Moderate-Dimensional Inferences on Quadratic Functionals in Ordinary Least Squares

Statistical inferences for quadratic functionals of linear regression parameter have found wide applications including signal detection, global testing, inferences of error variance and fraction of variance explained. Classical theory based on ordinary least squares estimator works perfectly in the low-dimensional regime, but fails when the parameter dimension $p_n$ grows proportionally to the sample size $n$. In some cases, its performance is not satisfactory even when $n\ge 5p_n$. The main contribution of this paper is to develop {\em dimension-adaptive} inferences for quadratic functionals when $\lim_{n\to \infty} p_n/n=\tau\in[0,1)$. We propose a bias-and-variance-corrected test statistic and demonstrate that its theoretical validity (such as consistency and asymptotic normality) is adaptive to both low dimension with $\tau = 0$ and moderate dimension with $\tau \in(0, 1)$. Our general theory holds, in particular, without Gaussian design/error or structural parameter assumption, and applies to a broad class of quadratic functionals covering all aforementioned applications. As a by-product, we find that the classical fixed-dimensional results continue to hold {\em if and only if} the signal-to-noise ratio is large enough, say when $p_n$ diverges but slower than $n$. Extensive numerical results demonstrate the satisfactory performance of the proposed methodology even when $p_n\ge 0.9n$ in some extreme cases. The mathematical arguments are based on the random matrix theory and leave-one-observation-out method.