For unbiased estimators, intuitively why $var(\widetilde{\beta})-var(\widehat{\beta})$ being p.s.d. means $\widehat{\beta}$ is more efficient?

by JTS365   Last Updated May 22, 2020 23:19 PM

For two scalar unbiased estimators $\widehat{\alpha}$ and $\widetilde{\alpha}$, we know that if one has smaller variance, then we say it is more efficient, which intuitively means that this estimator is more concentrated around true value (or has less dispersion). However, such an intuition seems to be lost in general for vector-valued estimators. For example, for vector-valued unbiased estimator $\widehat{\beta}$ and $\widetilde{\beta}$, we say that $\widehat{\beta}$ is more efficient than $\widetilde{\beta}$ if matrix $var(\widetilde{\beta})-var(\widehat{\beta})$ is positive semidefinite (p.s.d. in short), where $var(\widehat{\beta})$ and $var(\widetilde{\beta})$ denote variance covariance matrices.

I'm wondering how shall we intuitively interpret this positive semidefiniteness? Is there any intuitive connection between this positive semidefiniteness and $\widehat{\beta}$ being more concentrated just like for the scalar case? (Of course, if $var(\widehat{\beta})$ and $var(\widetilde{\beta})$ are both diagonal matrices, positive semidefiniteness means each element in $\widehat{\beta}$ has smaller variance. My question is about the more general case when they are not diagonal) Thanks!

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