Determine when to apply a finite-population correction to SDR variance estimates

Determine when a multiplicative finite-population correction should be applied to Successive Difference Replication variance estimates, particularly when the sampling fractions and domain fractions vary substantially across strata.

Background

Successive Difference Replication (SDR) variance estimation does not automatically incorporate the finite-population correction present in the corresponding stratified simple-random-sampling variance formula. The paper notes that Fay and Train, along with other authors, have suggested using a multiplicative correction in some circumstances, but does not establish a general rule for when that correction is appropriate.

The issue is especially difficult when the sampling fraction n_h/N_h and the domain fraction α_h differ markedly across strata. In such settings, the denominator of the paper’s relative-bias expression reflects the finite-population correction, but the heterogeneous stratum-specific fractions make the appropriate overall correction unclear.

References

SDR has no automatic finite-population correction, although formula (\ref{Vtru}) does. Fay and Train and other authors suggest that such a multiplicative correction to SDR variances estimates should be used in some circumstances, although it is not clear when.

— SDR Variance Estimates in Small Domains  (2608.17353 - Slud et al., 18 Aug 2026) in Remark 2 (label FPC), immediately following Proposition 1 in Section 5, “Theoretical Formula for SDR Bias in Stratified SRS Samples”