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We propose a new copula model that can be used with replicated spatial data. Unlike the multivariate normal copula, the proposed copula is based on the assumption that a common factor exists and affects the joint dependence of all measurements of the process. Moreover, the proposed copula can model tail dependence and tail asymmetry. The model is parameterized in terms of a covariance function that may be chosen from the many models proposed in the literature, such as the Matérn model. For some choice of common factors, the joint copula density is given in closed form and therefore likelihood estimation is very fast. In the general case, one-dimensional numerical integration is needed to calculate the likelihood, but estimation is still reasonably fast even with large data sets. We use simulation studies to show the wide range of dependence structures that can be generated by the proposed model with different choices of common factors. We apply the proposed model to spatial temperature data and compare its performance with some popular geostatistics models.

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Factor copula models involve latent variables that explain much of the dependence in the observed variables. Their log-likelihoods can involve one-dimensional or multidimensional integration. For the one-factor copula with weak residual dependence and for the oblique factor copula model, we show that, under some mild assumptions, proxy variables that are unweighted averages computed from the observed variables can be used for the latent variables when the dimension is large. Then alternative loglikelihoods without integrals can be used for parameter estimation. The proxy variables can help to select appropriate linking copulas in some factor copula models and to perform numerically faster maximum likelihood estimation of parameters. Simulation studies show that parameter estimates obtained using the proxy variable approach are close to those obtained using the maximum likelihood approach. The proxy variable approach is used to analyze a financial data set of stock returns in a single sector.