Closed form GLM cumulants and GLMM fitting with a SQUAR-EM-LA2 algorithm

We find closed form expressions for the standardized cumulants of generalized linear models. This reduces the complexity of their calculation from O(p6) to O(p2) operations which allows efficient construction of second-order saddlepoint approximations to the pdf of sufficient statistics. We adapt the result to obtain a closed form expression for the second-order Laplace approximation for a GLMM likelihood. Using this approximation, we develop a computationally highly efficient accelerated EM procedure, SQUAR-EM-LA2. The procedure is illustrated by fitting a GLMM to a well-known data set. Extensive simulations show the phenomenal performance of the approach. Matlab software is provided for implementing the proposed algorithm.