Rong Jiang and Keming Yu's contribution to the Discussion of ‘Estimating means of bounded random variables by betting’ by Waudby-Smith and Ramdas
本文是对一篇关于用赌博方法估计有界随机变量均值的论文的讨论，提出了三个评论：参数估计器数量过多可能导致误差叠加、方法能否推广到分位数以外的统计量、以及能否扩展到无界观测值。
We want to congratulate the authors on estimating means of bounded random variables in both with- and without-replacement settings. The authors constructed confidence intervals and time-uniform confidence sequences for the mean of a bounded random variable using test supermartingale technique. It deepens our understanding of the confidence sequence. Confidence sequence is one particular tool in sequential design that facilitates anytime-valid inference. In particular, confidence sequence is a sequence of confidence intervals that is valid at data-dependent stopping times. We offer three comments. First, the CtPrPl−EB in Theorem 2 involves (λt)t=1∞. The authors recommend the predictable plug-in (λtPrPl−EB)t=1∞ given by Whether so many parameter estimators λtPrPl−EB will lead to the superposition of errors and the failure of the method. The Hoeffding process (MtH(m))t=0∞ in equation (8) only need one λ. In particular, when t<100, is CtPrPl−EB still correct? See Figure 2, we can see the results are bed when t<100. Thus, whether Theorem 2 needs to add restrictions on t. Moreover, whether CtPrPl−EB is sensitive to c in λtPrPl−EB, and if so, how to select c, although 1/2 or 3/4 is recommended. Second, the author mentioned their test supermartingle can also be inverted to get a confidence sequence for any quantile. How about mode (Chernoff, 1964), expectile (Newey & Powell, 1987), and extremile (Daouia et al., 2019)? Third, the authors consider arbitrary distribution but bounded distribution which implies all moments exist and require a Chernoff-type assumption on the distribution resulting in O(logt/t) shrinkage rates for the confidence sequences. Recently, Wang and Ramdas (2023) show that employing Catoni’s estimator improves the rate to O(loglog2t/t) under weaker assumptions on the distribution ((1+δ)-th moment bound). We wonder if the methods and results can be generalised to unbounded observations, since the σ2-bounded-variance assumption (Wang & Ramdas, 2023) is more realistic and easier to verify.