Modelling pairs of Poissons and binomials with negative correlation
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arXiv
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| Format: | Preprint |
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2026
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| author | Hjort, Nils Lid |
| author_facet | Hjort, Nils Lid |
| contents | Suppose $f_1(x)$ and $f_2(y)$ are given marginals for pairs $(x,y)$. I consider the construction $f_1(x)f_2(y)\{ 1+αh_1(x)h_2(y) \}$, where $h_1$ and $h_2$ are seen as bounded adjustment functions, normalised to have means zero under $f_1$ and $f_2$. This defines a bivariate distribution for $(X,Y)$ with the specified marginal densities $f_1$ and $f_2$, with an interval of permissible values of $α$, both positive and negative; in particular, independence corresponds to an innter point in the adjustments parameter region. Applications to bivariate Poisson distributions, allowing both positive and negative correlation, are discussed. As illustration I provide a more accurate and extended analysis of a Poisson pairs dataset, pertaining to competing seeds and plants, for $n=958$ plots of soil, earlier analysed in the well-cited paper Lakshminarayana, Pandit, Rao, Srinivasa (1999). The general apparatus is also shown to work for negatively correlated binomials. Those methods are illustrated in a meta-analysis framework for two-by-two tables across different studies, pertaining to the Audit-C screening questionnaire for alcohol use disorders, where again negative correlation is demonstrated, between $X$, the number of correct `yes', and $Y$, the number of correct `no'. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17585 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Modelling pairs of Poissons and binomials with negative correlation Hjort, Nils Lid Methodology Statistics Theory Suppose $f_1(x)$ and $f_2(y)$ are given marginals for pairs $(x,y)$. I consider the construction $f_1(x)f_2(y)\{ 1+αh_1(x)h_2(y) \}$, where $h_1$ and $h_2$ are seen as bounded adjustment functions, normalised to have means zero under $f_1$ and $f_2$. This defines a bivariate distribution for $(X,Y)$ with the specified marginal densities $f_1$ and $f_2$, with an interval of permissible values of $α$, both positive and negative; in particular, independence corresponds to an innter point in the adjustments parameter region. Applications to bivariate Poisson distributions, allowing both positive and negative correlation, are discussed. As illustration I provide a more accurate and extended analysis of a Poisson pairs dataset, pertaining to competing seeds and plants, for $n=958$ plots of soil, earlier analysed in the well-cited paper Lakshminarayana, Pandit, Rao, Srinivasa (1999). The general apparatus is also shown to work for negatively correlated binomials. Those methods are illustrated in a meta-analysis framework for two-by-two tables across different studies, pertaining to the Audit-C screening questionnaire for alcohol use disorders, where again negative correlation is demonstrated, between $X$, the number of correct `yes', and $Y$, the number of correct `no'. |
| title | Modelling pairs of Poissons and binomials with negative correlation |
| topic | Methodology Statistics Theory |
| url | https://arxiv.org/abs/2605.17585 |