On multivariate associated kernels to estimate general density functions

Abstract

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.

Effects of associated kernels in nonparametric multiple regressions

“…Therefore, it can reach any point of the space which might be inaccessible with diagonal matrix. This type of kernel is called beta-Sarmanov kernel by Kokonendji and Somé (2015); see Sarmanov (1966) and also Lee (1996) for this construction of multivariate densities with correlation structure from independent components. Like Bertin and Klutnitchkoff (2014), the miminax properties of this bivariate beta kernel are also possible and more generally for associated kernels.…”

“…Recently, Libengué (2013) investigated several families of these univariate continuous kernels that he called univariate associated kernels; see also Kokonendji et al (2007), Kokonendji and Senga Kiéssé (2011), Zougab et al (2012) and Wansouwé et al (2014) for univariate discrete situations. A continuous multivariate version of these associated kernels have been studied by Kokonendji and Somé (2015) for density estimation.…”

Bayesian bandwidth selection in discrete multivariate associated kernel estimators for probability mass functions

“…The function K [j] x j ,h j is the jth discrete univariate associated kernel on the support S x j ,h j ⊆ Z. In principle, the estimator (6) is more appropriate for distributions without correlation in their components; see Bouezmarni and Roumbouts (2010) and also Kokonendji and Somé (2015) for continuous cases.…”

“…From Kokonendji and Somé (2015) it is known that, for both  f n from (1) and (6), we have  f n (x) ∈ [0, 1] for all x ∈ T d and …”

“…Now, we establish a simulation study to compare the performances of the Bayesian global approach against the classical method of cross-validation for bandwidth selection using estimator (6). Here, we consider the multivariate least squares cross-validation (LSCV) which is based on the minimization of the ISE; see Bouezmarni and Roumbouts (2010) and Kokonendji and Somé (2015) for more details. The criterion consists in choosing H that minimizes…”