Title: A projection pursuit framework for testing general high-dimensional hypothesis,
Abstract: This article develops a framework for testing general hypothesis in
high-dimensional models where the number of variables may far exceed the number
of observations. Existing literature has considered less than a handful of
hypotheses, such as testing individual coordinates of the model parameter.
However, the problem of testing general and complex hypotheses remains widely
open. We propose a new inference method developed around the hypothesis
adaptive projection pursuit framework, which solves the testing problems in the
most general case. The proposed inference is centered around a new class of
estimators defined as $l_1$ projection of the initial guess of the unknown onto
the space defined by the null. This projection automatically takes into account
the structure of the null hypothesis and allows us to study formal inference
for a number of long-standing problems. For example, we can directly conduct
inference on the sparsity level of the model parameters and the minimum signal
strength. This is especially significant given the fact that the former is a
fundamental condition underlying most of the theoretical development in
high-dimensional statistics, while the latter is a key condition used to
establish variable selection properties. Moreover, the proposed method is
asymptotically exact and has satisfactory power properties for testing very
general functionals of the high-dimensional parameters. The simulation studies
lend further support to our theoretical claims and additionally show excellent
finite-sample size and power properties of the proposed test.