Mean Estimation in Banach Spaces Under Infinite Variance and Martingale Dependence
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908278904061952 |
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| author | Whitehouse, Justin Chugg, Ben Martinez-Taboada, Diego Ramdas, Aaditya |
| author_facet | Whitehouse, Justin Chugg, Ben Martinez-Taboada, Diego Ramdas, Aaditya |
| contents | We consider estimating the shared mean of a sequence of heavy-tailed random variables taking values in a Banach space. In particular, we revisit and extend a simple truncation-based mean estimator first proposed by Catoni and Giulini. While existing truncation-based approaches require a bound on the raw (non-central) second moment of observations, our results hold under a bound on either the central or non-central $p$th moment for some $p \in (1,2]$. Our analysis thus handles distributions with infinite variance. The main contributions of the paper follow from exploiting connections between truncation-based mean estimation and the concentration of martingales in smooth Banach spaces. We prove two types of time-uniform bounds on the distance between the estimator and unknown mean: line-crossing inequalities, which can be optimized for a fixed sample size $n$, and iterated logarithm inequalities, which match the tightness of line-crossing inequalities at all points in time up to a doubly logarithmic factor in $n$. Our results do not depend on the dimension of the Banach space, hold under martingale dependence, and all constants in the inequalities are known and small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11271 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean Estimation in Banach Spaces Under Infinite Variance and Martingale Dependence Whitehouse, Justin Chugg, Ben Martinez-Taboada, Diego Ramdas, Aaditya Statistics Theory Probability Methodology We consider estimating the shared mean of a sequence of heavy-tailed random variables taking values in a Banach space. In particular, we revisit and extend a simple truncation-based mean estimator first proposed by Catoni and Giulini. While existing truncation-based approaches require a bound on the raw (non-central) second moment of observations, our results hold under a bound on either the central or non-central $p$th moment for some $p \in (1,2]$. Our analysis thus handles distributions with infinite variance. The main contributions of the paper follow from exploiting connections between truncation-based mean estimation and the concentration of martingales in smooth Banach spaces. We prove two types of time-uniform bounds on the distance between the estimator and unknown mean: line-crossing inequalities, which can be optimized for a fixed sample size $n$, and iterated logarithm inequalities, which match the tightness of line-crossing inequalities at all points in time up to a doubly logarithmic factor in $n$. Our results do not depend on the dimension of the Banach space, hold under martingale dependence, and all constants in the inequalities are known and small. |
| title | Mean Estimation in Banach Spaces Under Infinite Variance and Martingale Dependence |
| topic | Statistics Theory Probability Methodology |
| url | https://arxiv.org/abs/2411.11271 |