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There is gained interest in understanding statistical inference under possibly non-sparse high-dimensional models. For a given component of the regression coefficient, we show that the difficulty of the problem depends on the sparsity of the corresponding row of the precision matrix of the covariates, not the sparsity of the regression coefficients. We develop new concepts of uniform and essentially uniform non-testability that allow the study of limitations of tests across a broad set of alternatives. Uniform non-testability identifies a collection of alternatives such that the power of any test, against any alternative in the group, is asymptotically at most equal to the nominal size of the test. Implications of the new constructions include new minimax testability results that in sharp contrast to the existing results, do not depend on the sparsity of the regression parameters. We identify new tradeoffs between testability and feature correlation. In particular, we show that in models with weak feature correlations minimax lower bound can be attained by a test whose power has the parametric rate regardless of the size of the model sparsity.

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al.(2013). They also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Our results are successfully applied to

The central subspace of a pair of random variables $(y,x) \in \mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y \perp \hspace{-2mm} \perp x\mid P_{\mathcal{S}}x$. In this paper, we consider the minimax rate of estimating the central space of the multiple index models $y=f(\beta_{1}^{\tau}x,\beta_{2}^{\tau}x,...,\beta_{d}^{\tau}x,\epsilon)$ with at most $s$ active predictors where $x \sim N(0,I_{p})$. We first introduce a large class of models depending on the smallest non-zero eigenvalue $\lambda$ of $var(\mathbb{E}[x|y])$, over which we show that an aggregated estimator based on the SIR procedure converges at rate $d\wedge((sd+s\log(ep/s))/(n\lambda))$. We then show that this rate is optimal in two scenarios: the single index models; and the multiple index models with fixed central dimension $d$ and fixed $\lambda$. By assuming a technical conjecture, we can show that this rate is also optimal for multiple index models with bounded dimension of the central space. We believe that these (conditional) optimal rate results bring us meaningful insights of general SDR problems in high dimensions.

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