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.

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We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by \cite{Li:1991}. Under mild conditions, the asymptotic ratio ρ=limp/n is the phase transition parameter and the SIR estimator is consistent if and only if ρ=0. When dimension p is greater than n, we propose a diagonal thresholding screening SIR (DT-SIR) algorithm. This method provides us with an estimate of the eigen-space of the covariance matrix of the conditional expectation var(E[x|y]). The desired dimension reduction space is then obtained by multiplying the inverse of the covariance matrix on the eigen-space. Under certain sparsity assumptions on both the covariance matrix of predictors and the loadings of the directions, we prove the consistency of DT-SIR in estimating the dimension reduction space in high dimensional data analysis. Extensive numerical experiments demonstrate superior performances of the proposed method in comparison to its competitors.

Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve 1, 2. Understanding evolutionary game dynamics in structured populations remains difficult. Mathematical results are known for special structures in which all individuals have the same number of neighbours 3, 4, 5, 6, 7, 8. The general case, in which the number of neighbours can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class that suggests there is no efficient algorithm 9. Whether a simple solution for weak selection exists has remained unanswered. Here we provide a solution for weak selection that applies to any graph or network. Our method relies on calculating the coalescence times 10, 11 of random walks 12. We evaluate large numbers of diverse population structures for their propensity to favour cooperation. We study how small

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