Estimation of time series by Maximum Mean Discrepancy
Pierre Alquier, Jean-David Fermanian, Benjamin Poignard
TL;DR
This work develops two maximum mean discrepancy (MMD)–based minimum-distance estimators for dependent time series, enabling estimation when likelihoods are intractable by simulating from the model. It provides non-asymptotic bounds, consistency, and asymptotic normality under beta-mixing, and derives the asymptotic behavior for both the ideal and simulation-based estimators, including mixed-rate regimes. The authors implement gradient-based algorithms (ISMMD-sgd, PSMMD-sgd) with a Gaussian kernel and validate the approach through extensive simulations on SV, GARCH, ARMA, nonlinear MA, and Ricker models, highlighting robustness to misspecification and competitive or superior performance to standard likelihood-based methods. A data-driven lag selection method based on an MMD criterion is also proposed to adaptively choose the transform $y_t$. Overall, the paper provides a practical, theoretically grounded framework for simulation-based MMD inference in dependent settings with latent structure.
Abstract
We define two minimum distance estimators for dependent data by minimizing some approximated Maximum Mean Discrepancy distances between the true empirical distribution of observations and their assumed (parametric) model distribution. When the latter one is intractable, it is approximated by simulation, allowing to accommodate most dynamic processes with latent variables. We derive the non-asymptotic and the large sample properties of our estimators in the context of absolutely regular/beta-mixing random elements. Our simulation experiments illustrate the robustness of our procedures to model misspecification, particularly in comparison with alternative standard estimation methods.