Minimax estimation of partially-observed vector autoregressions

Multi-dimensional time series are a core ingredient of the statistical modeling toolkit, for which numerous estimation methods are known. But when observations are scarce or corrupted, the learning task becomes much harder. The question is: how much harder? In this paper, we study the properties of a partially-observed Vector AutoRegressive process, which is a state-space model endowed with a stochastic observation mechanism. Our goal is to estimate its transition matrix, but we only have access to a small and noisy subsample of the state components. We start by describing an estimator based on the Yule-Walker equation and the Dantzig selector, and we give an upper bound on its non-asymptotic error. Then, we provide a matching minimax lower bound, thus proving near-optimality of our estimator. The convergence rate we obtain sheds light on the role of several key parameters such as the sampling ratio, the amount of noise and the number of non-zero coefficients in the transition matrix, which can be dense in low dimension but is usually sparse in high dimension. These theoretical findings are commented and illustrated by numerical experiments on simulated data.