On the effect of estimating the error density in nonparametric deconvolution

It is quite common in the statistical literature on nonparametric deconvolution to assume that the error density is perfectly known. Since this seems to be unrealistic in many practical applications, we study the effect of estimating the unknown error density. We derive minimax rates of convergence and propose a modification of the usual kernel-based estimation scheme, which takes the uncertainty about the error density into account. A simulation study quantifies the possible gains by this new method in finite sample situations.

[1]  David Ruppert,et al.  Asymptotics for the Transformation Kernel Density Estimator , 1995 .

[2]  D. Ruppert,et al.  Taylor series approximations of transformation kernel density estimators , 1994 .

[3]  Jens Perch Nielsen,et al.  A simple bias reduction method for density estimation , 1995 .

[4]  M. Samiuddin,et al.  On nonparametric kernel density estimates , 1990 .

[5]  M. C. Jones,et al.  A Comparison of Higher-Order Bias Kernel Density Estimators , 1997 .

[6]  J. S. Marron,et al.  Variable window width kernel estimates of probability densities , 1988 .

[7]  M. Samiuddin,et al.  A new kernel density estimate , 1993 .

[8]  Ian Mckay.  A note on bias reduction in variable‐kernel density estimates , 1993 .

[9]  Ian Abramson.  On Bandwidth Variation in Kernel Estimates-A Square Root Law , 1982 .

[10]  M. C. Jones,et al.  Variable location and scale kernel density estimation , 1994 .

[11]  P. Hall.  On the bias of variable bandwidth curve estimators , 1990 .

[12]  M. C. Jones.  Variable kernel density estimates and variable kernel density estimates , 1990 .

[13]  Peter Hall,et al.  On local smoothing of nonparametric curve estimators , 1996 .

[14]  C. Quesenberry,et al.  A nonparametric estimate of a multivariate density function , 1965 .

[15]  D. W. Scott,et al.  Variable Kernel Density Estimation , 1992 .

[16]  David Ruppert,et al.  Bias reduction in kernel density estimation by smoothed empirical transformations , 1994 .

[17]  L. Breiman,et al.  Variable Kernel Estimates of Multivariate Densities , 1977 .

[18]  O. Hössjer.  Asymptotic bias and variance for a general class of varying bandwidth density estimators , 1996 .

[19]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[20]  David W. Scott,et al.  Multivariate Density Estimation: Theory, Practice, and Visualization , 1992, Wiley Series in Probability and Statistics.

[21]  Peter Hall.  On Plug-in Rules for Local Smoothing of Density Estimators , 1993 .