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Recent developments in flexible mean and quantile regression

In both, mean and quantile regression, flexible models are needed to capture the complexity of the underlying data process. Among flexible models encountered in a multivariate covariate regression setting are additive models, single-index models, and varying coefficient models. In this talk the focus is on flexible varying coefficient models. Quantile regression is an important tool for describing the characteristics of conditional distributions. Similarly to median versus mean estimates, median regression estimates are more robust against outliers in the response variable than mean regression estimates. Most of the talk focuses on quantile regression in varying coefficient models, using as a major working tool B-spline approximations. Important issues in quantile regression inference are: methods to prevent that the estimated quantile curves cross; homoscedasticity versus heteroscedasticity; and testing procedures for testing for specific shapes (constancy, monotonicity, convexity, ...) of coefficient functions. All, apart from the first, are also concerns in mean regression.