Local smoothing with robustness against outlying predictors

Abstract

Outlying pollutant concentration data are frequently observed in time series studies conducted to investigate the effects of atmospheric pollution on mortality/morbidity. These outliers may severely affect the estimation procedures and even generate unexpected results like a protective effect of pollution. Although robust methods have been proposed to downweight the effect of outliers in the response variable distribution, little has been done to handle outlying explanatory variable values. We consider a robust local polynomial smoothing technique which may be useful for such purposes. It is based on downweighting points with a small design density and may also be used as a diagnostic tool to identify outliers. Using data from a study conducted in São Paulo, Brazil, we show how an unexpected form of the relative risk curve of mortality attributable to pollution by SO2 obtained via nonrobust methods may be completely reversed when the proposed technique is employed. Copyright © 2004 John Wiley & Sons, Ltd.

On design-weighted local fitting and its relation to the Horvitz-Thompson estimator

“…The asymptotical result (17) confirms the statement by Hastie & Loader (1993), who called an endpoint 'the most informative observation' when fitting at this endpoint. Einbeck, André & Singer (2004) added that this holds only when this point can be considered as 'as reliable as in the interior'. Any kind of robust estimation implies that one is not willing to trust a certain group of data points (in this case the outlying predictors and its associated y−values), whereas the asymptotic result is -as HT -certainly based on full reliance on the information content of all data points, including outlying predictors.…”

“…The bandwidth g was selected for each simulated data set anew using Silverman's (1986, p. 48) bandwidth selector, as also proposed in Einbeck, André & Singer (2004). The asymptotic weights (the third boxplot each) turned out to be superior for both sample sizes and all three error criteria.…”

“…Here we concentrate on the case of a constant bandwidth h as in Einbeck, André & Singer (2004), who proposed to set k equal to some small positive integer, e.g. k = 1 or 2.…”

Utilização da regressão quantílica local na construção de curvas de referência

“…The construction of the reference curves envolves the estimation of the populational quantiles of respoílse variable (y) distribution, at di6erent values of pledictor vaiiable (X).With the objectivo to construct reference curves, tais work presents some nonparametric regressioii methods that fit condicional quantiles functions locally. Local quantile regression methods that are robust to outlying predictor variable values are deríved by adapting the methodology suggested in Einbeck et al (2004) in the local âtting of the regression function.These procedurcs a.re applied to a data set related to the cardiology área, with the objective to construct reference curves for the average cardiac frequency of healthful individuais as function of their age. The obtained resulta show that the reference curves constructed by the robust methods are more appropriate in the regions in which the design is sparse.…”

“…With the objectivo to construct reference curves, tais work presents some nonparametric regressioii methods that fit condicional quantiles functions locally. Local quantile regression methods that are robust to outlying predictor variable values are deríved by adapting the methodology suggested in Einbeck et al (2004) in the local âtting of the regression function.…”