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THE ANNALS of STATISTICS ISSN 0090-5364 (print) ISSN 2168-8966 (online) THE ANNALS of STATISTICS AN OFFICIAL JOURNAL OF THE INSTITUTE OF MATHEMATICAL STATISTICS Articles Testing for stationarity of functional time series in the frequency domain ALEXANDER AUE AND ANNE VAN DELFT 2505 On spike and slab empirical Bayes multiple testing ISMAËL CASTILLO AND ÉTIENNE ROQUAIN 2548 Theoretical and computational guarantees of mean field variational inference for communitydetection........... ANDERSON Y. Z HANG AND HARRISON H. ZHOU 2575 Minimax optimal sequential hypothesis tests for Markov processes MICHAEL FAUSS,ABDELHAK M. ZOUBIR AND H. VINCENT POOR 2599 Test of significance for high-dimensional longitudinal data ETHAN X. FANG,YANG NING AND RUNZE LI 2622 Geometrizing rates of convergence under local differential privacy constraints ANGELIKA ROHDE AND LUKAS STEINBERGER 2646 Additive regression with Hilbertian responses JEONG MIN JEON AND BYEONG U. PARK 2671 Nonparametric Bayesian estimation for multivariate Hawkes processes SOPHIE DONNET,VINCENT RIVOIRARD AND JUDITH ROUSSEAU 2698 Hypothesis testing for high-dimensional time series via self-normalization RUNMIN WANG AND XIAOFENG SHAO 2728 Variational analysis of constrained M-estimators JOHANNES O. ROYSET AND ROGER J-B WETS 2759 Which bridge estimator is the best for variable selection? SHUAIWEN WANG,HAOLEI WENG AND ARIAN MALEKI 2791 Permutation methods for factor analysis and PCA . .........EDGAR DOBRIBAN 2824 A general framework for Bayes structured linear models CHAO GAO,AAD W. VA N D E R VAART AND HARRISON H. ZHOU 2848 Asymptotic distribution and detection thresholds for two-sample tests based on geometric graphs..........................................BHASWAR B. BHATTACHARYA 2879 ControlledsequentialMonteCarlo.............. JEREMY HENG,ADRIAN N. BISHOP, GEORGE DELIGIANNIDIS AND ARNAUD DOUCET 2904 A framework for adaptive MCMC targeting multimodal distributions EMILIA POMPE,CHRIS HOLMES AND KRZYSZTOF ŁATUSZYNSKI´ 2930 Valid post-selection inference in model-free linear regression ARUN K. KUCHIBHOTLA,LAWRENCE D. BROWN,ANDREAS BUJA, JUNHUI CAI,EDWARD I. GEORGE AND LINDA H. ZHAO 2953 Inference for spherical location under high concentration DAV Y PAINDAVEINE AND THOMAS VERDEBOUT 2982 Semiparametric Bayesian causal inference . KOLYAN RAY AND AAD VAN DER VAART 2999 Relaxing the assumptions of knockoffs by conditioning DONGMING HUANG AND LUCAS JANSON 3021 Analytical nonlinear shrinkage of large-dimensional covariance matrices OLIVIER LEDOIT AND MICHAEL WOLF 3043 Coupled conditional backward sampling particle filter ANTHONY LEE,SUMEETPAL S. SINGH AND MATTI VIHOLA 3066 Asymptotic risk and phase transition of l1-penalized robust estimator. .HANWEN HUANG 3090 Vol. 48, No. 5—October 2020 THE ANNALS OF STATISTICS Vol. 48, No. 5, pp. 2505–3111 October 2020 INSTITUTE OF MATHEMATICAL STATISTICS (Organized September 12, 1935) The purpose of the Institute is to foster the development and dissemination of the theory and applications of statistics and probability. IMS OFFICERS President: Regina Y. Liu, Department of Statistics, Rutgers University, Piscataway, New Jersey 08854-8019, USA President-Elect: Krzysztof Burdzy, Department of Mathematics, University of Washington, Seattle, Washington 98195-4350, USA Past President: Susan Murphy, Department of Statistics, Harvard University, Cambridge, Massachusetts 02138-2901, USA Executive Secretary: Edsel Peña, Department of Statistics, University of South Carolina, Columbia, South Carolina 29208-001, USA Treasurer: Zhengjun Zhang, Department of Statistics, University of Wisconsin, Madison, Wisconsin 53706-1510, USA Program Secretary: Ming Yuan, Department of Statistics, Columbia University, New York, NY 10027-5927, USA IMS EDITORS The Annals of Statistics. Editors: Richard J. Samworth, Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WB, UK. Ming Yuan, Department of Statistics, Columbia University, New York, NY 10027, USA The Annals of Applied Statistics. Editor-in-Chief : Karen Kafadar, Department of Statistics, University of Virginia, Heidelberg Institute for Theoretical Studies, Charlottesville, VA 22904-4135, USA The Annals of Probability. Editor: Amir Dembo, Department of Statistics and Department of Mathematics, Stan- ford University, Stanford, California 94305, USA The Annals of Applied Probability. Editors: François Delarue, Laboratoire J. A. Dieudonné, Université de Nice Sophia-Antipolis, France-06108 Nice Cedex 2. Peter Friz, Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany and Weierstrass-Institut für Angewandte Analysis und Stochastik, 10117 Berlin, Germany Statistical Science. Editor: Sonia Petrone, Department of Decision