A robust test for the scale function in nonparametric regression

JORGE G. ADROVER, GRACIELA L. BOENTE Y MARCELO RUIZ

Viña del Mar

Congreso; Coloquio Latinoamericano de Probabilidad y Estadística Matemática, Viña del Mar (Chile) 26 al 30 de marzo de 2012; 2012

Consider n observations (Yi,xi), verifying a nonparametric regression model Yi = g(xi) +Uis(xi), where x1,...,xn are fixed design points in [0; 1], s  is an unknown scale function, g denotes the unknown regression function and the errors Ui are i.i.d. random variables with common distribution G. To allow for outliers and some other departures from a central distribution function F0, G is assumed to belong to a gross error epsilon-contamination neighborhood. G. Boente, M. Ruiz and R. Zamar (2010) define robust estimators for s(x) without estimating simultaneously the regression function g(x). More precisely, they consider kernel-based M-estimators hat(s)(x), based on successive diferences of the responses Yi´s,which turn out to be consistent and asymptotically normal under general conditions.As in any nonparametric setting, the rate of convergence of the estimators depends on the smoothing parameter. In this talk, robust tests for hypothesis involving the scale component are introduced. We first consider the simple hypothesis H0 : s(x) = s0(x); forall x in (0, 1), where s0(x) is a known function. The test statistic is given by a goodness of fit type statistics test.Its asymptotic behavior under the null hypothesis and under contigous alternatives is investigated. The finite sample properties are investigated by means of a simulation study, and the test is compared with some nonrobust tests for scale.