Moderate-Dimensional Inferences on Quadratic Functionals in Ordinary Least Squares

Statistical inferences on quadratic functionals of linear regression parameter have found wide applications including signal detection, one/two-sample global testing, inference of fraction of variance explained and genetic co-heritability. Conventional theory based on ordinary least squares estimator works perfectly in the fixed-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 $n\geq 5p_n$. The main contribution of this paper is to illustrate that signal-to-noise ratio (SNR) plays a crucial role in the moderate-dimensional inferences where $\lim_{n\to\infty} p_n/n = \tau\in (0, 1)$. In the case of weak SNR, as often occurred in the moderate-dimensional regime, both bias and variance need to be corrected in the traditional inference procedures. The amount of correction mainly depends on SNR and $\tau$, and could be fairly large as $\tau\to1$. However, the classical fixed-dimensional results continue to hold if and only if SNR is large enough, say when $p_n$ diverges but slower than $n$. Our general theory holds, in particular, without Gaussian design/error or structural parameter assumption, and apply to a broad class of quadratical functionals, covering all aforementioned applications. The mathematical arguments are based on random matrix theory and leave-one-out method. Extensive numerical results demonstrate the satisfactory performances of the proposed methodology even when $p_n\geq 0.9n$ in some extreme case.