# M-type penalized splines with auxiliary scale estimation

## TL;DR

This paper studies M-type penalized spline estimators with auxiliary scale estimation, demonstrating they achieve optimal convergence rates and are robust to atypical observations, supported by theoretical analysis and empirical examples.

## Contribution

It provides the first theoretical analysis of M-type penalized spline estimators with auxiliary scale estimation, showing they match the convergence rates of classical methods.

## Key findings

- M-type penalized splines achieve the same convergence rates as least-squares methods.
- They are robust to atypical observations in practical data.
- Using a small number of knots is theoretically justified.

## Abstract

Penalized spline smoothing is a popular and flexible method of obtaining estimates in nonparametric regression but the classical least-squares criterion is highly susceptible to model deviations and atypical observations. Penalized spline estimation with a resistant loss function is a natural remedy, yet to this day the asymptotic properties of M-type penalized spline estimators have not been studied. We show in this paper that M-type penalized spline estimators achieve the same rates of convergence as their least-squares counterparts, even with auxiliary scale estimation. We further find theoretical justification for the use of a small number of knots relative to the sample size. We illustrate the benefits of M-type penalized splines in a Monte-Carlo study and two real-data examples, which contain atypical observations.