Benjamini and Yekutieli (2001) consider adjustments for multiple testing under positive regression dependence (PRDS) to control the false-discovery rate.
Definition 1 Let \(D\) be an increasing set, meaning that \(x \in D\) implies \(y \in D\) for any \(y \geq x\). A vector \(\mathbf{X}\) is PRDS if, for each index \(i \in I_0\), \(\Pr(\boldsymbol{X} \in D \mid X_i=x)\) is nondecreasing in \(x\).
Benjamini and Yekutieli (2001) consider test statistics which have a multivariate Gaussian distribution where all components positively correlated. The proposition below establishes positive regression dependence for quadratic forms of such Gaussian vectors using properties of their underlying Wishart distribution.
Proposition 1 Let \(\boldsymbol{X} \sim \mathsf{No}_p(\boldsymbol{0}_p, \boldsymbol{\Sigma})\) be a random vector with \(\boldsymbol{\Sigma}\) regular, \(n>p\) and such that \(\boldsymbol{\Sigma}_{ij} > 0\) for \(1 \leq i, j \leq p\). Then, \(\boldsymbol{X}^\top\boldsymbol{X}\) is positive regression dependent.
Proof. We use throughout the proof standard properties of the Wishart distribution; the reader is referred to Rao (1973) for references.
Consider the quadratic form \(\boldsymbol{W}=\boldsymbol{X}^\top\boldsymbol{X} \sim \mathsf{Wi}_p(n, \boldsymbol{\Sigma}, \mathbf{O})\). We partition the matrix \(\boldsymbol{W}\) into \[\begin{align*} \boldsymbol{W}=\begin{pmatrix} W_{11} & \boldsymbol{W}_{12} \\ \boldsymbol{W}_{21} & \boldsymbol{W}_{22} \end{pmatrix} \end{align*}\] where \(\boldsymbol{W}_{12}=\boldsymbol{W}_{21}^\top\) is a \((p-1) \times 1\) matrix and \(\boldsymbol{W}_{22}\) is \((p-1)\times (p-1).\) Then, \[\begin{align*} \boldsymbol{W}_{2|1} \coloneqq \boldsymbol{W}_{22}-W_{11}^{-1}\boldsymbol{W}_{21}\boldsymbol{W}_{12} \sim \mathcal{W}_{p-1}\left(n-1, \boldsymbol{\Sigma}_{2|1}, \mathbf{O}\right), \end{align*}\] where \(\boldsymbol{\Sigma}_{2|1}=\boldsymbol{\Sigma}_{22}-\sigma_{11}^{-1}\boldsymbol{\Sigma}_{21}\boldsymbol{\Sigma}_{12}.\) Furthermore, conditional on \(W_{11}=w_{11},\) \(\boldsymbol{W}_{12}\) is independent of \(\boldsymbol{W}_{2|1}\) and its distribution is \(\mathsf{No}_p(w_{11}\boldsymbol{\Sigma}_{21}\boldsymbol{\Sigma}_{11}^{-1}, w_{11}\boldsymbol{\Sigma}_{2|1})\).
The quadratic form \(\boldsymbol{W}_{21}\boldsymbol{W}_{12}\) conditional on \(W_{11}=w_{11}\) follows a non-central Wishart distribution, \(\mathcal{W}_{p-1}(1, w_{11}\boldsymbol{\Sigma}_{2|1}, w_{11}^2\sigma_{11}^{-2}\boldsymbol{\Sigma}_{21}\boldsymbol{\Sigma}_{12})\).
Conditional independence also implies that \[\begin{align*} \mathsf{E}(\boldsymbol{W}_{22} \mid W_{11}=w_{11})&=\mathsf{E}(\boldsymbol{W}_{2|1}+w_{11}^{-1}\boldsymbol{W}_{21}\boldsymbol{W}_{12} \mid W_{11}=w_{11}) \\&= n \boldsymbol{\Sigma}_{2|1} + w_{11}\sigma_{11}^{-2}\boldsymbol{\Sigma}_{21}\boldsymbol{\Sigma}_{12}. \end{align*}\] The conditional mean entry-wise increases as a function of \(w_{11}\) given that each argument of \(\boldsymbol{\Sigma}_{12}\) is positive by assumption.
References
Citation
@online{belzile2017,
author = {Belzile, Léo},
title = {Positive Regression Dependence for Quadratic Forms of
{Gaussians}},
date = {2017-02-15},
url = {https://lbelzile.bitbucket.io/posts/positive-regression-dep},
langid = {en}
}