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Abstract Summary The accurate estimation of prediction errors in time series is an important problem. It immediately affects the accuracy of prediction intervals but also the quality of a number of widely used time series model selection criteria such as AIC and others. Except for simple cases, however, it is difficult or even infeasible to obtain exact analytical expressions for one-step and multi-step predictions. This may be one of the reasons that, unlike in the independent case (see Efron, 2004), until today there has been no fully established methodology for time series prediction error estimation. Starting from an approximation to the bias-variance decomposition of the squared prediction error, this work is therefore concerned with the estimation of prediction errors in both univariate and multivariate stationary time series. In particular, several estimates are developed for a general class of predictors that includes most of the popular linear, nonlinear, parametric and nonparametric time series models used in practice, where causal invertible ARMA and nonparametric AR processes are discussed as lead examples. Simulation results indicate that the proposed estimators perform quite well in finite samples. The estimates may also be used for model selection when the purpose of modeling is prediction. 

We are interested in testing general linear hypotheses in a high-dimensional multivariate linear regression model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. A family of rotation-invariant tests is proposed that involves a flexible spectral shrinkage scheme applied to the sample error covariance matrix. The asymptotic normality of the test statistic under the null hypothesis is derived in the setting where dimensionality is comparable to sample sizes, assuming the existence of certain moments for the observations. The asymptotic power of the proposed test is studied under various local alternatives. The power characteristics are then utilized to propose a data-driven selection of the spectral shrinkage function. As an illustration of the general theory, we construct a family of tests involving ridge-type regularization and suggest possible extensions to more complex regularizers. A simulation study is carried out to examine the numerical performance of the proposed tests.