标题： 挑战经验均值和经验方差：偏差研究 显示英文标题

标题： Challenging the empirical mean and empirical variance: a deviation study

摘要： 我们提出了新的M估计量，用于实值随机变量的均值和方差，基于PAC-Bayes界。 我们分析了这些估计量在样本分布具有有界方差或有界方差和有界峰度时的非渐近最小最大性质。 在这些弱假设下，允许重尾分布，我们证明了经验均值的最坏情况偏差是次优的。 我们确实证明了对于任何置信水平，存在某个M估计量，其偏差与高斯统计样本的经验均值的偏差处于同一数量级，即使统计样本是重尾的。 实验表明，这些新估计量的表现甚至优于我们的界限所预测，显示出在非高斯样本分布（如两个高斯测度的混合）中，所有概率水平的偏差分位数函数都低于经验均值。 显示英文摘要

摘要： We present new M-estimators of the mean and variance of real valued random variables, based on PAC-Bayes bounds. We analyze the non-asymptotic minimax properties of the deviations of those estimators for sample distributions having either a bounded variance or a bounded variance and a bounded kurtosis. Under those weak hypotheses, allowing for heavy-tailed distributions, we show that the worst case deviations of the empirical mean are suboptimal. We prove indeed that for any confidence level, there is some M-estimator whose deviations are of the same order as the deviations of the empirical mean of a Gaussian statistical sample, even when the statistical sample is instead heavy-tailed. Experiments reveal that these new estimators perform even better than predicted by our bounds, showing deviation quantile functions uniformly lower at all probability levels than the empirical mean for non Gaussian sample distributions as simple as the mixture of two Gaussian measures.