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We consider the problem of finding confidence intervals for the risk of forecasting the future of a stationary, ergodic stochastic process, using a model estimated from the past of the process. We show that a bootstrap procedure provides valid confidence intervals for the risk, when the data source is sufficiently mixing, and the loss function and the estimator are suitably smooth. Autoregressive (AR(d)) models estimated by least squares obey the necessary regularity conditions, even when mis-specified, and simulations show that the finite- sample coverage of our bounds quickly converges to the theoretical, asymptotic level. As an intermediate step, we derive sufficient conditions for asymptotic independence between empirical distribution functions formed by splitting a realization of a stochastic process, of independent interest.
For testing hypothesis on the covariance operator of functional time series, we suggest to use the full functional information and to avoid dimension reduction techniques. The limit distribution follows from the central limit theorem of the weak conv
We analyse the reconstruction error of principal component analysis (PCA) and prove non-asymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give orac
We derive generalization error bounds for traditional time-series forecasting models. Our results hold for many standard forecasting tools including autoregressive models, moving average models, and, more generally, linear state-space models. These n
Prediction for high dimensional time series is a challenging task due to the curse of dimensionality problem. Classical parametric models like ARIMA or VAR require strong modeling assumptions and time stationarity and are often overparametrized. This
We consider a bivariate time series $(X_t,Y_t)$ that is given by a simple linear autoregressive model. Assuming that the equations describing each variable as a linear combination of past values are considered structural equations, there is a clear m