user2016

Likelihood of conditional auto-regression (CAR) models is expensive to compute even for a moderate data size around 1000 and it is usually not in closed form with latent variables. In this work we approximate the likelihood by Monte Carlo methods and propose two algorithms for optimising the Monte Carlo likelihood. The algorithms search for the maximum of the Monte Carlo likelihood and by taking the Monte Carlo error into account, the algorithms appear to be stable regardless the initial parameter value. Both algorithms are implemented in R and the iterative procedures are fully automatic with user-specified parameters to control the Monte Carlo simulation and convergence criteria.nnWe first demonstrate the use of the algorithms by simulated CAR data on a $20 \times 20$ torus. Then methods were applied to a data from forest restoration experiment with around 7000 trees arranged in transects in study plots. The growth rate of trees was modelled by a linear mixed effect model with CAR spatial error and CAR random effects. A approximation to the MLE was found by our proposed algorithms in a reasonable computational time.