In this article, we propose a conditional model estimator (cmmse) for the design-based mean squared error (dMSE) of a small area mean estimator under the basic unit level model. The mean squared error dMSE refers to the variability of a small area estimator over all possible sample selections. It is different from the model mean squared error (mMSE), traditionally used to measure the efficiency in small area estimation problems. For known model parameters, Rao, Rubin-Bleuer & Estevao [Rao et al., Survey Methodology 2018; 44, 151–166] showed that dMSE depends on two quadratic finite population parameters. A design estimator of dMSE, denoted as dmse, is obtained by substituting the quadratic parameters with their corresponding design unbiased estimators. Rao, Rubin-Bleuer & Estevao [Rao et al., Survey Methodology 2018; 44, 151–166] proposed a composite MSE estimator (cmse) based on both the design and the model. This estimator is defined as a weighted average between the design-based dmse and a model-based estimator (mmse). Given known variance components, we obtain a new formula for dMSE that accounts for the estimation of the fixed model coefficients. Our conditional model MSE estimator cmmse is obtained by replacing the quadratic finite population parameters by their best predictions under the model, in the new formula of dMSE. Properties of the proposed estimator are studied in terms of design bias, relative root mean squared error, coverage rate and a score function of the confidence intervals.

中文翻译：

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嵌套误差线性回归模型下基于小面积均值模型的估计量的基于设计的均方误差估计

在本文中，我们针对基本单元级模型下小面积均值估计器的基于设计的均方误差 (dMSE) 提出了条件模型估计器 (cmmse)。均方误差 dMSE 是指小面积估计量在所有可能的样本选择上的可变性。它不同于传统上用于衡量小面积估计问题的效率的模型均方误差 (mMSE)。对于已知的模型参数，Rao、Rubin-Bleuer 和 Estevao [Rao 等人，Survey Methodology 2018；[44, 151–166] 表明 dMSE 取决于两个二次有限总体参数。dMSE 的设计估计量，表示为 dmse，是通过用相应的设计无偏估计量替换二次参数而获得的。Rao、Rubin-Bleuer 和 Estevao [Rao 等人，2018 年调查方法；44, 151–166] 提出了一种基于设计和模型的复合 MSE 估计器 (cmse)。该估计量定义为基于设计的 dmse 和基于模型的估计量 (mmse) 之间的加权平均值。给定已知的方差分量，我们获得了一个新的 dMSE 公式，它解释了固定模型系数的估计。我们的条件模型 MSE 估计器 cmmse 是通过在 dMSE 的新公式中用模型下的最佳预测替换二次有限总体参数来获得的。在设计偏差、相对均方根误差、覆盖率和置信区间的得分函数方面研究了所提出的估计器的特性。