  We construct a density estimator in the bivariate uniform deconvolution
model. For this model we derive four inversion formulas to express the
bivariate density that we want to estimate in terms of the bivariate density of
the observations. By substituting a kernel density estimator of the density of
the observations we then get four different estimators. Next we construct an
asymptotically optimal convex combination of these four estimators. Expansions
for the bias, variance, as well as asymptotic normality, are derived. Some
simulated examples are presented.
