Shrinkage and visualization of multivariate Gram-Charlier approximations

Gy. H. Terdik${}^a$, and E. Taufer${}^b$

${}^a$Universityof Debrecen, ${}^b$University of Trento

For univariate densities, the fourth-order Gram Charlier (GC) approximation is expressed through skewness- and kurtosis-driven Hermite corrections to a Gaussian baseline. Since truncation may produce negative values even when the true cumulants are used, ensuring positivity requires that the GC correction factor remain non-negative on a prescribed central interval. To address violations, we consider two shrinkage strategies: a one-parameter scaling of the full correction term, and a two-parameter scaling that separately shrinks the skewness and kurtosis components. The latter is formulated as a convex quadratic projection problem over a feasible set defined by linear inequalities on a dense grid, yielding a unique solution.

The idea is extended to the multivariate setting. Three nested correction schemes are proposed: global one-parameter scaling, $K$-parameter scaling by Hermite order, and a finest-grain distinct-parameter scaling acting on the distinct tensor coefficients of multivariate Hermite terms. By rewriting the multivariate correction in terms of distinct tensor entries and associated weights, the positivity constraints become linear, and the resulting shrinkage problem can be solved efficiently by convex optimization tools such as CVXR ([1]).

In addition, the work outlines visualization strategies for comparing multivariate densities and their approximations, including spiral-based representations in two dimensions and cone-based restrictions in three dimensions. These provide interpretable geometric views of fitted densities and help assess the effects of shrinkage in practice.

Keywords: Multivariate density estimation; convex optimization; geometric visualization.