Asymptotic mixed normality of maximum likelihood estimator for Ewens--Pitman partition
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916717558497280 |
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| author | Koriyama, Takuya Matsuda, Takeru Komaki, Fumiyasu |
| author_facet | Koriyama, Takuya Matsuda, Takeru Komaki, Fumiyasu |
| contents | This paper investigates the asymptotic properties of parameter estimation for the Ewens--Pitman partition with parameters $0<α<1$ and $θ>-α$. Especially, we show that the maximum likelihood estimator (MLE) of $α$ is $n^{α/2}$-consistent and converges to a variance mixture of normal distributions, where the variance is governed by the Mittag-Leffler distribution. Moreover, we show that a proper normalization involving a random statistic eliminates the randomness in the variance. Building on this result, we construct an approximate confidence interval for $α$. Our proof relies on a stable martingale central limit theorem, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_01949 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotic mixed normality of maximum likelihood estimator for Ewens--Pitman partition Koriyama, Takuya Matsuda, Takeru Komaki, Fumiyasu Statistics Theory Probability This paper investigates the asymptotic properties of parameter estimation for the Ewens--Pitman partition with parameters $0<α<1$ and $θ>-α$. Especially, we show that the maximum likelihood estimator (MLE) of $α$ is $n^{α/2}$-consistent and converges to a variance mixture of normal distributions, where the variance is governed by the Mittag-Leffler distribution. Moreover, we show that a proper normalization involving a random statistic eliminates the randomness in the variance. Building on this result, we construct an approximate confidence interval for $α$. Our proof relies on a stable martingale central limit theorem, which is of independent interest. |
| title | Asymptotic mixed normality of maximum likelihood estimator for Ewens--Pitman partition |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2207.01949 |