Benjamini and Yekutieli (2001) consider adjustments for multiple testing under positive regression dependence (PRDS) to control the false-discovery rate.
Definition 1 Let be an increasing set, meaning that implies for any . A vector is PRDS if, for each index , is nondecreasing in .
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 be a random vector with regular, and such that for . Then, 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 . We partition the matrix into where is a matrix and is Then, where Furthermore, conditional on is independent of and its distribution is .
The quadratic form conditional on follows a non-central Wishart distribution, .
Conditional independence also implies that The conditional mean entry-wise increases as a function of given that each argument of is positive by assumption.
References
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.
Citation
BibTeX 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}
}