The robust focused information criterion for strong mixing stochastic processes with $$\mathscr {L}^{2}$$ L 2 -differentiable parametric densities

Claeskens and Hjort (J Am Stat Assoc 98(464):900–916, 2003) constructed the focused information criterion (FIC) using maximum likelihood estimators to facilitate the contextual selection of probability models for independently distributed observations. We generalize these results to the case of stationary, strong mixing stochastic processes exhibiting outliers when the “true” finite dimensional distribution of the process lies in the contamination neighbourhood of the assumed parametric model for the process. Given the natural filtration of a stochastic process, we obtain the FIC using robust M-estimators having bounded conditional influence functions. We utilize Le Cam’s contiguity lemmas to tract the parametric form of the asymptotic bias of M-estimators under model misspecification induced by additive outliers. The local asymptotic normality is established assuming the finite dimensional parametric density of the process to be $$\mathscr {L}^{2}$$ L 2 -differentiable (differentiable in quadratic mean). As a result, our theory is also applicable for constructing FIC for moderately irregular models outside the exponential family such as Laplace and related densities. We apply our results to derive the robust FIC for simultaneous selection of the order and the innovation density of asymmetric Laplace autoregressive processes observed with outliers. We demonstrate our theory with the focused modeling of United States Dollar to Indian Rupee currency exchange rates exhibiting outliers post Indian demonetization.