对称及偏斜对称分布族下测量误差模型的稳健估计与诊断研究

Measurement error (ME) is widely found in various types of real data, and is one of the hot research topic in statistics today. In the analysis of such data, it is likely to induce inference bias due to the misspecification of the distribution. However, this problem can be avoided or alleviated when the symmetrical and skew-symmetrical distribution classes are used to modeling. This project will study statistical inferences and diagnostics of the ME model under symmetrical and skew-symmetrical distributions. We aim to establish consistent forms of the estimations and diagnostic metrics for the ME model under the distribution classes, in order to obtain more robust, reliable statistical analysis results. Firstly, we propose to model the complicated ME data by the symmetrical distributions. Based on the hierarchical structure of scale mixtures of normal distributions, the EM, regression calibration, simulation and extrapolation estimation methods will be constructed. The asymptotic properties of the estimators will also be given. Secondly, hypothesis tests, such as heteroscedasticity, interest parameters and the symmetry of the distribution will be studied, together with the asymptotic properties and powers of the test statistics. Based on the Q distance and K-L divergence, the influence analysis will also be derived. Furthermore, the statistical inference and diagnostic theory will be extended to the skew-symmetric ME model to construct the corresponding estimation methods and diagnostic metrics. The project will help researchers analyze complex data containing ME more scientifically.

测量误差广泛存在于各类实际数据中，是当今统计学研究热点之一。对于该类数据，若依指定分布建模则很可能产生分布误判下的推断偏差，基于对称和偏斜对称分布族建模能够有效地避免或缓解这一问题。本项目拟研究对称和偏斜对称分布族下测量误差模型的统计推断及诊断问题，旨在建立分布族下测量误差模型的一致估计式和诊断度量，以期获得更为稳健、可靠的统计分析结果。本项目首先基于对称分布对复杂测量误差数据进行模型构建，并利用正态尺度混合分布的多层结构建立EM估计、回归校正和模拟外推等估计方法，导出估计量的渐近性质；其次，研究模型的异方差检验、兴趣参数检验和分布对称性检验方法，给出检验统计量的渐近性质和检验功效，并基于Q距离和K-L距离建立影响分析度量；此外，将对称分布测量误差模型的统计推断和诊断理论发展至偏斜对称分布模型，构建相应的估计理论和诊断度量。本项研究将有助于研究者更为科学地分析含有测量误差的复杂数据。

测量误差广泛存在于各类实际数据。针对此类数据，传统的参数建模方法往往是基于指定的分布假设，难以避免分布误判问题；而非参数建模方法往往会损失数据信息，且在衔接后续的统计分析上缺乏优势。本项目在对称分布和偏斜对称分布族框架下，对几类测量误差模型进行了系统的统计分析，获得了基于分布族的稳健估计、检验、预测和诊断方法。主要研究内容及取得的结果包括：针对不同的复杂测量误差模型，建立了回归校正估计、EM算法估计和Bayes估计等有效算法，并通过模拟比较了不同估计方法的效率；研究了几类模型的诊断问题，包括等式或不等式约束条件下的约束估计和检验，修正似然比统计量等；研究了偏斜分布族下测量误差模型的参数估计和预测方法；基于重尾随机过程对一类函数型数据进行了建模、估计和预测分析。本项目的研究具有很好的理论和实际意义，基于分布族的统计推断和诊断方法能够有效地避免或缓解分布误判问题，同时具备降低异常点干扰的稳健性；所有算法均可以提供相应的Matlab或R代码，方便实际应用者更为科学地分析含测量误差的复杂数据。