Index: The Book of Statistical ProofsStatistical Models ▷ Multivariate normal data ▷ Transformed general linear model ▷ Derivation of the distribution

Theorem: Let there be two general linear models of measured data $Y$

$\label{eq:glm1} Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma)$ $\label{eq:glm2} Y = X_t \Gamma + E_t, \; E_t \sim \mathcal{MN}(0, V, \Sigma_t)$

and a matrix $T$ transforming $X_t$ into $X$:

$\label{eq:X-Xt-T} X = X_t \, T \; .$

Then, the transformed general linear model is given by

$\label{eq:tglm} \hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}(0, U, \Sigma)$

where the covariance across rows is $U = ( X_t^\mathrm{T} V^{-1} X_t )^{-1}$.

Proof: The linear transformation theorem for the matrix-normal distribution states:

$\label{eq:matn-ltt} X \sim \mathcal{MN}(M, U, V) \quad \Rightarrow \quad Y = AXB + C \sim \mathcal{MN}(AMB+C, AUA^\mathrm{T}, B^\mathrm{T}VB) \; .$

The weighted least squares parameter estimates for \eqref{eq:glm2} are given by

$\label{eq:glm2-wls} \hat{\Gamma} = ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} Y \; .$

Using \eqref{eq:glm1} and \eqref{eq:matn-ltt}, the distribution of $Y$ is

$\label{eq:Y-dist} Y \sim \mathcal{MN}(X B, V, \Sigma)$

Combining \eqref{eq:glm2-wls} with \eqref{eq:Y-dist}, the distribution of $\hat{\Gamma}$ is

$\label{eq:G-dist} \begin{split} \hat{\Gamma} &\sim \mathcal{MN}\left( \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] X B, \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] V \left[ V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1} \right], \Sigma \right) \\ &\sim \mathcal{MN}\left( ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t \, T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \\ &\sim \mathcal{MN}\left( T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \; . \end{split}$

This can also be written as

$\label{eq:tglm-qed} \hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}\left( 0, ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right)$

which is equivalent to \eqref{eq:tglm}.

Sources:

Metadata: ID: P265 | shortcut: tglm-dist | author: JoramSoch | date: 2021-10-21, 15:03.