Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices

Multivariate kernel density estimation is an important technique in exploratory data analysis. Its utility relies on its ease of interpretation, especially by graphical means. The crucial factor which determines the performance of kernel density estimation is the bandwidth matrix selection. Research in finding optimal bandwidth matrices began with restricted parameterizations of the bandwidth matrix which mimic univariate selectors. Progressively these restrictions were relaxed to develop more flexible selectors. In this paper, we propose the first plug-in bandwidth selector with the unconstrained parameterizations of both the final and pilot selectors. Up till now, the development of unconstrained pilot selectors was hindered by the traditional vectorization of higher-order derivatives which lead to increasingly intractable matrix algebraic expressions. We resolve this by introducing an alternative vectorization which gives elegant and tractable expressions. This allows us to quantify the asymptotic and finite sample properties of unconstrained pilot selectors. For target densities with intricate structure (such as multimodality), our unconstrained selectors show the most improvement over the existing plug-in selectors.