# Deconvolution from repeated measurements with non-symmetric errors

vendredi 6 juin 2014, 9h30 - 10h30

Consider a density deconvolution problem: The random variable $X$ of interest is observed with an additional additive error $\eps$, independent of $X$. The target is to recover the density $f$ of $X$.

A vast amount of literature is available on the case where $n$ independent copies
$Y_j=X_j+\eps_j, \ j=1, \cdots, n$
of the contaminated random variable $Y=X+\eps$ are available and
the distribution of the errors is known. However, it is often not realistic in applications to assume prefect knowledge of the error distribution. If the distribution of the noise is assumed to be unknown, some additional information on the errors is required to make the problem identifiable. Many publications have dealt with the case that an additional sample of the pure noise, independent of the $Y_j$, is available.

We focus on the panel data model, where $X$ is observed repeatedly, with independent errors:
$Y_{j,k}=X_j +\eps_{j,k},\ j=1, \cdots, n; \ k=1,2.$
This model has recently been investigated under the additional assumption that the distribution of the errors is symmetric.

In the present talk, we focus on the situation where the symmetry assumption on the noise need no longer be satisfied. It turns out that the problem is structurally more complicated in the non-symmetric case and a completely different estimation procedure is required.

We construct estimators and derive non-parametric risk bounds and rates of convergence. It turns out that we can dispose of most of the extremely restrictive assumptions which have been imposed in earlier publications on the subject and, at the same time, substantially improve the rates of convergence.

We discuss strategies for the adaptive choice of the bandwidth, present some data examples and compare our results to the standard estimators for the symmetric-error case.