Statistical modeling and inference for toroidal and cylindrical data under multivariate wrapped and partially wrapped Gaussian distributions

This dissertation addresses methodological and computational challenges in the analysis of multivariate directional data, focusing on toroidal and cylindrical domains. Standard likelihood-based inference in these settings often suffers from poor scalability or low accuracy, particularly in high dimensions or with high dispersion. We propose two novel frameworks tailored to address application-specific needs while improving computational efficiency. The first contribution introduces an Indirect Inference estimator for multivariate Wrapped and Partially Wrapped Normal distributions. The method avoids direct likelihood evaluation by matching simulated summaries from an auxiliary Gaussian model to the observed data. It achieves estimation performances comparable to EM and CEM methods, while offering a drastic reduction in runtime, particularly in higher dimensions. In high-variance univariate cases, it also outperforms standard methods in terms of bias and stability. The second contribution develops a copula-based spatial model for cylindrical data, combining a wrapped Gaussian marginal for direction, a log-normal marginal for intensity, and a parametric copula, enabling asymmetric tail dependence. Covariates are incorporated both in the non-stationary Gaussian random fields defining the margins and in the copula parameter. Applied to wind direction and speed data, the model demonstrates improved fit and predictive performance. Model comparison is guided by DIC and WAIC, while a cylindrical CRPS is introduced to assess probabilistic forecast accuracy. Together, these contributions advance inference for complex directional data by uniting statistical flexibility with computational tractability.