Efficient estimation of high-dimensional multivariate normal copula models with discrete spatial responses

TL;DR

This paper compares the distributional transform and maximum simulated likelihood methods for estimating high-dimensional multivariate normal copula models with discrete spatial responses, highlighting efficiency and bias issues.

Contribution

It provides a detailed analysis of the distributional transform's limitations and introduces a nearly efficient maximum simulated likelihood approach for high-dimensional discrete data.

Findings

Distributional transform leads to biased estimates with increased discretization.

Maximum simulated likelihood achieves near-optimal efficiency.

Application to spatial count data demonstrates substantial efficiency gains.

Abstract

The distributional transform (DT) is amongst the computational methods used for estimation of high-dimensional multivariate normal copula models with discrete responses. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the small-sample and asymptotic efficiency of the method for estimating high-dimensional multivariate normal copula models with univariate Bernoulli, Poisson, and negative binomial margins, and show that the DT approximation leads to biased estimates when there is more discretisation. For a high-dimensional discrete response, we implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi Monte Carlo methods. Efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified high-dimensional multivariate normal copula models. Both methods are illustrated with spatially aggregated count data sets, and it is shown that there is a substantial gain on efficiency via the maximum simulated likelihood method.