A Fourier Approach to Nonparametric Deconvolution of a Density Estimate

SUMMARY We consider the problem of constructing a nonparametric estimate of a probability density function h from independent random samples of observations from densities a and f, when a represents the convolution of h and f. Our approach is based on truncated Fourier inversion, in which the truncation point plays the role of a smoothing parameter. We derive the asymptotic mean integrated squared error of the estimate and use this formula to suggest a simple practical method for choosing the truncation point from the data. Strikingly, when the smoothing parameter is chosen in this way then in many circumstances the estimator behaves, to first order, as though the true f were known. Simulated data sets are used to demonstrate that the method can give good results in practice.