Normal integral representation for the joint survival function of the cumulative sums of the components of multinomial random vectors
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914613437661184 |
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| author | Ouimet, Frédéric |
| author_facet | Ouimet, Frédéric |
| contents | This paper presents a multivariate normal integral representation for the joint survival function of the cumulative sums of the components of any multinomial random vector at interior lattice points. This result can be viewed as a multivariate analog of Equation (7) in Carter and Pollard (2004), whose proof starts from the beta integral representation of binomial survival probabilities and uses Laplace's method to improve Tusnády's inequality. Our findings are based on a crucial relationship between the joint survival function of the cumulative sums of the components of any multinomial random vector and a Dirichlet probability over a corresponding cumulative-sum region. The main motivation is that such an explicit formula may eventually help streamline the conditional quantile-transformation arguments used in the multivariate KMT approximation of Einmahl (1989), a connection left for future work. We provide numerical checks of the identity for $d = 2,3,4,5$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_18509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normal integral representation for the joint survival function of the cumulative sums of the components of multinomial random vectors Ouimet, Frédéric Statistics Theory Probability 62E17, 62H10, 62H12, 62E20 This paper presents a multivariate normal integral representation for the joint survival function of the cumulative sums of the components of any multinomial random vector at interior lattice points. This result can be viewed as a multivariate analog of Equation (7) in Carter and Pollard (2004), whose proof starts from the beta integral representation of binomial survival probabilities and uses Laplace's method to improve Tusnády's inequality. Our findings are based on a crucial relationship between the joint survival function of the cumulative sums of the components of any multinomial random vector and a Dirichlet probability over a corresponding cumulative-sum region. The main motivation is that such an explicit formula may eventually help streamline the conditional quantile-transformation arguments used in the multivariate KMT approximation of Einmahl (1989), a connection left for future work. We provide numerical checks of the identity for $d = 2,3,4,5$. |
| title | Normal integral representation for the joint survival function of the cumulative sums of the components of multinomial random vectors |
| topic | Statistics Theory Probability 62E17, 62H10, 62H12, 62E20 |
| url | https://arxiv.org/abs/2406.18509 |