Department of Mathematics

Two dimensional uniform kernel deconvolution

**Bert van Es**

(UvA)

13 May 2009

Download abstract here.

In a general deconvolution model we have a sample of *n* independent *X*_{i}
which are equal to the sum of independent and unknown *Y*_{i}
and *Z*_{i}. So *X*_{i}=Y_{i}+Z_{i}. We assume that the *Z*_{i} have a known
distribution. The aim is to estimate the probability density *f* of
the *Y*_{i} from this sample of *X*_{i}.

Since the density of the observed *X*_{i} is equal to the convolution
of the densities of the *Y*_{i} and *Z*_{i} 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 *Z*_{i}. However, it fails in the model where the known density
of the *Z*_{i} 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 *X*_{i}, *Y*_{i} and *Z*_{i} are two
dimensional random vectors and where the distribution of the *Z*_{i}
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