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\(M\)-type penalized splines with auxiliary scale estimation. (English) Zbl 1460.62052
Summary: Penalized spline regression is a popular and flexible method of obtaining estimates in nonparametric models 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 illustrate the benefits of \(M\)-type penalized splines in a Monte-Carlo study and two real-data examples, which contain atypical observations.
MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G35 Nonparametric robustness
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