Extensions of smoothing via taut strings

Suppose that we observe independent random pairs (X1,Y1), (X2,Y2), …, (Xn,Yn). Our goal is to estimate regression functions such as the conditional mean or β–quantile of Y given X, where 00 is some tuning parameter. This framework is extended further to include binary or Poisson regression, and to include localized total variation penalties. The latter are needed to construct estimators adapting to inhomogeneous smoothness of f. For the general framework we develop noniterative algorithms for the solution of the minimization problems which are closely related to the taut string algorithm (cf. Davies and Kovac, 2001). Further we establish a connection between the present setting and monotone regression, extending previous work by Mammen and van de Geer (1997). The algorithmic considerations and numerical examples are complemented by two consistency results.