Robust bandwidth selectors in semiparametric partly linear regression models

Consider a semiparametric partly linear model, with response variable $y$ and covariates $x_1,\dots, x_p$ and $t$. This model can be a suitable choice when one suspects that the response depends linearly on $x$, but that it is nonlinearly related to $t$. Least square estimators have been studied by several authors. All these estimators, as nonparametric estimators, depend on a smoothing parameter that should be chosen by the practitioner. As it is well known, large bandwidths produce estimators with small variance but high bias, while small values produce more wiggly curves. This trade-off between bias and variance lead to several proposals to select the smoothing parameter, such as cross-validation procedures and plug-in methods. It is well known that, both in linear regression and in nonparametric regression, least squares estimators can be seriously affected by anomalous data. The same statement holds for partly linear models. To avoid that problem, Bianco and Boente (2003) considered a three-step robust estimate for the regression parameter and the regression function. In this talk, we will introduce a robust plug-in selector for the bandwidth, under a partly linear model with fixed design which converges to the optimal one and leads to robust data-driven estimates of the regression function and the regression parameter. Our plug-in proposal is based on nonparametric robust estimates of the $j$-th derivatives, which extends the proposals given when $j=2$. We define an empirical influence measure for data-driven bandwidth selectors and, through it, we study the sensitivity of the plug-in selector. We use a Monte Carlo study to compare the performance of the classical approach and of the resistant selectors under normality and contamination. It appears that the robust selector compares favourably to its competitor, despite the need to select a pilot bandwidth. When combined with the three-step procedure proposed by Bianco and Boente (2003), it leads to robust data-driven estimates both of the regression function and the regression parameter.