Statistical inference in finite mixture of regressions and mixture-of-experts models in high-dimensional spaces, and varying coefficient finite mixture of regression models

导读：近年来，我们见证了从生物学、医学到工程学、社会科学和计量经济学等不同科学研究领域中大规模数据（通常被称为大数据）的兴起。在分析此类数据时，一个常见的统计问题是将感兴趣的响应变量建模为大量特征的一个小子集的函数。这被称为特征选择问题。除了噪声积累和虚假相关外，高维数据中未观察到的异质性使特征选择问题更加困难。有限混合回归（FMR）和混合专家（MOE）是捕获数据异质性的强大统计模型。本文的第一部分主要研究FMR/MOE中的特征选择、估计和后选择推理问题。第二部分涉及变系数有限混合回归（VC-FMR）模型，其中回归系数作为指标变量（如时间）的平滑函数而变化。例如，在市场细分研究中，消费者对产品的偏好通常会随着时间和不同的细分市场而变化。VC-FMR模型为这种涉及异构功能数据的现象建模提供了一种自然的工具。然而，这些相对较新的模型的方法和计算工具在很大程度上尚未开发。***目的：我的研究项目的重点是开发可靠的统计方法和计算效率高的算法，用于估计、特征选择和选择后推理，如假设检验和高维FMR/MOE的置信区间。我研究的另一个重点是VC-FMR中的估计和特征选择。我的长期目标集中在复杂的时间序列数据和高维异构和依赖数据。***方法：研究基于LASSO/SCAD的高维FMR/MOE模型参数估计和特征选择的正则化技术。本文将研究坐标下降型期望最大化（EM）算法的数值计算。选择后推理，如稀疏FMR/MOE中参数的假设检验和置信区间将基于样本分裂技术进行探索。将基于正则化局部核似然的方法用于VC-FMR模型的功能参数估计和特征选择。****影响：我提出的研究计划将解决高维FMR/MOE模型以及VC-FMR模型中未解决的统计问题，并为更广泛的统计受众感兴趣的实际问题提供解决方案。提出的方法可以立即用于解决诸如生物学、工程学、健康科学和市场研究等领域的科学问题。**

INTRODUCTION: In recent years, we have witnessed the rise of large scale data, colloquially referred to as big data, in different fields of scientific research ranging from biology and medicine to engineering, the social sciences and econometrics. A common statistical problem of interest in the analysis of such data is to model a response variable of interest as a function of a small subset of a large number of features. This is referred to as a feature selection problem. In addition to noise accumulation and spurious correlation, unobserved heterogeneity in high-dimensional data makes the feature selection problem even harder. Finite mixture of regressions (FMR) and mixture-of-experts (MOE) are powerful statistical models for capturing heterogeneity in data. The first part of this proposal focuses on feature selection, estimation and post-selection inference problems in FMR/MOE. The second part concerns varying coefficient finite mixture of regression (VC-FMR) models in which regression coefficients change as smooth functions of an index variable such as time. For example, in market segmentation research, consumer preferences for products often change over time and across different market segments. VC-FMR models provide a natural tool for modeling such phenomena which involves heterogeneous functional data. However, methodological and computational tools for these relatively new models are largely unexplored. ***OBJECTIVES: An emphasis of my research program is on developing sound statistical methodology and computationally efficient algorithms for estimation, feature selection, and also post-selection inference such as hypothesis testing and confidence intervals in FMR/MOE in high dimensions. Another focal point of my research concerns estimation and feature selection in VC-FMR. My longer term objectives focus on complex time series data and high-dimensional heterogeneous and dependent data. ***METHODS: I will study the regularization techniques LASSO/SCAD for simultaneous parameter estimation and feature selection in FMR/MOE models in high dimensions. Coordinate descent-type expectation-maximization (EM) algorithms will be investigated for numerical computations. Post-selection inference such as hypothesis testing and confidence intervals for parameters in sparse FMR/MOE will be explored based on sample splitting techniques. Regularized local kernel likelihood-based methods will be used for functional parameter estimation and feature selection in VC-FMR models. ****IMPACT: My proposed research program will address unresolved statistical issues in FMR/MOE models in high dimensions as well as in VC-FMR models, and offer solutions to practical problems of interest to a broader statistical audience. The proposed methods could then immediately be used to solve scientific problems in areas such as biology, engineering, the health sciences, and marketing research.    **