Catoni-style Confidence Sequences under Infinite Variance
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In this paper, we provide an extension of confidence sequences for settings where the variance of the data-generating distribution does not exist or is infinite. Confidence sequences furnish confidence intervals that are valid at arbitrary data-dependent stopping times, naturally having a wide range of applications. We first establish a lower bound for the width of the Catoni-style confidence sequences for the finite variance case to highlight the looseness of the existing results. Next, we derive tight Catoni-style confidence sequences for data distributions having a relaxed bounded p^th-moment, where p ∈ (1,2], and strengthen the results for the finite variance case of p =2. The derived results are shown to better than confidence sequences obtained using Dubins-Savage inequality.
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