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Summary Modern statistical methods for multivariate time series rely on the eigendecomposition of matrix-valued functions such as time-varying covariance and spectral density matrices. The curse of indeterminacy or misidentification of smooth eigenvector functions has not received much attention. We resolve this important problem and recover smooth trajectories by examining the distance between the eigenvectors of the same matrix-valued function evaluated at two consecutive points. We change the sign of the next eigenvector if its distance with the current one is larger than the square root of 2. In the case of distinct eigenvalues, this simple method delivers smooth eigenvectors. For coalescing eigenvalues, we match the corresponding eigenvectors and apply an additional signing around the coalescing points. We establish consistency and rates of convergence for the proposed smooth eigenvector estimators. Simulation results and applications to real data confirm that our approach is needed to obtain smooth eigenvectors. 

Summary We develop a uniform test for detecting and dating the integrated or mildly explosive behaviour of a strictly stationary generalized autoregressive conditional heteroskedasticity (GARCH) process. Namely, we test the null hypothesis of a globally stable GARCH process with constant parameters against the alternative that there is an ‘abnormal’ period with changed parameter values. During this period, the parameter-value change may lead to an integrated or mildly explosive behaviour of the volatility process. It is assumed that both the magnitude and the timing of the breaks are unknown. We develop a double-supreme test for the existence of breaks, and then provide an algorithm to identify the periods of changes. Our theoretical results hold under mild moment assumptions on the innovations of the GARCH process. Technically, the existing properties for the quasi-maximum likelihood estimation in the GARCH model need to be reinvestigated to hold uniformly over all possible periods of change. The key results involve a uniform weak Bahadur representation for the estimated parameters, which leads to weak convergence of the test statistic to the supreme of a Gaussian process. Simulations in the Appendix show that the test has good size and power for reasonably long time series. We apply the test to the conventional early-warning indicators of both the financial market and a representative of the emerging Fintech market, i.e., the Bitcoin returns. 

This paper considers the recursive estimation of quantiles using the stochastic gradient descent (SGD) algorithm with Polyak-Ruppert averaging. The algorithm offers a compu- tationally and memory efficient alternative to the usual empirical estimator. Our focus is on studying the non-asymptotic behavior by providing exponentially decreasing tail prob- ability bounds under mild assumptions on the smoothness of the density functions. This novel non-asymptotic result is based on a bound of the moment generating function of the SGD estimate. We apply our result to the problem of best arm identification in a multi-armed stochastic bandit setting under quantile preferences. 

Summary In this paper, we develop a systematic theory for high-dimensional analysis of variance in multivariate linear regression, where the dimension and the number of coefficients can both grow with the sample size. We propose a new U-type statistic to test linear hypotheses and establish a high-dimensional Gaussian approximation result under fairly mild moment assumptions. Our general framework and theory can be used to deal with the classical one-way multivariate analysis of variance, and the nonparametric one-way multivariate analysis of variance in high dimensions. To implement the test procedure, we introduce a sample-splitting-based estimator of the second moment of the error covariance and discuss its properties. A simulation study shows that our proposed test outperforms some existing tests in various settings. 

Abstract Motivated from sequential detection of transient signals in high dimensional data stream, we first study the performance of EWMA and MA charts for detecting a transient signal in a single sequence in terms of the power of detection under the constraint of false detecting probability in the stationary state. Satisfactory approximations are given for the false detection probability and the power of detection. Comparison of EWMA, MA, and CUSUM charts shows that both charts are quite competitive. A multivariate EWMA procedure is considered by using the squared sum of individual EWMA processes and a fairly accurate approximation for the false detection probability is also given. To increase the power of detection, we use the Min‐δ procedure by truncating the estimated weak signals. Dow Jones 30 industrial stock prices are used for illustration.