Sub-Gaussian estimators of the mean of a random vector (1702.00482v1)

Abstract: We study the problem of estimating the mean of a random vector $X$ given a sample of $N$ independent, identically distributed points. We introduce a new estimator that achieves a purely sub-Gaussian performance under the only condition that the second moment of $X$ exists. The estimator is based on a novel concept of a multivariate median.

• The paper presents a new estimator for random vector mean achieving sub-Gaussian performance with heavy tails, requiring only the second moment.
• The estimator achieves performance bounds similar to standard sub-Gaussian methods, requiring only the existence of the covariance matrix.
• This robust estimator applies to fields like economics and machine learning with heavy tails; future work should optimize computational efficiency.

Sub-Gaussian Estimators of the Mean of a Random Vector

The paper presents a novel approach to estimating the mean of a random vector, focusing on achieving a sub-Gaussian performance even in situations where the distribution is not necessarily sub-Gaussian and may display heavy tails. In the paper, the authors introduced a new estimator that presumes solely the existence of the second moment of the random vector. This advancement rests on defining a new construct of multivariate median, based on the concept of "median-of-means tournaments".

Overview

The research addresses the challenge of estimating the mean vector $[u = \text{E}X]$ of a $d$-dimensional random vector $X$, utilizing $N$ independent and identically distributed samples. In classical statistical practice, the quality of an estimator is often gauged by its mean squared error, where the sample mean offers a baseline risk measure. However, it is not necessarily optimal, particularly in the presence of heavy tails.

The authors propose an estimator that achieves sub-Gaussian performance bounds reminiscent of the performance seen with sample means under ideal conditions, such as standard normal distributions. This feat was achieved without assuming the distribution itself is sub-Gaussian. Under the sole requirement that the covariance matrix exists, they demonstrate that their estimator performs as efficiently in terms of the bounds it observes, like traditional sub-Gaussian estimators with stricter assumptions.

Notable Results

The paper proves that their proposed estimator, under minimal conditions, can achieve performance bounds comparable to those derived under sub-Gaussian distribution assumptions. Specifically, the authors delineate that for the estimator $\hat{u}_N$, with probability at least $1 - \delta$,

$$|\hat{u}_N - u| \leq C \left(\sqrt{\frac{\text{Tr}(E)}{N}} + \sqrt{\frac{\lambda_{\text{max}} \log (2/\delta)}{N}}\right),$$

where $\lambda_{\text{max}}$ is the maximum eigenvalue of the covariance matrix $E$, and $C$ is a universal constant. This finding marks a significant stride in advancing estimators that provide robust performance in settings compounded by heavy-tailed distributions.

Theoretical and Practical Implications

The findings suggest profound implications for a diverse array of applications where robust mean estimation in multi-dimensional settings is needed, and the distribution tails are heavy. The research could impact fields such as economics and machine learning, where distributions often exhibit such characteristics.

By relying less heavily on stringent distributional prerequisites, the estimator enhances the robustness and reliability of statistical analysis, thereby enhancing decision-making processes reliant on these statistical measurements. Moreover, this work extends to infinite-dimensional spaces, offering further utility in applications dealing with high-dimensional or functional data.

Future Directions

Future research could explore optimization approaches to improve computational efficiency in implementing this estimator. Although the theoretical performance is promising, practical computation, especially in high dimensions, presents hurdles yet to be surmounted. Additionally, studying variations or adaptations to this estimator that further simplify computation without compromising performance might prove fruitful.

Further exploration into the "median-of-means tournaments" might yield insights that can enhance other statistical estimators' robustness and applicability under diverse conditions. Consideration of more comprehensive scenarios, potentially incorporating additional distributional assumptions or constraints, could enrich the estimator’s utility.

Through this work, the authors offer a notable contribution toward understanding and implementing mean estimation techniques resilient to the challenges posed by heavy-tailed distributions, broadening the horizon for applied statistical and data analytic methodologies.