报告题目：The Valid Regions of Gram-Charlier Densities: A Revisit

报告人：林炜

报告时间：2024年6月12日下午14：00-15：00

报告地点：腾讯会议号：276-510-295

摘要：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.

报告人简介：林炜，双博士学位，浙江大学数学理学博士学位，Universityof Otago金融学博士学位。现任杭州师范大学数学学院统计系助理教授，硕士研究生导师。研究领域有：衍生品定价、半参数化密度函数逼近。在《Journal of Futures Markets》、《Journal of Computational and Applied Mathematics》、《Review of Derivatives Research》、《Journal of Mathematical Analysis and Applications》、《中国科学：数学》、等期刊发表多篇论文。