Why stratification may hurt, & how much

There are circumstances under which stratified sampling is worse than simple random sampling, even if the allocation of the sample sizes is optimal. This phenomenon was discovered more than sixty years ago, but is not as widely known as one might expect. We provide it with lower and upper bounds for its badness as well as with an explanation.

💡 Research Summary

The paper revisits a little‑known but long‑standing paradox in survey sampling: even when the allocation of sample sizes across strata follows the optimal Neyman rule, a stratified design can yield a higher variance than a simple random sample (SRS) of the same total size. The authors first formalize the problem by partitioning a finite population into (K) strata, each with size (N_i), mean (\mu_i) and variance (\sigma_i^2). Under optimal allocation the stratum sample sizes are (n_i \propto N_i\sigma_i). The variance of the overall mean estimator under stratification is then

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