Scalable Likelihood Inference for Student-t Copula Count Time Series

Count time series often exhibit extremal dependence that may not be adequately captured by Gaussian copula models. We develop a likelihood-based framework for count-valued time series using Student-t copulas with latent ARMA dependence. The latent process is constructed through a scale-mixture representation of a Gaussian ARMA process, preserving the second-order dependence structure while introducing tail dependence and greater persistence of extreme events. Likelihood inference requires evaluating high-dimensional truncated multivariate t probabilities, which is computationally demanding under heavy tails and strong serial dependence. To address this challenge, we develop scalable likelihood approximations tailored to the time series structure. In particular, we formulate a time series version of minimax exponential tilting for multivariate t probabilities, termed Time Series Minimax Exponential Tilting (TMET), which exploits the exact conditional representation of the latent ARMA process. The resulting algorithm reduces computational complexity from cubic to near-linear in the series length while retaining the high accuracy of minimax exponential tilting. For comparison, we also extend two widely used Gaussian copula approximations—the continuous extension (CE) method and the Geweke–Hajivassiliou–Keane (GHK) simulator—to the Student-t copula setting. Simulation studies show that TMET outperforms CE and GHK, particularly under strong dependence, heavy tails, and low-count regimes. The framework also supports predictive inference and residual diagnostics. An application to weekly rotavirus counts illustrates how the Student-t copula provides a flexible extension of the Gaussian copula while retaining stable inference even when tail dependence is weak or absent.