Sciences, Università Bocconi, 20100 Milano MI, Italy The IMS Bulletin. Editor: Vlada Limic, UMR 7501 de l’Université de Strasbourg et du CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France The Annals of Statistics [ISSN 0090-5364 (print); ISSN 2168-8966 (online)], Volume 48, Number 5, October 2020. Published bimonthly by the Institute of Mathematical Statistics, 3163 Somerset Drive, Cleveland, Ohio 44122, USA. Periodicals postage paid at Cleveland, Ohio, and at additional mailing offices. POSTMASTER: Send address changes to The Annals of Statistics, Institute of Mathematical Statistics, Dues and Subscriptions Office, 9650 Rockville Pike, Suite L 2310, Bethesda, Maryland 20814-3998, USA. Copyright © 2020 by the Institute of Mathematical Statistics Printed in the United States of America The Annals of Statistics 2020, Vol. 48, No. 5, 2505–2547 https://doi.org/10.1214/19-AOS1895 © Institute of Mathematical Statistics, 2020 TESTING FOR STATIONARITY OF FUNCTIONAL TIME SERIES IN THE FREQUENCY DOMAIN BY ALEXANDER AUE1 AND ANNE VAN DELFT2 1Department of Statistics, University of California, [email protected] 2Fakultät für Mathematik, Ruhr-Universität Bochum, [email protected] Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much in- creased pace. This article contributes to the research in this area by proposing a new stationarity test for functional time series based on frequency domain methods. The proposed test statistics is based on joint dimension reduction via functional principal components analysis across the spectral density oper- ators at all Fourier frequencies, explicitly allowing for frequency-dependent levels of truncation to adapt to the dynamics of the underlying functional time series. The properties of the test are derived both under the null hypothesis of stationary functional time series and under the smooth alternative of locally stationary functional time series. The methodology is theoretically justified through asymptotic results. Evidence from simulation studies and an appli- cation to annual temperature curves suggests that the test works well in finite samples. REFERENCES ANTONIADIS,A.andSAPATINAS, T. (2003). Wavelet methods for continuous-time prediction using Hilbert- valued autoregressive processes. J. Multivariate Anal. 87 133–158. MR2007265 https://doi.org/10.1016/ S0047-259X(03)00028-9 AUE,A.andHORVÁTH, L. (2013). Structural breaks in time series. J. Time Series Anal. 34 1–16. MR3008012 https://doi.org/10.1111/j.1467-9892.2012.00819.x AUE,A.,DUBART NORINHO,D.D.andHÖRMANN, S. (2015). On the prediction of stationary functional time series. J. Amer. Statist. Assoc. 110 378–392. MR3338510 https://doi.org/10.1080/01621459.2014.909317 AUE,A.,RICE,G.andSÖNMEZ, O. (2018). Detecting and dating structural breaks in functional data without dimension reduction. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80 509–529. MR3798876 https://doi.org/10.1111/ rssb.12257 AUE,A.andVA N DELFT, A. (2019). Supplement to “Testing for stationarity of functional time series in the frequency domain.” https://doi.org/10.1214/19-AOS1895SUPP. BANDYOPADHYAY,S.,JENTSCH,C.andSUBBA RAO, S. (2017). A spectral domain test for stationarity of spatio-temporal data. J. Time Series Anal. 38 326–351. MR3611747 https://doi.org/10.1111/jtsa.12222 BANDYOPADHYAY,S.andSUBBA RAO, S. (2017). A test for stationarity for irregularly spaced spatial data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 95–123. MR3597966 https://doi.org/10.1111/rssb.12161 BESSE,P.,CARDOT,H.andSTEPHENSON, D. (2000). Autoregressive forecasting of some functional climatic variations. Scand. J. Stat. 27 673–687. BOSQ, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics 149. Springer, New York. MR1783138 https://doi.org/10.1007/978-1-4612-1154-9 BRILLINGER, D. (1981). Time Series: Data Analysis and Theory. McGraw Hill, New York. BRILLINGER,D.R.andROSENBLATT, M. (1967). Asymptotic theory of estimates of k-th order spectra. In Spectral Analysis Time Series (Proc. Advanced Sem., Madison, Wis., 1966) 153–188. John Wiley, New York. MR0211566 CREMERS,H.andKADELKA, D. (1986). On weak convergence of integral functionals of stochastic pro- E cesses with applications to processes taking paths in Lp . Stochastic Process. Appl. 21 305–317. MR0833957 https://doi.org/10.1016/0304-4149(86)90102-X MSC2020 subject classifications. Primary 62G99, 62H99; secondary 62M10, 62M15, 91B84. Key words and phrases. Frequency domain methods, functional data analysis, locally stationary processes, spectral analysis. DAHLHAUS, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37. MR1429916 https://doi.org/10.1214/aos/1034276620 DETTE,H.,PREUSS,P.andVETTER, M. (2011). A measure of
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