  Multivariate associated kernel estimators, which depend on both target point
and bandwidth matrix, are appropriate for partially or totally bounded
distributions and generalize the classical ones as Gaussian. Previous studies
on multivariate associated kernels have been restricted to product of
univariate associated kernels, also considered having diagonal bandwidth
matrices. However, it is shown in classical cases that for certain forms of
target density such as multimodal, the use of full bandwidth matrices offers
the potential for significantly improved density estimation. In this paper,
general associated kernel estimators with correlation structure are introduced.
Properties of these estimators are presented; in particular, the boundary bias
is investigated. Then, the generalized bivariate beta kernels are handled with
more details. The associated kernel with a correlation structure is built with
a variant of the mode-dispersion method and two families of bandwidth matrices
are discussed under the criterion of cross-validation. Several simulation
studies are done. In the particular situation of bivariate beta kernels, it is
therefore pointed out the very good performance of associated kernel estimators
with correlation structure compared to the diagonal case. Finally, an
illustration on real dataset of paired rates in a framework of political
elections is presented.
