Stationary birth-death processes generating inflation-deflation distributions: Avoiding the issue of dominance
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918508265209856 |
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| author | Skulpakdee, Wanrudee Hunkrajok, Mongkol |
| author_facet | Skulpakdee, Wanrudee Hunkrajok, Mongkol |
| contents | A mixture of two or more count distributions has become deeply embedded in the analysis of excess counts, often relative to the stationary (equilibrium) distributions of birth-death processes such as the geometric, Poisson, Poisson-Lindley (PL), negative binomial (NB), hyper-Poisson (HP), and Conway-Maxwell-Poisson (CMP) distributions. However, the mechanism by which excess counts arise--namely, through modifications of the birth and death rates in the base distributions--has not yet been directly examined in the research literature. All well-known inflation mixture distributions are, in fact, parameterizations of the stationary distributions of birth-death processes. Thus, although the resulting distributions share the same shapes, they arise from distinct mechanisms and are not equivalent in regression analyses. This paper focuses on inflation-deflation stationary distributions arising from modified birth-death processes that form an exponential family and introduces two types of such distributions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17764 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stationary birth-death processes generating inflation-deflation distributions: Avoiding the issue of dominance Skulpakdee, Wanrudee Hunkrajok, Mongkol Methodology A mixture of two or more count distributions has become deeply embedded in the analysis of excess counts, often relative to the stationary (equilibrium) distributions of birth-death processes such as the geometric, Poisson, Poisson-Lindley (PL), negative binomial (NB), hyper-Poisson (HP), and Conway-Maxwell-Poisson (CMP) distributions. However, the mechanism by which excess counts arise--namely, through modifications of the birth and death rates in the base distributions--has not yet been directly examined in the research literature. All well-known inflation mixture distributions are, in fact, parameterizations of the stationary distributions of birth-death processes. Thus, although the resulting distributions share the same shapes, they arise from distinct mechanisms and are not equivalent in regression analyses. This paper focuses on inflation-deflation stationary distributions arising from modified birth-death processes that form an exponential family and introduces two types of such distributions. |
| title | Stationary birth-death processes generating inflation-deflation distributions: Avoiding the issue of dominance |
| topic | Methodology |
| url | https://arxiv.org/abs/2605.17764 |