《order determination for large dimensional matrices》

香港浸会大学数学系/北京师范大学统计学院朱力行教授莅临我院指导

2017年12月25日下午两点半，应广州大学经济与统计学院和岭南统计科学研究中心的邀请，香港浸会大学数学系/北京师范大学统计学院朱力行教授在行政东后座310作了题为“order determination for large dimensional matrices”的讲座——暨“羊城讲坛”第二十八讲，旨在进一步提高年轻学者及研究生对研究的理解。此次讲座由李元教授主持，相关专业的师生参加了此次讲座。本报告系统地研究了具有固定和发散维数的维数降维理论;对于收敛于其极限模型的局部替代模型，其预测的协变量较少，讨论何时可以一致地估计预测协变量的数目;对于超高维因子模型，研究了常用因子个数的估计一致性。对该方法的有限样本性能进行了数值研究。

摘要：Popularly used eigendecomposition-based criteria such as BIC type, ratio estimation and principal component-based criterion often underestimate model dimensionality for regressions or the number of factors for factor models. This longstanding problem is caused by the existence of one or two dominating eigenvalues compared to other nonzero eigenvalues. To alleviate this difficulty, we propose a thresholding double ridge ratio criterion such that the true dimension can be better identified. Unlike all existing eigendecomposition-based criteria, this criterion can define consistent estimate without requiring the uniqueness of minimum and can then handle possible multiple local minima scenarios. This generic strategy would be readily applied to other dimensionality or order determination problems. In this paper, we systematically investigate, for general sufficient dimension reduction theory, the dimensionality determination with fixed and divergent dimensions; for local alternative models that converge to its limiting model with fewer projected covariates, discuss when the number of projected covariates can be consistently estimated, when cannot; and for ultra-high dimensional factor models, study the estimation consistency for the number of common factors. Numerical studies are conducted to examine the finite sample performance of the method.