Mini-Workshop: Frontiers in Quantile Regression

Quantiles play an essential role in modern statistics, as emphasized by the fundamental work of Parzen (1978) and Tukey (1977). Quantile regression was introduced by Koenker and Bassett (1978) as a complement to least squares estimation (LSE) or maximum likelihood estimation (MLE) and leads to far-reaching extensions of ”classical” regression analysis by estimating families of conditional quantile surfaces, which describe the relation between a one-dimensional response y and a high dimensional predictor x. Since its introduction quantile regression has found great attraction in mathematical and applied statistics because of its natural interpretability and robustness, which yields attractive applications in such important areas as medicine, economics, engineering and environmental modeling. Although classical quantile regression theory is very well developed, the implicit definition of quantile regression still yields many new mathematical challenges such as multivariate, censored and longitudinal data, which were discussed during the workshop. Mathematics Subject Classification (2000): 62G10, 62G08, 62G30. Introduction by the Organisers The workshopFrontiers of quantile regression, organised by Victor Chernozhukov (Boston), Holger Dette (Bochum), Xuming He (Ann Arbor) and Roger Koenker (Champaign) was held 25 November – 1 December 2012. This meeting was well attended by 16 participants with broad geographic representation from all continents. During the workshop all mathematical aspects of the recent development in quantile regression analysis were discussed. A particular focus was on Multivariate quantile regression where several new concepts were presented by the 3340 Oberwolfach Report 56/2012 participants, including Bahadur representations, asymptotic normality and uniform convergence of the corresponding estimates. Other talks discussed quantile regression for longitudinal data and random effect models with applications in functional data analysis and biostatistics and the definition of new spectra of stationary time series via quantile regression methods. Several speakers presented their results on variable selection in high-dimensional quantile regression models, especially under the framework of “large p small n paradigm (here p refers to the dimension of the parameter to be estimated and n denotes the sample size). It was shown that useful model identification is possible when sparsity of the model is expected to hold. Two other speakers discussed quantile regression methods for censored data. Specifically, the following research fields in quantile regression were discussed during the workshop. (1) Multivariate quantile regression (2) Quantile regression for longitudinal data and random effect models (3) Bayesian analysis in quantile regression (4) Variable selection in high-dimensional quantile regression models (5) Quantile regression for censored data (6) Quantile regression in time series The workshop stimulated intensive discussions between all participants and new developments in various subfields of quantile regression analysis. For example, the problem of quantile regression for multivariate was discussed in three talks from different perspectives. Similarly, in the context of stationary time series a spectral theory will be developed, which avoids the existence of any moments.