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Official Journal of the Bernoulli Society for Mathematical Statistics And Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty Five Number Two May 2019 ISSN: 1350-7265 CONTENTS MAMMEN, E., VAN KEILEGOM, I. and YU, K. 793 Expansion for moments of regression quantiles with applications to nonparametric testing GRACZYK, P. and MAŁECKI, J. 828 On squared Bessel particle systems EVANS, R.J. and RICHARDSON, T.S. 848 Smooth, identifiable supermodels of discrete DAG models with latent variables CHAE, M. and WALKER, S.G. 877 Bayesian consistency for a nonparametric stationary Markov model BELOMESTNY, D., PANOV, V. and WOERNER, J.H.C. 902 Low-frequency estimation of continuous-time moving average Lévy processes ZEMEL, Y. and PANARETOS, V.M. 932 Fréchet means and Procrustes analysis in Wasserstein space CASTRO, R.M. and TÁNCZOS, E. 977 Are there needles in a moving haystack? Adaptive sensing for detection of dynamically evolving signals DAHLHAUS, R., RICHTER, S and WU, W.B. 1013 Towards a general theory for nonlinear locally stationary processes CHEN, X., CHEN, Z.-Q., TRAN, K. and YIN, G. 1045 Properties of switching jump diffusions: Maximum principles and Harnack inequalities BARBOUR, A.D., RÖLLIN, A. and ROSS, N. 1076 Error bounds in local limit theorems using Stein’s method BECHERER, D., BILAREV, T. and FRENTRUP, P. 1105 Stability for gains from large investors’ strategies in M1/J1 topologies OATES, C.J., COCKAYNE, J., BRIOL, F.-X. and GIROLAMI, M. 1141 Convergence rates for a class of estimators based on Stein’s method IRUROZKI, E., CALVO, B. and LOZANO, J.A. 1160 Mallows and generalized Mallows model for matchings LEE, J.H. and SONG, K. 1189 Stable limit theorems for empirical processes under conditional neighborhood dependence LEDERER, J., YU, L. and GAYNANOVA, I. 1225 Oracle inequalities for high-dimensional prediction (continued) The papers published in Bernoulli are indexed or abstracted in Current Index to Statistics, Mathematical Reviews, Statistical Theory and Method Abstracts-Zentralblatt (STMA-Z), and Zentralblatt für Mathematik (also avalaible on the MATH via STN database and Compact MATH CD-ROM). A list of forthcoming papers can be found online at http://www. bernoulli-society.org/index.php/publications/bernoulli-journal/bernoulli-journal-papers Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty Five Number Two May 2019 ISSN: 1350-7265 CONTENTS (continued) CAMPBELL, T., HUGGINS, J.H., HOW, J.P. and BRODERICK, T. 1256 Truncated random measures YU, Z., LEVINE, M. and CHENG, G. 1289 Minimax optimal estimation in partially linear additive models under high dimension LIESE, F., MEISTER, A. and KAPPUS, J. 1326 Strong Gaussian approximation of the mixture Rasch model ROUEFF, F. and VON SACHS, R. 1355 Time-frequency analysis of locally stationary Hawkes processes AHN, S.W. and PETERSON, J. 1386 Quenched central limit theorem rates of convergence for one-dimensional random walks in random environments DURIEU, O. and WANG, Y. 1412 From random partitions to fractional Brownian sheets MAURER, A. 1451 A Bernstein-type inequality for functions of bounded interaction ZHOU, C., HAN, F., ZHANG, X.-S. and LIU, H. 1472 An extreme-value approach for testing the equality of large U-statistic based correlation matrices OLSSON, J. and DOUC, R. 1504 Numerically stable online estimation of variance in particle filters SEPEHRI, A. 1536 New tests of uniformity on the compact classical groups as diagnostics for weak-∗ mixing of Markov chains BRETON, J.-C., CLARENNE, A. and GOBARD, R. 1568 Macroscopic analysis of determinantal random balls Volume 25 Number 2 May 2019 Pages 793–1601 ISI/BS Volume 25 Number 2 May 2019 ISSN 1350-7265 BERNOULLI Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Aims and Scope BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. Bernoulli Society for Mathematical Statistics and Probability The Bernoulli Society was founded in 1973. It is an autonomous Association of the International Sta- tistical Institute, ISI. According to its statutes, the object of the Bernoulli Society is the advancement, through international contacts, of the sciences of probability (including the theory of stochastic pro- cesses) and mathematical statistics and of their applications to all those