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Abstract:
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To estimate a regression when the errors have a non-identity covariance matrix, we usually turn first to generalized least squares (GLS). GLS proves to be computationally challenging in the very simple set-up of unbalanced crossed random-effects models that we study here. Even for the two-factor random-effects model, with an explicit covariance structure, the computational cost for likelihood evaluation is O(N^3/2). Numerous Bayesian algorithms take super-linear time in order to obtain good parameter estimates. A variant of the Gibbs sampler(Roberts et al) achieves linear-complexity when all levels of a factor are observed equally often, which is very rare. We develop statistical methods to do inference for large crossed random effects structure. The method we develop is statistically efficient and is computable in O(N) work for the linear mixed model. The speed of convergence depends on a certain matrix norm. Under a much more general set up, we exhibit the theoretical convergence of our algorithm in simulated datasets. When running on a real dataset from Stitch Fix, we show the efficiency of the method can range from 1 to 50 in terms of the variance of the estimated coefficients.
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