A limit formula and a series expansion for the bivariate Normal tail probability
This work presents a limit formula for the bivariate Normal tail probability. It only requires the larger threshold to grow indefinitely, but otherwise has no restrictions on how the thresholds grow. The correlation parameter can change and possibly depend on the thresholds. The formula is applic...
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| Format: | Journal Article |
| Language: | English |
| Published: |
2024
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| Online Access: | https://hdl.handle.net/10356/178833 |
| _version_ | 1811688513164279808 |
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| author | Au, Siu-Kui |
| author2 | School of Civil and Environmental Engineering |
| author_facet | School of Civil and Environmental Engineering Au, Siu-Kui |
| author_sort | Au, Siu-Kui |
| collection | NTU |
| description | This work presents a limit formula for the bivariate Normal tail probability.
It only requires the larger threshold to grow indefinitely, but otherwise has no
restrictions on how the thresholds grow. The correlation parameter can change
and possibly depend on the thresholds. The formula is applicable regardless of Salvage’s
condition. Asymptotically, it reduces to Ruben’s formula and Hashorva’s
formula under the corresponding conditions, and therefore can be considered
a generalisation. Under a mild condition, it satisfies Plackett’s identity on the
derivative with respect to the correlation parameter. Motivated by the limit formula,
a series expansion is also obtained for the exact tail probability using
derivatives of the univariate Mill’s ratio. Under similar conditions for the limit
formula, the series converges and its truncated approximation has a small remainder
term for large thresholds. To take advantage of this, a simple procedure is
developed for the general case by remapping the parameters so that they satisfy
the conditions. Examples are presented to illustrate the theoretical findings. |
| first_indexed | 2024-10-01T05:33:24Z |
| format | Journal Article |
| id | ntu-10356/178833 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T05:33:24Z |
| publishDate | 2024 |
| record_format | dspace |
| spelling | ntu-10356/1788332024-07-12T15:33:35Z A limit formula and a series expansion for the bivariate Normal tail probability Au, Siu-Kui School of Civil and Environmental Engineering Mathematical Sciences Bivariate normal probability Hashorva’s formula Plackett’s identity Ruben’s formula Salvage’s condition This work presents a limit formula for the bivariate Normal tail probability. It only requires the larger threshold to grow indefinitely, but otherwise has no restrictions on how the thresholds grow. The correlation parameter can change and possibly depend on the thresholds. The formula is applicable regardless of Salvage’s condition. Asymptotically, it reduces to Ruben’s formula and Hashorva’s formula under the corresponding conditions, and therefore can be considered a generalisation. Under a mild condition, it satisfies Plackett’s identity on the derivative with respect to the correlation parameter. Motivated by the limit formula, a series expansion is also obtained for the exact tail probability using derivatives of the univariate Mill’s ratio. Under similar conditions for the limit formula, the series converges and its truncated approximation has a small remainder term for large thresholds. To take advantage of this, a simple procedure is developed for the general case by remapping the parameters so that they satisfy the conditions. Examples are presented to illustrate the theoretical findings. Ministry of Education (MOE) Submitted/Accepted version The research presented in this paper is supported by Academic Research Fund Tier 1 (RG68/22) from the Ministry of Education, Singapore. 2024-07-11T00:52:03Z 2024-07-11T00:52:03Z 2024 Journal Article Au, S. (2024). A limit formula and a series expansion for the bivariate Normal tail probability. Statistics and Computing, 34, 152-. https://dx.doi.org/10.1007/s11222-024-10466-w 0960-3174 https://hdl.handle.net/10356/178833 10.1007/s11222-024-10466-w 34 152 en RG68/22 Statistics and Computing © 2024 The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature. All rights reserved. This article may be downloaded for personal use only. Any other use requires prior permission of the copyright holder. The Version of Record is available online at http://doi.org/10.1007/s11222-024-10466-w. application/pdf |
| spellingShingle | Mathematical Sciences Bivariate normal probability Hashorva’s formula Plackett’s identity Ruben’s formula Salvage’s condition Au, Siu-Kui A limit formula and a series expansion for the bivariate Normal tail probability |
| title | A limit formula and a series expansion for the bivariate Normal tail probability |
| title_full | A limit formula and a series expansion for the bivariate Normal tail probability |
| title_fullStr | A limit formula and a series expansion for the bivariate Normal tail probability |
| title_full_unstemmed | A limit formula and a series expansion for the bivariate Normal tail probability |
| title_short | A limit formula and a series expansion for the bivariate Normal tail probability |
| title_sort | limit formula and a series expansion for the bivariate normal tail probability |
| topic | Mathematical Sciences Bivariate normal probability Hashorva’s formula Plackett’s identity Ruben’s formula Salvage’s condition |
| url | https://hdl.handle.net/10356/178833 |
| work_keys_str_mv | AT ausiukui alimitformulaandaseriesexpansionforthebivariatenormaltailprobability AT ausiukui limitformulaandaseriesexpansionforthebivariatenormaltailprobability |