Control-variate and polynomial-maximization (PMM) estimators are optimized at a single fixed distribution, yet they are increasingly proposed to strengthen hypothesis tests, which decide between two regions of a parameter family. We give a closed-form criterion for when this transfer succeeds. For an H0-centered augmentation of a target moment statistic with null-optimized weight vector K0, the alternative-side expectation equals the target plus K0^T mu_a,H1, where mu_a,H1 is the alternative-side mean of the augmenting basis. Null-variance reduction therefore transfers without bias only under the orthogonality condition K0^T mu_a,H1 = 0; requiring each augmenting function to remain mean-zero is sufficient but not necessary. We instantiate the criterion on the recently proposed Wiedermann-Shi third-order cumulant test for measurement-error independence. A second-order PMM correction is unbiased and lower-variance under the null (relative efficiency >= 1 in all 36 conditions; aggregated mean ARE values 1.23-5.16; Type-I 0.04-0.09), yet provably inconsistent under the alternative: the antisymmetric polynomial auxiliaries acquire nonzero means, attenuating the target by a closed-form factor and costing 7-52 percentage points of power, worst where the test is strongest and worsening under heavy tails. A fourth-order variant reduces variance (ratio 1.127) but fails a nuisance guard (rejection 0.295 versus 0.10). We derive a reusable alternative-consistency acceptance gate for variance-reduced test statistics.

翻译：控制变量法与多项式最大化（PMM）估计量虽针对单一固定分布优化，却常被用于增强需在参数族两区域间作出判定的假设检验。本文给出了这种迁移成功与否的闭式准则。对于以零假设为中心的增广目标矩统计量，其权重向量经零优化设为K0时，备择假设下的期望值等于目标统计量加上K0^T mu_a,H1，其中mu_a,H1为增广基在备择假设侧的均值。因此，仅在正交条件K0^T mu_a,H1 = 0成立时，零假设方差缩减才能无偏迁移；要求每个增广函数均保持零均值是充分但不必要条件。我们将该准则应用于近期提出的Wiedermann-Shi三阶累积量检验（用于测量误差独立性）。二阶PMM校正虽在零假设下实现无偏且更低方差（全部36种条件下相对效率≥1；聚合平均ARE值1.23-5.16；第一类错误率0.04-0.09），但在备择假设下被证明不一致：反对称多项式辅助量获得非零均值，使目标统计量衰减闭式因子，导致统计功效损失7-52个百分点，其中检验最强时最严重，且厚尾分布下进一步恶化。四阶变体虽降低方差（比率1.127），却无法通过干扰项检验（拒绝率0.295 vs 0.10）。我们推导出方差缩减检验统计量的备择假设一致性可复用准入门限。