Improved Small Domain Estimation via Compromise Regression Weights

Shrinkage estimates of small domain parameters typically use a combination of a noisy “direct” estimate that only uses data from a specific small domain and a more stable regression estimate. When the regression model is misspecified, estimation performance for the noisier domains can suffer due to substantial shrinkage toward a poorly estimated regression surface. In this article, we introduce a new class of robust, empirically-driven regression weights that target estimation of the small domain means under potential misspecification of the global regression model. Our regression weights are a convex combination of the model-based weights associated with the best linear unbiased predictor (BLUP) and those associated with the observed best predictor (OBP). The mixing parameter in this convex combination is found by minimizing a novel, unbiased estimate of the mean-squared prediction error for the small domain means, and we label the associated small domain estimates the “compromise best predictor,” or CBP. Using a data-adaptive mixture for the regression weights enables the CBP to preserve the robustness of the OBP while retaining the main advantages of the EBLUP whenever the regression model is correct. We demonstrate the use of the CBP in an application estimating gait speed in older adults. Supplementary materials for this article are available online.

小域参数的收缩估计通常结合仅使用特定小域自身数据的含噪“直接”估计，与稳定性更优的回归估计共同构建。当回归模型设定错误时，由于向估计效果不佳的回归曲面产生过度收缩，噪声水平较高的小域的估计性能可能会大幅受损。本文提出一类全新的、经验驱动的稳健回归权重，用于在全局回归模型存在潜在设定错误的场景下，实现小域均值的估计。所提出的回归权重是两类权重的凸组合：一类是与最佳线性无偏预测（best linear unbiased predictor, BLUP）相关的模型基权重，另一类是与观测最佳预测（observed best predictor, OBP）相关的权重。该凸组合中的混合参数，通过最小化小域均值均方预测误差的一种新型无偏估计量进行求解，我们将由此得到的小域估计量命名为“折中最佳预测（compromise best predictor, CBP）”。通过为回归权重采用数据自适应混合策略，折中最佳预测（CBP）能够在回归模型设定正确时，保留经验最佳线性无偏预测（Empirical Best Linear Unbiased Predictor, EBLUP）的核心优势，同时兼具观测最佳预测的稳健性。最后，我们通过一项老年人步态速度估计的应用案例，展示了折中最佳预测的具体使用方法。本文的补充材料可在线获取。