Estimating the Maximum Expected Value: An Analysis of (Nested) Cross Validation and the Maximum Sample Average

TL;DR

This paper analyzes the bias and variance of estimators for the maximum expected value, focusing on the maximum sample average and cross validation, highlighting their limitations and problem-dependent performance.

Contribution

It provides a theoretical analysis of the bias, variance, and consistency of common estimators for maximum expected value, including bounds and insights into their trade-offs.

Findings

No unbiased estimator exists for the maximum expected value.

Cross validation can reduce variance but may introduce large bias.

Estimator performance is highly problem-dependent.

Abstract

We investigate the accuracy of the two most common estimators for the maximum expected value of a general set of random variables: a generalization of the maximum sample average, and cross validation. No unbiased estimator exists and we show that it is non-trivial to select a good estimator without knowledge about the distributions of the random variables. We investigate and bound the bias and variance of the aforementioned estimators and prove consistency. The variance of cross validation can be significantly reduced, but not without risking a large bias. The bias and variance of different variants of cross validation are shown to be very problem-dependent, and a wrong choice can lead to very inaccurate estimates.