  A modified gamma kernel should not be automatically preferred to the standard
gamma kernel, especially for univariate convex densities with a pole at the
origin. In the multivariate case, multiple combined gamma kernels, defined as a
product of univariate standard and modified ones, are here introduced for
nonparametric and semiparametric smoothing of unknown orthant densities with
support $[0,\infty)^d$. Asymptotical properties of these multivariate
associated kernel estimators are established. Bayesian estimation of adaptive
bandwidth vectors using multiple pure combined gamma smoothers, and in
semiparametric setup, are exactly derived under the usual quadratic function.
The simulation results and four illustrations on real datasets reveal very
interesting advantages of the proposed combined approach for nonparametric
smoothing, compare to both pure standard and pure modified gamma kernel
versions, and under integrated squared error and average log-likelihood
criteria.
