[R-sig-ME] lmer estimates for highly unbalanced anova

[Asaf Weinstein]

Dear lme4 community,

I am doing some theoretical research on a problem involving a two-way
random effects model with an unbalanced design.
I am wondering if the behavior of the ML estimates for the variance
components (theta, in the notation of lme4 documentation) was studied under
highly unbalanced design, e.g., the ratio lambda := max(K_ij)/min(K_ij) is
very large, where K_ij = number of observations in the ij cell (i=1,..,R,
j=1,...,C).
More specifically, I am wondering if anybody has studied the asymptotic
behavior of the MLE when R,C-->\infty and also lambda -->\infty at some
rate, say o([RC]^{\alpha}).
I am suspecting that the performance of the ML estimator may deteriorate
when the design matrix (=matrix with element (i,j) equal to K_ij) is
ill-conditioned, i.e., lambda is very large; but I am curious whether this
can be confirmed.
I would appreciate very much any comment or reference that might be
relevant.

Thank you very much,
Asaf

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