Restricted maximum likelihood from error contrasts

Assume AC and a Gaussian family of the form in Gaussian branch of the linear mixed model, indexed by a nonempty parameter set $\mathrm{\Theta }$. For each $\theta \in \mathrm{\Theta }$, let $G_{\theta }$ and $R_{\theta }$ be the latent and residual covariance matrices in $V_{\theta }=Z{G}_{\theta }{Z}^T+R_{\theta }\succ 0.$ Let the fixed design $X\in {\mathbb{R}}^{n\times p}$ have full column rank with $p<n$. Fix an error-contrast matrix $K\in {\mathbb{R}}^{n\times (n-p)}$ as in Error-contrast matrix for a full-rank fixed design, independently of $\theta$. By A Gaussian error contrast has the induced multivariate-normal law, the contrast vector $K^{T}Y$ has law $N_{n-p}(0,K^T{V}_{\theta }K)$ and its law does not depend on the fixed-effect parameter $\beta$. The local result The REML error-contrast covariance is positive definite shows that this covariance is positive definite, and A positive-definite multivariate normal law has its Gaussian density gives its Gaussian density. The restricted likelihood for this fixed contrast basis is $L_R(\theta ;y)=(2\pi {)}^{-(n-p)\mathrm{/}2}\det (K^T{V}_{\theta }K{)}^{-1\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}(-\frac{1}{2}(K^{T}y{)}^T(K^T{V}_{\theta }K{)}^{-1}(K^{T}y))\mathrm{.}$ This item defines the likelihood for a fixed, parameter-independent choice of contrast coordinates; it makes no claim about likelihood comparisons across different fixed-effect designs.

The local column-basis convention is $K\in {\mathbb{R}}^{n\times (n-p)}$ with columns spanning $\mathrm{col}(X{)}^{\perp }$, so $K^{T}Y$ is defined. The restriction $p<n$ supplies a nonzero contrast space; when $p=n$, this definition makes no variance-likelihood claim.