Sub-Gaussian mean estimators (1509.05845v1)

Abstract: We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a non-asymptotic point of view. In particular, we define estimators with a sub-Gaussian behavior even for certain heavy-tailed distributions. We also prove various impossibility results for mean estimators.

An Analytical Evaluation of Sub-Gaussian Mean Estimators

This paper, authored by Luc Devroye, Matthieu Lerasle, Gábor Lugosi, and Roberto I. Oliveira, presents a thorough investigation into the estimation of means from independent and identically distributed (i.i.d.) samples, with a specific focus on sub-Gaussian performance, even for distributions with heavy tails. The core aim is to balance accurate mean estimation with robustness against large deviations. The authors also discuss various impossibility results, raising awareness about the limitations inherent in mean estimation.

Frameworks for Mean Estimation

The paper opens with the identification of the classical problem of estimating the mean of a distribution. The Standard Empirical Mean (SEM) is traditionally employed for this purpose, which performs adequately under normality and finite variance assumptions. However, its reliance on the Central Limit Theorem (CLT) falls short when applied to heavy-tailed distributions with finite but potentially large-going second moments. The authors propose sub-Gaussian estimators as robust alternatives that could potentially extend the sem's optimality in terms of deviation bounds.

Sub-Gaussian Estimators

Central to the paper is the exploration of what the authors term as sub-Gaussian estimators — estimators that offer non-asymptotic guarantees with constraints on their deviation probabilities from the true mean, while still preserving Gaussian-like tail behaviors. For these estimators, they define the problem space concerning specific distribution classes:

• $P_2$: Distributions with finite second moments.
• $P_2^{\sigma^2}$: Distributions with known variance.
• $P_{\text{krt} \leq \kappa}$: Distributions with bounded kurtosis.

The authors demonstrate that under a range of distributions, there do exist sub-Gaussian estimators which perform in a manner that is superior to or on par with the traditional empirical mean. The constructions of these estimators often involve adaptive methods utilizing data-driven truncations and aggregation techniques, such as the median of means.

Impossibility Results

Possibly the most striking findings of this paper include several impossibility results that suggest inherent limitations under broad distributional classes. For instance, the authors demonstrate that without a known variance or bounded kurtosis, no mean estimator can universally achieve sub-Gaussian performance. They also show that for particular distribution families, such as scaled Bernoulli and Laplace, certain performance thresholds cannot be surpassed, highlighting the contrasts between achievable and theoretical bounds within mean estimation.

Regularity Conditions and Practical Constructions

The authors develop a notion of $k$-regularity, encapsulating assumptions more robust than simple bounds on moments to facilitate sub-Gaussian estimation. They examine symmetries and higher moment conditions, precisely quantifying how these assumptions can lead to feasible estimator construction. For distributions with restricted kurtosis, constructions approach the theoretical optimum concerning deviation rates.

The authors provide formalized proofs, demonstrating instances and conditions where sub-Gaussian estimators not only exist but perform optimally given certain trade-offs or additional distributional insights. Their strategy aligns with empirical means under slight modifications, such as median of means modalities, empowering practical statistical learning imperatives.

Implications and Future Directions

While constructing robust estimators is a non-trivial and computationally challenging task, the findings underscore potential improvements in statistical methodologies, particularly in machine learning and data science applications dealing with non-Gaussian data. The implications extend to empirical risk minimization contexts where optimization of loss functions requires considerations of sub-Gaussian behavior.

Future research directions could explore deriving truly sub-Gaussian confidence intervals and extending these concepts to vectorial or functional data. Additionally, practical considerations such as efficiency trade-offs, robustness to higher-dimensional settings, and online or iterative learning paradigms respond to contemporary data-driven needs.

In conclusion, this paper represents a significant foray into the nuanced requirements for mean estimation beyond traditional paradigms, offering both a theoretic framework and practical techniques poised for further exploration and application in statistics and related fields.