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.
multivariate
Author
Published

Wednesday, February 15, 2017

Benjamini and Yekutieli () 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 xD implies yD for any yx. A vector X is PRDS if, for each index iI0, Pr(XDXi=x) is nondecreasing in x.

Benjamini and Yekutieli () 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 XNop(0p,Σ) be a random vector with Σ regular, n>p and such that Σij>0 for 1i,jp. Then, XX is positive regression dependent.

Proof. We use throughout the proof standard properties of the Wishart distribution; the reader is referred to Rao () for references.

Consider the quadratic form W=XXWip(n,Σ,O). We partition the matrix W into W=(W11W12W21W22) where W12=W21 is a (p1)×1 matrix and W22 is (p1)×(p1). Then, W2|1:=W22W111W21W12Wp1(n1,Σ2|1,O), where Σ2|1=Σ22σ111Σ21Σ12. Furthermore, conditional on W11=w11, W12 is independent of W2|1 and its distribution is Nop(w11Σ21Σ111,w11Σ2|1).

The quadratic form W21W12 conditional on W11=w11 follows a non-central Wishart distribution, Wp1(1,w11Σ2|1,w112σ112Σ21Σ12).

Conditional independence also implies that E(W22W11=w11)=E(W2|1+w111W21W12W11=w11)=nΣ2|1+w11σ112Σ21Σ12. The conditional mean entry-wise increases as a function of w11 given that each argument of Σ12 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}
}
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/.