Suppose we have the following mixed model:

$y = \boldsymbol{X \beta} + \boldsymbol{Z \gamma} + \epsilon$


$\mathbf{X}$ is the fixed effect design matrix.

$\boldsymbol{Z}$ is the design matrix of RVs.

$\boldsymbol{\gamma}$ is the vector of random effect parameters.

$\epsilon$ Is not assumed to be independent and homogenous.

In my SAS notes when they calculate the solution to the beta coefficients in this mixed model, we see the following notation.

$\hat{\boldsymbol{\beta}} = (\boldsymbol{X'\hat{V}^{-1}X})^{-} \boldsymbol{X'\hat{V}y}$


$\hat{\boldsymbol{V}} = \boldsymbol{Z \hat{\boldsymbol{G}}Z' + \hat{R}} $

But im unsure about the notation $\boldsymbol{X^{-}}$. What does the minus in the exponent of the matrix $(\boldsymbol{X'\hat{V}^{-1}X})^{-}$ mean?. Or is simply an error and should be the inverse?

  • 1
    $\begingroup$ It's a generalized inverse, used here because $\boldsymbol{X'\hat{V}^{-1}X}$ may not be invertible. $\endgroup$ Nov 24 at 18:37

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