Positive regression dependence for quadratic forms of Gaussians

A short proof that quadratic form of a Gaussian vector verify positive regression dependence when the entries of their covariance matrix are positive.

Wednesday, February 15, 2017

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.


Benjamini, Yoav, and Daniel Yekutieli. 2001. “The Control of the False Discovery Rate in Multiple Testing Under Dependency.” The Annals of Statistics 29 (4): 1165–88. https://doi.org/10.1214/aos/1013699998.
Rao, Calyampudi Radhakrishna. 1973. Linear Statistical Inference and Its Applications. 2nd ed. New York, NY: Wiley. https://doi.org/10.1002/9780470316436.


BibTeX citation:
  author = {Belzile, Léo},
  title = {Positive Regression Dependence for Quadratic Forms of
  date = {2017-02-15},
  url = {https://lbelzile.bitbucket.io/posts/positive-regression-dep},
  langid = {en}
For attribution, please cite this work as:
Belzile, Léo. 2017. “Positive Regression Dependence for Quadratic Forms of Gaussians.” February 15, 2017. https://lbelzile.bitbucket.io/posts/positive-regression-dep.