Department of Statistics, Lanzhou University of Finance and Economics, Lanzhou, Gansu, China.
Department of Statistics Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai, China.
PLoS One. 2023 Nov 16;18(11):e0288639. doi: 10.1371/journal.pone.0288639. eCollection 2023.
Unit level model is one of the classical models in small area estimation, which plays an important role with unit information data. Empirical Bayesian(EB) estimation, as the optimal estimation under normal assumption, is the most commonly used parameter estimation method in unit level model. However, this kind of method is sensitive to outliers, and EB estimation will lead to considerable inflation of the mean square error(MSE) when there are outliers in the responses yij. In this study, we propose a robust estimation method for the unit-level model with outliers based on the minimum density power divergence. Firstly, by introducing the minimum density power divergence function, we give the estimation equation of the parameters of the unit level model, and obtain the asymptotic distribution of the robust parameters. Considering the existence of tuning parameters in the robust estimator, an optimal parameter selection algorithm is proposed. Secondly, empirical Bayesian predictors of unit and area mean in finite populations are given, and the MSE of the proposed robust estimators of small area means is given by bootstrap method. Finally, we verify the superior performance of our proposed method through simulation data and real data. Through comparison, our proposed method can can solve the outlier situation better.
单位水平模型是小区域估计中的经典模型之一，它在使用单位信息数据时起着重要作用。经验贝叶斯（EB）估计作为正态假设下的最优估计，是单位水平模型中最常用的参数估计方法。然而，这种方法对异常值很敏感，当响应 yij 中存在异常值时，EB 估计会导致均方误差（MSE）的显著膨胀。在这项研究中，我们提出了一种基于最小密度幂离差的带有异常值的单位水平模型的稳健估计方法。首先，通过引入最小密度幂离差函数，给出了单位水平模型参数的估计方程，并得到了稳健参数的渐近分布。考虑到稳健估计中的调整参数的存在，提出了一种最优参数选择算法。其次，给出了有限总体中单位和区域均值的经验贝叶斯预测，并通过自举法给出了小区域均值的拟议稳健估计的 MSE。最后，通过模拟数据和真实数据验证了我们所提出方法的优越性能。通过比较，我们提出的方法可以更好地解决异常值情况。
