Handy formulas for binomial moments
Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications. This paper introduces n...
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| Format: | Article |
| Language: | English |
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2024-07-01
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| Series: | Modern Stochastics: Theory and Applications |
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| Online Access: | https://www.vmsta.org/doi/10.15559/24-VMSTA260 |
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| _version_ | 1841549990490537984 |
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| author | Maciej Skorski |
| author_facet | Maciej Skorski |
| author_sort | Maciej Skorski |
| collection | DOAJ |
| description | Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications.
This paper introduces novel formulas for binomial moments in the form of polynomials in the variance rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in Python’s symbolic algebra package.
The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct asymptotically sharp estimates for central binomial moments are established, improving upon previously known partial results. |
| format | Article |
| id | doaj-art-313830faa9124632a8f4b3883f2481a4 |
| institution | Kabale University |
| issn | 2351-6046 2351-6054 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | VTeX |
| record_format | Article |
| series | Modern Stochastics: Theory and Applications |
| spelling | doaj-art-313830faa9124632a8f4b3883f2481a42025-01-10T11:16:09ZengVTeXModern Stochastics: Theory and Applications2351-60462351-60542024-07-01121274110.15559/24-VMSTA260Handy formulas for binomial momentsMaciej Skorski0University of Warsaw, PolandDespite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications. This paper introduces novel formulas for binomial moments in the form of polynomials in the variance rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in Python’s symbolic algebra package. The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct asymptotically sharp estimates for central binomial moments are established, improving upon previously known partial results.https://www.vmsta.org/doi/10.15559/24-VMSTA260binomial distributionhigh-order momentsmoment asymptoticssymbolic algebra |
| spellingShingle | Maciej Skorski Handy formulas for binomial moments Modern Stochastics: Theory and Applications binomial distribution high-order moments moment asymptotics symbolic algebra |
| title | Handy formulas for binomial moments |
| title_full | Handy formulas for binomial moments |
| title_fullStr | Handy formulas for binomial moments |
| title_full_unstemmed | Handy formulas for binomial moments |
| title_short | Handy formulas for binomial moments |
| title_sort | handy formulas for binomial moments |
| topic | binomial distribution high-order moments moment asymptotics symbolic algebra |
| url | https://www.vmsta.org/doi/10.15559/24-VMSTA260 |
| work_keys_str_mv | AT maciejskorski handyformulasforbinomialmoments |