基于γ散度的单元水平模型小域稳健估计

Robust Small Area Estimation for Unit Level Model with γ Divergence

在基于抽样调查数据对总体参数进行估计的方法中,小域估计方法能够借助于辅助信息对小样本乃至无样本区域的参数进行有效的估计,并被广泛应用于抽样估计领域。单元水平模型作为小域估计的基本模型之一,是处理单元级别数据估计的有力工具之一。在单元水平模型的应用条件中,需假定区域随机误差和模型随机误差均服从正态分布。然而,在抽样调查中,满足这一条件的调查数据是很少的,尤其是在观测数据中出现离群值时。不满足正态性假设条件下的小域估计量会产生较大的偏差和均方误,因此有必要研究针对正态性假设和离群观测值不敏感的稳健估计方法。通过引入γ散度和γ似然函数,构建了基于单元水平模型的小域稳健估计方法,得到了模型参数的稳健估计和小域目标变量的稳健估计。与现有的稳健估计方法相比,所提新方法能更好地处理区域随机误差和模型随机误差非正态的情形,对于目标变量存在离群观测的情形,具有更好的稳健性,估计均方误更小。在利用模拟数据进行验证中,比较了不同误差分布情形下几类常用估计方法得到的估计量的均方误差,并进一步探究了随着污染分布的方差和比率变化,所得估计量的均方误差变化情形。最后,通过应用于经典的小域估计数据,进一步验证了所提新方法的可行性。

In the methods of estimating population parameters based on sample survey data,small area estimation methods are able to effectively estimate parameters for small samples and even non-sampled areas with the help of auxiliary information,and are widely used in the field of sample estimation.The unit level model,as one of the basic models for small domain estimation,is one of the powerful tools for dealing with the estimates of data at the unit level.In the applications of the unit level model,it is assumed that both the area random error and the model random error follow normal distributions.However,it is rare for survey data to satisfy this condition in a sample survey,especially when there are outliers in the observations.Small area estimators which do not satisfy the normality assumptions can result in large biases and mean squared errors.Therefore,there is a need to investigate robust estimation methods that are insensitive to the normality assumption and outlier observations.The robust estimation method of unit level model in small area estimation is established by introducingγdivergence andγlikelihood function,and the robust estimation of model parameters and target variables in small area is obtained.Compared with the existing robust estimation methods,the proposed method can better deal with the case of non-normal area random error and model random error,and the proposed method has better robustness and smaller mean square error of the estimator when there is outlier observations of the target variable.In this paper,simulation data are used to verify the proposed estimation method.Comparing the mean square error of the estimators obtained by several common estimation methods under different error distributions,and further exploring the change of the mean square error of the estimators as the variance and proportion of the contaminated distribution change.Finally,the proposed method is applied to classical small area estimation data to further verify the feasibility of the proposed method.