# Comparing two variance matrices

I am looking for bibliographical reference for comparing two variance matrices with he following criterion:

$\text{Var}[X] \geq \text{Var}[Y] \quad \text{if} \quad \text{Var}[X]-\text{Var}[Y] \succeq 0$

This means that $X$ has greater variance than $Y$ if the difference of their variance matrices is positive semi-definite.

Any references to books or articles with citations that explain why this is a valid and common criterion to compare variability of random vectors is appreciated. If that is not possible any help on how to search for it would be ok.

• Where did you encounter such a criterion? – kjetil b halvorsen Jul 21 '14 at 16:06
• @kjetil, I first saw of it in Wikipedia when studing the Gauss Markov theorem which asses that least squares is BLUE. I have used on my thesis and nobody complained. It makes sense because for any fixed $\alpha$, Var[$\alpha^t X$]$\geq$Var[$\alpha^t Y$]. – Manuel Jul 21 '14 at 16:16

• Thanks @kjetil, do you know any reference in the statistical literature that makes use of Loewner order. I found Some applications of Loewner's ordering on symmetric matrices Annals of the Institute of Statistical Mathematics, 1967, Volume 19, Number 1, Page 245 Minoru Siotani'' may be useful but I don't have acces for it. – Manuel Jul 22 '14 at 13:06