Universiteit Utrecht

Department of Mathematics

Two dimensional uniform kernel deconvolution

Bert van Es

13 May 2009

Download abstract here.

In a general deconvolution model we have a sample of n independent Xi which are equal to the sum of independent and unknown Yi and Zi. So Xi=Yi+Zi. We assume that the Zi have a known distribution. The aim is to estimate the probability density f of the Yi from this sample of Xi.

Since the density of the observed Xi is equal to the convolution of the densities of the Yi and Zi one can derive a density estimator of f by Fourier inversion and kernel estimation of the density of the observations. This approach has proven to be useful in many deconvolution models, i.e. different known distributions of the Zi. However, it fails in the model where the known density of the Zi is uniform. This model is usually called uniform deconvolution.

We will present an alternative method based on kernel density estimation and a different, non Fourier, type of inversion of the convolution operator in this model. Following earlier work for the one dimensional model, cf Van Es 2002, we will use the same approach in the two dimensional model where the Xi, Yi and Zi are two dimensional random vectors and where the distribution of the Zi is uniform on the unit square.

We will derive expansions for the bias and variance and present some simulated examples.

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Alexandra Babenko