Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.00073 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914566617694208 |
|---|---|
| author | Kim, Taehun Lee, Jung Chan Choi, ByoungSeon |
| author_facet | Kim, Taehun Lee, Jung Chan Choi, ByoungSeon |
| contents | The positivity of the Gram-Charlier probability density function has been a subject of extensive study for decades. Since Barton and Dennis (1952) introduced numerical positivity conditions, no analytic closed-form expression was available until Kwon (2019, 2022) proposed analytic solutions for the valid region of Gram-Charlier densities. Despite the significance of the analytical solutions, the expressions remain algebraically complex. As these conditions for the Gram-Charlier densities are determined by a quartic polynomial, it is essential to investigate its positivity. In this work, necessary and sufficient conditions for the positivity of a quartic polynomial are derived through a separation method. Based on these conditions, more concise analytic expressions for the positivity of the Gram-Charlier density are proposed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00073 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Separation Method for Quartic Positivity and the Valid Region of Gram-Charlier densities Kim, Taehun Lee, Jung Chan Choi, ByoungSeon Symbolic Computation Primary 62E17, Secondary 26C05, 60E05 The positivity of the Gram-Charlier probability density function has been a subject of extensive study for decades. Since Barton and Dennis (1952) introduced numerical positivity conditions, no analytic closed-form expression was available until Kwon (2019, 2022) proposed analytic solutions for the valid region of Gram-Charlier densities. Despite the significance of the analytical solutions, the expressions remain algebraically complex. As these conditions for the Gram-Charlier densities are determined by a quartic polynomial, it is essential to investigate its positivity. In this work, necessary and sufficient conditions for the positivity of a quartic polynomial are derived through a separation method. Based on these conditions, more concise analytic expressions for the positivity of the Gram-Charlier density are proposed. |
| title | A Separation Method for Quartic Positivity and the Valid Region of Gram-Charlier densities |
| topic | Symbolic Computation Primary 62E17, Secondary 26C05, 60E05 |
| url | https://arxiv.org/abs/2603.00073 |