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)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反应和项目无反应。当所有调查变量都丢失或没有足够的可用信息时，单位无响应，而当一些但不是所有调查变量有缺失值时，则发生项目无响应。权重调整程序通常用于处理单位无响应，而归罪通常用于处理项目无响应。处理无反应的主要目的是减少无反应偏差，当受访者和非受访者在调查变量方面不同时就会发生这种偏差。 在实践中，调查统计人员也面临着影响单位的问题。有影响的单位被正确记录，并代表价值相似的其他人口单位。它们在样本中的存在往往会使经典估计非常不稳定。当收集的变量分布高度倾斜或某些单元具有较大的设计权重时，就会出现有影响的单元。如果估计量对有影响力的单位的存在不太敏感，那么它就被称为稳健的。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，目标参数不是平均实值，而是平均函数。例如，人们可能对估计固定时间间隔内大量用户的平均用电量曲线感兴趣。我们建议研究在存在缺失数据的情况下均值曲线的估计问题。我们打算通过最近邻估计的方法，建立基于观测数据和估算数据的估计量的理论性质。此外，一些单位可能会有很大的影响，这可能会使均值曲线的估计值及其方差都非常不稳定。我们计划开发稳健估计程序，研究其理论性质，并将所提出的方法应用于实际数据。 近年来，稳健的小区域估计受到了极大的关注。大多数研究都集中在感兴趣的连续特征上。文献中已经提出了几种基于线性混合模型(LMM)的经验最佳线性无偏预报器的稳健版本。在实践中，许多变量是绝对的，而不是连续的。因此，基于LMM的方法并不适用。目的是提出一个基于广义LMM的稳健小区域估计的统一框架，以便对任何类型的变量都能很容易地获得稳健预报器。 实际上，目标参数可以是复数参数；例如，分位数。在缺失数据的情况下，双稳健过程已被广泛研究。如果一个估计过程在无响应模型或归责模型被正确指定的情况下保持一致，则称该估计过程是双重稳健的。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，我们建议将最近提出的多重稳健性概念推广到有限总体抽样。在实践中，可以拟合多个无响应模型和多个归因模型，每个模型涉及协变量的不同子集，并且可能涉及不同的链接函数。如果对于倾向性分数或兴趣特征的这些多个模型中的任何一个被正确地指定，则称估计量是乘法稳健的。我们还计划开发多个稳健的方差估计器，如果这些多个模型中的任何一个是正确的，那么这些估计器对于真实的方差保持一致。

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.