:10.1080/02331889008802238
Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040–1053.
Mathematical Reviews (MathSciNet): MR673642
Digital Object Identifier: doi:10.1214/aos/1176345969
Project Euclid: euclid.aos/1176345969
van de Geer, S. (1995). Asymptotic normality in mixture models. ESAIM Probab. Statist. 1 17–33.
Mathematical Reviews (MathSciNet): MR1382516
Digital Object Identifier: doi:10.1051/ps:1997101
van Es, B., Spreij, P. and van Zanten, H. (2003). Nonparametric volatility density estimation. Bernoulli 9 451–465.
Mathematical Reviews (MathSciNet): MR1997492
Digital Object Identifier: doi:10.3150/bj/1065444813
Project Euclid: euclid.bj/1065444813
Zhang, C. H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806–830.
Mathematical Reviews (MathSciNet): MR1056338
Digital Object Identifier: doi:10.1214/aos/1176347627
Project Euclid: euclid.aos/1176347627

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