  In this paper, we consider the nonparametric estimation of the multivariate
probability density function and its partial derivative with a support on
$[0,\infty)$. To this end we use the class of kernel estimators with asymmetric
gamma kernel functions. The gamma kernels are nonnegative. They change their
shape depending on the position on the semi-axis and are robust to the boundary
bias problem. We investigate the mean integrated squared error (MISE) assuming
dependent data with strong mixing and find the optimal bandwidth of the kernel
as a minimum of the MISE. We derive the bias, the variance and the covariance
of the density and of its partial derivative.
