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Asymptotics for linear mixed models

Attention Conservation Notice: This is probably well known in some circles

Suppose you have a (parametric, Normal) linear mixed model \[Y=X\beta+Zb+\epsilon\] where \(\epsilon\) are iid \(N(0,\sigma^2)\) and \(b\) are \(N(0, \sigma^2V(\theta))\). Write \(\Xi\) for the marginal covariance matrix of \(Y\): \[\Xi = \mathrm{cov}[Y]=\sigma^2(I+Z^TVZ)\]

The loglikelihood can be written \[\ell=-\frac{1}{2}\log\left|\Xi\right|-\frac{1}{2}\sum_{i,j} (Y_i-X_i\beta)^T\Xi^{-1}(Y_j-X_j\beta).\]

We might want to treat this as a \(M\)-estimation problem and use the asymptotic behaviour of \(\ell(\beta,\sigma^2,\theta)\) plus the smoothness of \(\ell\) to deduce the asymptotic behaviour of the parameter estimates. A complication is that the loglikelihood isn’t a sum and most of our limit theorems are about sums.

The loglikelihood is, however, a quadratic form in Gaussian variables. This has a known distribution: it’s a linear combination of \(n\) non-central \(\chi^2_1\) variables, where the coefficients of the linear combination are the eigenvalues of \[J(\beta,\sigma^2,\theta)=\Xi^{-1}(\beta,\sigma,\theta)\Xi(\beta_0,\sigma_0,\theta_0).\] Here the first \(\Xi\) is the matrix in the middle of the quadratic form (evaluated at the current parameters) and the second is the actual covariance matrix of \(Y-X\beta\) (evaluated at the true parameters).

Since \(\ell\) is a sum of independent non-central \(\chi^2\), it is asymptotically Normal as long as the coefficients and non-centrality parameters satisfy a Lindeberg condition. This condition will certainly be satisfied if the parameter values are close enough to the true values; at the true values \(J\) is the identity matrix and the non-centrality parameters are all zero.

If \(\ell\) is asymptotically Normally distributed and the true values of the parameters aren’t at the boundary of the parameter space, we can hope to use the smoothness of \(\ell\) to derive asymptotic Normality of the parameter estimates in the usual sort of way.

The assumption that the model is correctly specified isn’t really needed; if the data come from a mixed model different from the one being fitted, the loglikelihood will still be a quadratic form in \(Y-\mu\). We’ll need to worry more about the Lindeberg condition on the non-centrality parameters and eigenvalues since the matrix \(J\) need not be all that close to the identity. We’ll also have the usual issues about what even do the parameters mean if the model is misspecified.

Interestingly, the Normality assumption seems much more important. It’s harder to say anything general about quadratic forms in non-Normal variables, because decorrelating non-Normal variables need not leave them even approximately independent. More theoretical machinery seems to be needed to tackle non-Normal \(\epsilon\) and \(b\).