Inference in the presence of influential units and nonresponse for functional and non-functional survey data

在大多数（如果不是全部的话）调查中不可避免地会出现无回应的情况。从本质上讲，调查统计学家区分了单位无反应和项目无反应。当所有的调查变量都缺失或可用信息不足时，就会出现单位无响应，而当一些但不是所有的调查变量都缺失值时，就会出现项目无响应。权重调整程序通常用于处理单位无反应，而归算程序通常用于处理项目无反应。处理无反应时的主要目标是减少无反应偏差，这种偏差发生在受访者和非受访者在调查变量方面不同的情况下。在实践中，调查统计人员也面临有影响力单位的问题。有影响力的单位被正确地记录下来，并代表了价值相似的其他人口单位。它们在样本中的存在往往使经典估计量非常不稳定。当所收集的变量分布高度偏斜或某些单位具有较大的设计权重时，就会出现有影响的单位。如果一个估计器对有影响的单位的存在不太敏感，就说它是稳健的。鲁棒估计有偏，但其均方误差小于非鲁棒估计。在某些情况下，目标参数不是平均实值，而是平均函数。例如，人们可能对估计大量消费者在固定时间间隔内的平均用电量曲线感兴趣。我们提出研究在数据缺失的情况下均值曲线的估计问题。我们打算用最近邻插值的方法建立基于观测数据和输入数据的估计器的理论性质。此外，一些单位可能影响很大，这可能使平均曲线的估计量及其方差都非常不稳定。我们计划开发稳健的估计程序，研究它们的理论性质，并将所提出的方法应用于实际数据。鲁棒小面积估计近年来受到广泛关注。大多数研究都集中在兴趣的连续特征上。基于线性混合模型（LMM）的经验最佳线性无偏预测器的几个鲁棒版本已经在文献中提出。在实践中，许多变量是分类的，而不是连续的。因此，基于lmm的方法不适合。目的是提出一种基于广义lmm的鲁棒小面积估计的统一框架，以便对任何类型的变量都可以容易地获得鲁棒预测因子。在实际应用中，目标参数可能是一个复杂参数；例如，分位数。在数据缺失的情况下，双鲁棒程序已被广泛研究。如果一个估计过程在正确指定非响应模型或输入模型的情况下保持一致，则称为双鲁棒。到目前为止，文献主要集中在估计总体均值上。我们计划开发复杂参数的双重保护估计程序，并建立它们的理论性质。最后，我们建议将最近的一个称为多重鲁棒性的概念扩展到有限总体抽样。实际上，可能会拟合多个非响应模型和多个imputation模型，每个模型都涉及不同的协变量子集和可能不同的链接函数。如果正确指定了倾向得分或兴趣特征的多个模型中的任何一个是一致的，则称估计量具有多重鲁棒性。我们还计划开发多个稳健方差估计器，如果这些多个模型中的任何一个是正确的，则这些估计器对真实方差保持一致。

Nonresponse inevitably occurs in most, if not all, surveys. Essentially, survey statisticians distinguish unit nonresponse from item nonresponse. Unit nonresponse occurs when all the survey variables are missing or not enough usable information is available, whereas item nonresponse occurs when some but not all the survey variables have missing values. Weight adjustment procedures are generally used to treat unit nonresponse, whereas imputation is generally used to handle item nonresponse. The main objective when treating nonresponse is the reduction of the nonresponse bias, which occurs when respondents and nonrespondents are different with respect to the survey variables. In practice, surveys statisticians also face the problem of influential units. Influential units are correctly recorded and represent other population units similar in value. Their presence in the sample tends to make the classical estimators very unstable. Influential units occur when the distribution of the variables being collected is highly skewed or when some units have a large design weight. An estimator is said to be robust if it is not too sensitive to the presence of influential units. Robust estimators are biased but their mean square error is smaller than that of non-robust estimators. In some situations, the target parameter is not a mean real value but a mean function. For example, one may be interested in estimating the mean electricity consumption curve of a large number of consumers in a fixed time interval. We propose to study the problem of estimating the mean curve in the presence of missing data. We intend to establish the theoretical properties of estimators based on observed data and imputed data by means of nearest-neighbour imputation. Also, some units may be highly influential, which can make both the estimator of the mean curve and its variance very unstable. We plan to develop robust estimation procedures, study their theoretical properties and apply the proposed methods to real data. Robust small area estimation has received considerable attention in recent years. Most research has focussed on continuous characteristics of interest. Several robust versions of the empirical best linear unbiased predictor based on linear mixed models (LMM) have been proposed in the literature. In practice, many variables are categorical rather than continuous. As a result, methods based on LMMs are not suited. The objective is to propose a unified framework for robust small area estimation based on generalized LMMs so that robust predictors can be readily obtained for any type of variable. In practice, the target parameter may be a complex parameter; e.g., a quantile. Doubly robust procedures have been widely studied in the context of missing data. An estimation procedure is said to be doubly robust if it remains consistent if either the nonresponse model or the imputation model is correctly specified. So far, the literature has focussed on estimating a population mean. We plan to develop doubly protected estimation procedures for complex parameters and establish their theoretical properties. Finally, we propose to extend a recent concept called multiple robustness to finite population sampling. In practice, multiple nonresponse models and multiple imputation models may be fitted, each involving different subsets of covariates and possibly different link functions. An estimator is said to be multiply robust if it is consistent if any one of those multiple models, for either the propensity score or the characteristic of interest, is correctly specified. We also plan to develop multiply robust variance estimators that remain consistent for the true variance if any one of those multiple models is correct.