PIMS-UAlberta Distinguished Lecture: Lixing Zhu

This talk describes our attempt to attack two longstanding problems in the model dimensionality (order) when criteria that are based on eigen-decomposition of target matrices are used in practice. First, due to the existence of some dominating eigenvalues compared to other nonzero eigenvalues, the true dimensionality is often underestimated. Second, the estimation accuracy of any existing method often relies on the uniqueness of minimum/maximum of the criterion. Yet, it is often not the case particularly for the models that converge to a limit with smaller dimensionality. To alleviate these difficulties, we propose a thresholding double ridge ratio criterion. Unlike all the existing eigen-decomposition based criteria, this criterion can define a consistent estimate even when there are several local minima. This generic strategy is readily applied to many fields. As the examples, we give the details about dimension reduction in regressions with fixed and divergent dimensions; about when the number of projected covariates can be consistently estimated, when cannot if a sequence of regression models converges to a limiting model with fewer projected covariates; about ultra-high dimensional factor models. It can also applied to spiked population models. Numerical studies are conducted to examine the finite sample performance of the method.