Improved Catoni-Type Confidence Sequences for Estimating the Mean When the Variance Is Infinite
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913493743042560 |
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| author | Wei, Chengfu Stoyanov, Jordan Chen, Yiming Chen, Zijun |
| author_facet | Wei, Chengfu Stoyanov, Jordan Chen, Yiming Chen, Zijun |
| contents | We consider a discrete time stochastic model with infinite variance and study the mean estimation problem as in Wang and Ramdas (2023). We refine the Catoni-type confidence sequence (abbr. CS) and use an idea of Bhatt et al. (2022) to achieve notable improvements of some currently existing results for such model.
Specifically, for given $α\in (0, 1]$, we assume that there is a known upper bound $ν_α > 0$ for the $(1 + α)$-th central moment of the population distribution that the sample follows. Our findings replicate and `optimize' results in the above references for $α= 1$ (i.e., in models with finite variance) and enhance the results for $α< 1$. Furthermore, by employing the stitching method, we derive an upper bound on the width of the CS as $\mathcal{O} \left(((\log \log t)/t)^{\fracα{1+α}}\right)$ for the shrinking rate as $t$ increases, and $\mathcal{O}(\left(\log (1/δ)\right)^{\frac{α}{1+α}})$ for the growth rate as $δ$ decreases. These bounds are improving upon the bounds found in Wang and Ramdas (2023). Our theoretical results are illustrated by results from a series of simulation experiments. Comparing the performance of our improved $α$-Catoni-type CS with the bound in the above cited paper indicates that our CS achieves tighter width. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Catoni-Type Confidence Sequences for Estimating the Mean When the Variance Is Infinite Wei, Chengfu Stoyanov, Jordan Chen, Yiming Chen, Zijun Statistics Theory Probability 60G42, 62F25, 62L12 We consider a discrete time stochastic model with infinite variance and study the mean estimation problem as in Wang and Ramdas (2023). We refine the Catoni-type confidence sequence (abbr. CS) and use an idea of Bhatt et al. (2022) to achieve notable improvements of some currently existing results for such model. Specifically, for given $α\in (0, 1]$, we assume that there is a known upper bound $ν_α > 0$ for the $(1 + α)$-th central moment of the population distribution that the sample follows. Our findings replicate and `optimize' results in the above references for $α= 1$ (i.e., in models with finite variance) and enhance the results for $α< 1$. Furthermore, by employing the stitching method, we derive an upper bound on the width of the CS as $\mathcal{O} \left(((\log \log t)/t)^{\fracα{1+α}}\right)$ for the shrinking rate as $t$ increases, and $\mathcal{O}(\left(\log (1/δ)\right)^{\frac{α}{1+α}})$ for the growth rate as $δ$ decreases. These bounds are improving upon the bounds found in Wang and Ramdas (2023). Our theoretical results are illustrated by results from a series of simulation experiments. Comparing the performance of our improved $α$-Catoni-type CS with the bound in the above cited paper indicates that our CS achieves tighter width. |
| title | Improved Catoni-Type Confidence Sequences for Estimating the Mean When the Variance Is Infinite |
| topic | Statistics Theory Probability 60G42, 62F25, 62L12 |
| url | https://arxiv.org/abs/2409.04198 |