# Strange notation in beta parameter estimation of mixed models

Suppose we have the following mixed model:

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

Where:

$$\mathbf{X}$$ 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.

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

Where:

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

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?

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