Estimation of distributions, moments and quantiles in deconvolution problems

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.

[1]  C. Butucea.  Deconvolution of supersmooth densities with smooth noise , 2004 .

[2]  Jianqing Fan.  On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems , 1991 .

[3]  B. Vidakovic,et al.  Adaptive wavelet estimator for nonparametric density deconvolution , 1999 .

[4]  Peter J. Diggle,et al.  A Fourier Approach to Nonparametric Deconvolution of a Density Estimate , 1993 .

[5]  H. Zanten,et al.  Nonparametric volatility density estimation , 2001, math/0107135.

[6]  P. Hall,et al.  Optimal Rates of Convergence for Deconvolving a Density , 1988 .

[7]  Cun-Hui Zhang.  Fourier Methods for Estimating Mixing Densities and Distributions , 1990 .

[8]  Estimation of integrated squared density derivatives from a contaminated sample , 2002 .

[9]  Ja-Yong Koo.  Logspline Deconvolution in Besov Space , 1999 .

[10]  Density deconvolution based on wavelets with bounded supports , 2002 .

[11]  Michael H. Neumann,et al.  On the effect of estimating the error density in nonparametric deconvolution , 1997 .

[12]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[13]  L. Devroye.  Consistent deconvolution in density estimation , 1989 .

[14]  Deconvolution kernel estimator for mean transformation with ordinary smooth error , 2003 .

[15]  D. Papanastassiou,et al.  Estimating the Distribution Function of a Stationary Process Involving Measurement Errors , 2001 .

[16]  P. Hall,et al.  Bootstrap Confidence Regions for Functional Relationships in Errors-in- Variables Models , 1993 .

[17]  R. Carroll,et al.  Deconvolving kernel density estimators , 1987 .

[18]  Jianqing Fan,et al.  Adaptively Local One-Dimensional Subproblems with Application to a Deconvolution Problem , 1993 .

[19]  C. Hesse,et al.  Optimal iterative density deconvolution , 2004 .

[20]  Ja-Yong Koo,et al.  Wavelet deconvolution , 2002, IEEE Trans. Inf. Theory.

[21]  Aurore Delaigle,et al.  On optimal kernel choice for deconvolution , 2006 .

[22]  Sara van de Geer,et al.  Asymptotic Normality in Mixture Models , 1997 .

[23]  Irène Gijbels,et al.  Bootstrap bandwidth selection in kernel density estimation from a contaminated sample , 2004 .

[24]  Irène Gijbels,et al.  Practical bandwidth selection in deconvolution kernel density estimation , 2004, Comput. Stat. Data Anal..

[25]  G. Jongbloed.  Exponential deconvolution: two asymptotically equivalent estimators , 1998 .

[26]  Cristina Butucea,et al.  Sharp Optimality in Density Deconvolution with Dominating Bias. II , 2008 .

[27]  Jianqing Fan,et al.  Global Behavior of Deconvolution Kernel Estimates , 1989 .

[28]  P. Groeneboom,et al.  Density estimation in the uniform deconvolution model , 2003 .

[29]  Christian H. Hesse,et al.  Data-driven deconvolution , 1999 .

[30]  Clifford B. Cordy,et al.  Deconvolution of a Distribution Function , 1997 .