The sample mean is often used to aggregate different unbiased estimates of a parameter, producing a final estimate that is unbiased but possibly high-variance. This paper introduces the Bayesian median of means, an aggregation rule that roughly interpolates between the sample mean and median, resulting in estimates with much smaller variance at the expense of bias. While the procedure is non-parametric, its squared bias is asymptotically negligible relative to the variance, similar to maximum likelihood estimators. The Bayesian median of means is consistent, and concentration bounds for the estimator's bias and @math error are derived, as well as a fast non-randomized approximating algorithm. The performances of both the exact and the approximate procedures match that of the sample mean in low-variance settings, and exhibit much better results in high-variance scenarios. The empirical performances are examined in real and simulated data, and in applications such as importance sampling, cross-validation and bagging.
An important part of the legacy of Evarist Gine is his fundamental contributions to our understanding of U-statistics and U-processes. In this paper we discuss the estimation of the mean of multivariate functions in case of possibly heavy-tailed distributions. In such situations, reliable estimates of the mean cannot be obtained by usual U-statistics. We introduce a new estimator, based on the so-called median-of-means technique. We develop performance bounds for this new estimator that generalizes an estimate of Arcones and Gine (1993), showing that the new estimator performs, under minimal moment conditions, as well as classical U-statistics for bounded random variables. We discuss an application of this estimator to clustering.