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Title: Order determination for large-dimensional matrices Authors:  Lixing Zhu - Hong Kong Baptist University (Hong Kong) [presenting]
Abstract: In sufficient dimension reduction field, a long-standing problem is under-determination of the structural dimension of the central subspace when the criteria that are based on eigendecomposition of target matrices are used. 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 eigendecomposition-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 applications, 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 approximate factor models.