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.

  • $\begingroup$ Where did you encounter such a criterion? $\endgroup$ Jul 21, 2014 at 16:06
  • $\begingroup$ @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[$\alpha^t Y$]. $\endgroup$
    – Manuel
    Jul 21, 2014 at 16:16
  • $\begingroup$ Curious if you found such citations? $\endgroup$
    – Diya
    Feb 3, 2021 at 13:19

1 Answer 1


This order is the standard order on positive-definite matrices. It is sometimes called the "Loewner order", or the cone ordering in the cone of positive-definite matrices. Googling those terms will give many hits. A book dedicated to inequalities and orders is Marshall, Olkin, Arnold: "Inequalities: Theory of Majorization and Its Applications", chapter 14 D (second edition).

  • $\begingroup$ Go to google scholar and search flor "mathematical statistics loewner order" $\endgroup$ Jul 23, 2014 at 2:24

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