  Understanding statistical inference under possibly non-sparse
high-dimensional models has gained much interest recently. 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. Implications of the new constructions include
new minimax testability results that, in sharp contrast to the current 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 $\sqrt{n}$ rate, regardless of the size
of the model sparsity.