aspects of human endeavour which are directed towards the increase of natural knowledge and the welfare of mankind. Meetings: http://www.bernoulli-society.org/index.php/meetings The Society holds a World Congress every four years; more frequent meetings, coordinated by the So- ciety’s standing committees and often organised in collaboration with other organisations, are the Eu- ropean Meeting of Statisticians, the Conference on Stochastic Processes and their Applications, the CLAPEM meeting (Latin-American Congress on Probability and Mathematical Statistics), the Euro- pean Young Statisticians Meeting, and various meetings on special topics – in the physical sciences in particular. The Society, as an association of the ISI, also collaborates with other ISI associations in the organization of the biennial ISI World Statistics Congresses (formerly ISI Sessions). Executive Committee The Society is headed by an Executive Committee. As of December 2018 the Executive Committee con- sists of: President: Susan Murphy (USA); President Elect: Claudia Klüppelberg (Germany); Past Presi- dent: Sara van de Geer (Switzerland); Treasurer: Lynne Billard (USA); Scientific Secretary: Byeong U. Park (South Korea); Membership Secretary: Leonardo Rolla (Argentina); Past Membership Secretary: Mark Podolskij (Denmark); Publication Secretary: Herold Dehling (Germany); ISI Director: Ada van Krimpen (Netherlands). Further, the Society has a twelve member Council and a number of standing committees to carry out the tasks outlined above. Final authority is the general assembly of members of the Society, meeting at least biennially at the ISI World Statistics Congresses. The papers published in Bernoulli are indexed or abstracted in Current Index to Statistics, Math- ematical Reviews, Statistical Theory and Method Abstracts-Zentralblatt (STMA-Z), Thomson Sci- entific and Zentralblatt für Mathematik (also available on the MATH via STN database and Com- pact MATH CD-ROM). A list of forthcoming papers can be found online at http://www.bernoulli- society.org/index.php/publications/bernoulli-journal/bernoulli-journal-papers ©2019 International Statistical Institute/Bernoulli Society All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the Publisher. In 2019 Bernoulli consists of 4 issues published in February, May, August and November. Bernoulli 25(2), 2019, 793–827 https://doi.org/10.3150/17-BEJ986 Expansion for moments of regression quantiles with applications to nonparametric testing ENNO MAMMEN1, INGRID VAN KEILEGOM2 and KYUSANG YU3 1Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidel- berg, Germany. E-mail: [email protected] 2ORSTAT, KU Leuven, Naamsestraat 69, 3000 Leuven, Belgium. E-mail: [email protected] 3Department of Applied Statistics, Konkuk University, Seoul 143-701, Korea. E-mail: [email protected] We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya– Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requires the calculation of moments of Nadaraya–Watson quan- tile regression estimators. This calculation is done by application of higher order Edgeworth expansions. Keywords: Bahadur expansions; goodness-of-fit tests; kernel smoothing; nonparametric regression; nonparametric testing; quantiles References [1] Aït-Sahalia, Y., Fan, J. and Peng, H. (2009). Nonparametric transition-based tests for jump diffusions. J. Amer. Statist. Assoc. 104 1102–1116. MR2750239 [2] Angrist, J., Chernozhukov, V. and Fernández-Val, I. (2006). Quantile regression under misspecifica- tion, with an application to the U.S. wage structure. Econometrica 74 539–563. MR2207400 [3] Bhattacharya, R.N. and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. Wiley Series in Probability and Mathematical Statistics Wiley, New York. MR0436272 [4] Bierens, H.J. and Ginther, D. (2001). Integrated conditional moment testing of quantile regression models. Empir. Econ. 26 307–324. [5] Chao, S.-K., Volgushev, S. and Cheng, G. (2017). Quantile processes for semi and nonparametric regression. Electron. J. Stat. 11 3272–3331. MR3708539 [6] Chaudhuri, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur repre- sentation. Ann. Statist. 19 760–777. MR1105843 [7] Conde-Amboage, M., Sánchez-Sellero, C. and González-Manteiga, W. (2015).
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