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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2407.06522 |
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| _version_ | 1866912509654466560 |
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| author | AL-Najafi, Amenah Tirnakli, Ugur Nelson, Kenric P. |
| author_facet | AL-Najafi, Amenah Tirnakli, Ugur Nelson, Kenric P. |
| contents | Heavy-tailed distributions are infamously difficult to estimate because their moments tend to infinity as the shape of the tail decay increases. Nevertheless, this study shows the utilization of a modified group of moments for estimating a heavy-tailed distribution. These modified moments are determined from powers of the original distribution. The nth-power distribution is guaranteed to have finite moments up to n-1. Samples from the nth-power distribution are drawn from n-tuple Independent Approximates, which are the set of independent samples grouped into n-tuples and sub-selected to be approximately equal to each other. We show that Independent Approximates are a maximum likelihood estimator for the generalized Pareto and the Student's t distributions, which are members of the family of coupled exponential distributions. We use the first (original), second, and third power distributions to estimate their zeroth (geometric mean), first, and second power-moments respectively. In turn, these power-moments are used to estimate the scale and shape of the distributions. A least absolute deviation criteria is used to select the optimal set of Independent Approximates. Estimates using higher powers and moments are possible though the number of n-tuples that are approximately equal may be limited. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06522 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Independent Approximates provide a maximum likelihood estimate for heavy-tailed distributions AL-Najafi, Amenah Tirnakli, Ugur Nelson, Kenric P. Methodology Heavy-tailed distributions are infamously difficult to estimate because their moments tend to infinity as the shape of the tail decay increases. Nevertheless, this study shows the utilization of a modified group of moments for estimating a heavy-tailed distribution. These modified moments are determined from powers of the original distribution. The nth-power distribution is guaranteed to have finite moments up to n-1. Samples from the nth-power distribution are drawn from n-tuple Independent Approximates, which are the set of independent samples grouped into n-tuples and sub-selected to be approximately equal to each other. We show that Independent Approximates are a maximum likelihood estimator for the generalized Pareto and the Student's t distributions, which are members of the family of coupled exponential distributions. We use the first (original), second, and third power distributions to estimate their zeroth (geometric mean), first, and second power-moments respectively. In turn, these power-moments are used to estimate the scale and shape of the distributions. A least absolute deviation criteria is used to select the optimal set of Independent Approximates. Estimates using higher powers and moments are possible though the number of n-tuples that are approximately equal may be limited. |
| title | Independent Approximates provide a maximum likelihood estimate for heavy-tailed distributions |
| topic | Methodology |
| url | https://arxiv.org/abs/2407.06522 |