Abstract
Density estimation plays an important and fundamental role in finance, pattern recognition, machine learning, and statistics. Based on derivatives of a Gaussian density, a Gram–Charlier series presents an infinite expansion. Its truncated series is often used in many fields to approximate probability density functions. Although the expansions are convenient, there are constrained regions on the value of the cumulants (or moments) that admit a valid (nonnegative) probability density function. Lin and Zhang (2022)’s paper focuses on Gram–Charlier densities to show how the valid region of higher cumulants can be numerically implemented by semidefinite programming, which ensures that a series truncated at a cumulant of an arbitrary even order represents a valid probability density. This paper is the further exploration into the same problem. First, we use the representation theorem of such polynomials as sum of squares on the Gram–Charlier density to show how to develop the corresponding convex optimization problem for its valid region. Second, we provide the valid skewness–kurtosis regions of Gram–Charlier densities only up to the sixteenth-order because the semidefinite programming fails to calculate these regions when the order is above that. Third, we explore the valid region of the fourth-order Gram–Charlier defined on an arbitrary finite domain [−q,q] but not the field R of real numbers. Our analysis proves that the ranges of skewness and kurtosis can be broadened with the finite domains, which earn a wider application. Furthermore, the impact of the length of finite domains 2q on valid regions has been shown.