## Electronic Journal of Statistics

### Extensions of smoothing via taut strings

#### Abstract

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 0<β<1. In order to achieve this we minimize criteria such as, for instance, $$\sum_{i=1}^n ρ(f(X_i) - Y_i) + λ⋅\mathrm{TV}(f)$$ among all candidate functions f. Here ρ is some convex function depending on the particular regression function we have in mind, TV(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.

#### Article information

Source
Electron. J. Statist., Volume 3 (2009), 41-75.

Dates
First available in Project Euclid: 28 January 2009

https://projecteuclid.org/euclid.ejs/1233176790

Digital Object Identifier
doi:10.1214/08-EJS216

Mathematical Reviews number (MathSciNet)
MR2471586

Zentralblatt MATH identifier
1326.62087

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G35: Robustness

#### Citation

Dümbgen, Lutz; Kovac, Arne. Extensions of smoothing via taut strings. Electron. J. Statist. 3 (2009), 41--75. doi:10.1214/08-EJS216. https://projecteuclid.org/euclid.ejs/1233176790

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