 Cross-fitted combination of unbiased sample means

Split independent observational blocks into two folds. Use one fold to estimate the combination weight for the two sample means in the other fold, and conversely. Each held-out combination is an unbiased estimator conditionally on its training fold because its two held-out sample means have the same expectation. A deterministic average of the two combinations remains unbiased. Under consistency of the moment estimates and balanced fold sizes, the weights approach those of the minimum-variance combination of unbiased sample means. Paired observations must stay together when splitting or bootstrapping so their covariance is preserved.