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答复的情况。从本质上讲，调查统计学家区分单位无应答和项目无应答。当所有的调查变量都缺失或没有足够的可用信息时，就会发生单位无应答，而当一些但不是所有的调查变量都有缺失值时，就会发生项目无应答。权重调整程序通常用于处理单位无应答，而插补通常用于处理项目无应答。处理无应答的主要目标是减少无应答偏倚，这种偏倚发生在应答者和无应答者在调查变量方面不同时。在实践中，调查统计人员也面临着有影响力的单位的问题。正确记录影响单位，并代表其他数值相似的总体单位。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元具有较大的设计权重时，会出现影响单元。如果一个估计量对有影响的单位的存在不太敏感，那么它就是稳健的。稳健估计是有偏的，但其均方误差小于非稳健估计。在某些情况下，目标参数不是平均真实的值，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电力消耗曲线感兴趣。我们建议研究的问题估计的平均曲线在缺失数据的存在。我们打算建立的理论性质的估计的基础上观察到的数据和插补数据的最近邻插补。此外，某些单位可能具有高度影响力，这可能使均值曲线及其方差的估计量非常不稳定。我们计划开发强大的估计程序，研究其理论特性，并将所提出的方法应用到真实的数据。稳健的小面积估计近年来受到了相当大的关注。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的几个鲁棒版本已在文献中提出。在实践中，许多变量是分类的，而不是连续的。因此，基于LIFE的方法不适合。我们的目标是提出一个统一的框架，强大的小区域估计的基础上广义LIFE，使强大的预测可以很容易地获得任何类型的变量。在实践中，目标参数可以是复杂参数;例如，分位数在缺失数据的背景下，已经广泛研究了双稳健程序。如果正确指定了无响应模型或插补模型，估计过程保持一致，则称该估计过程具有双重稳健性。到目前为止，文献集中于估计总体均值。我们计划开发复参数的双重保护估计程序，并建立其理论性质。最后，我们提出了一个新的概念，称为多重鲁棒性有限的人口抽样。在实践中，可以拟合多个无应答模型和多个插补模型，每个模型涉及协变量的不同子集和可能不同的链接函数。一个估计量被认为是多重稳健的，如果它是一致的，如果这些多个模型中的任何一个，对于倾向得分或感兴趣的特征，是正确指定的。我们还计划开发多重稳健方差估计器，如果这些多个模型中的任何一个是正确的，则该估计器对于真实方差保持一致。

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.