31,419 research outputs found

    Uniform disconnectedness and Quasi-Assouad Dimension

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    The uniform disconnectedness is an important invariant property under bi-Lipschitz mapping, and the Assouad dimension dimAX<1\dim _{A}X<1 implies the uniform disconnectedness of XX. According to quasi-Lipschitz mapping, we introduce the quasi-Assouad dimension dimqA\dim _{qA} such that dimqAX<1\dim _{qA}X<1 implies its quasi uniform disconnectedness. We obtain dimBXdimqAXdimAX\overline{\dim } _{B}X\leq \dim _{qA}X\leq \dim _{A}X and compute the quasi-Assouad dimension of Moran set

    Nonparametric Independence Screening in Sparse Ultra-High Dimensional Additive Models

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    A variable screening procedure via correlation learning was proposed Fan and Lv (2008) to reduce dimensionality in sparse ultra-high dimensional models. Even when the true model is linear, the marginal regression can be highly nonlinear. To address this issue, we further extend the correlation learning to marginal nonparametric learning. Our nonparametric independence screening is called NIS, a specific member of the sure independence screening. Several closely related variable screening procedures are proposed. Under the nonparametric additive models, it is shown that under some mild technical conditions, the proposed independence screening methods enjoy a sure screening property. The extent to which the dimensionality can be reduced by independence screening is also explicitly quantified. As a methodological extension, an iterative nonparametric independence screening (INIS) is also proposed to enhance the finite sample performance for fitting sparse additive models. The simulation results and a real data analysis demonstrate that the proposed procedure works well with moderate sample size and large dimension and performs better than competing methods.Comment: 48 page
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