Robust Estimation of Error Scale in Nonparametric Regression Models

G. BOENTE, I. GHEMENT, M. RUIZ Y R. ZAMAR

Buenos Aires

Congreso; International Conference on Robust Statistics (ICORS 2007); 2007

We consider the problem of robust estimation of the scale in nonparametric regression models. The model of interest can be expressed as Yi = g(xi) + Ui(xi), i = 1, . . . ,n, (1) where the Yi’s are observed responses, the xi’s are fixed design points in the interval (0, 1) , g(·) is an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. where the Yi’s are observed responses, the xi’s are fixed design points in the interval (0, 1) , g(·) is an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. where the Yi’s are observed responses, the xi’s are fixed design points in the interval (0, 1) , g(·) is an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. an unknown, smooth regression curve and the smooth function (·) represents the unknown scale function to be robustly estimated. The Ui’s are i.i.d. unobservable random errors with common distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distribution function G belonging to an -contaminated neighborhood of a nominal model F0. distributi