STATISTICAL INFERENCE WITH F-STATISTICS WHEN FITTING SIMPLE MODELS TO HIGH-DIMENSIONAL DATA

Abstract We study linear subset regression in the context of high-dimensional overall model $y = \vartheta +\theta ' z + \epsilon $ with univariate response y and a d -vector random regressors , independent $\epsilon . Here, “high-dimensional” means that number available explanatory variables is much larger than n observations. consider simple submodels where regressed on set p given by $x M'z$ for some $d \times p$ matrix M full rank $p < n$ The corresponding model, is, $y=\alpha +\beta x e$ usually justified imposing appropriate restrictions unknown parameter $\theta model; otherwise, this can be grossly misspecified sense relevant may have been omitted. In paper, we establish asymptotic validity standard F -test surrogate $\beta an sense, even when misspecified, without any whatsoever assuming Gaussian data.