Extensions of smoothing via taut strings

Abstract

Suppose that we observe independent random pairs $(X_1,Y_1)$, $(X_2,Y_2)$, >..., $(X_n,Y_n)$. Our goal is to estimate regression functions such as the conditional mean or $β$--quantile of $Y$ given $X$, where $0<β<1$. In order to achieve this we minimize criteria such as, for instance, $$ \sum_{i=1}^n ρ(f(X_i) - Y_i) + λ\cdot \mathop TV\nolimits (f) $$ among all candidate functions $f$. Here $ρ$ is some convex function depending on the particular regression function we have in mind, $\mathop {\rm TV}\nolimits (f)$ stands for the total variation of $f$, and $λ>0$ 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.