Robust estimation of nonparametric function via addition sequence

In this paper, we propose a robust method for the estimation of regression function. By symmetric addition, we change platykurtic errors into leptokurtic errors; and then estimate the nonparametric function by the local polynomial least absolute deviation regression. Different from the local polynomial least squares estimator, the new estimator is robust for outliers and heavy-tailed errors even if the error variance does not exist; different from the usual local polynomial least absolute deviation estimator and the composite quantile regression estimator, it does not depend on the finite density values at chosen quantile points, but relies on the expectation of the error density. To improve the finite sample performance, two bias-reduced versions are further proposed under different smoothness conditions. For the equidistant designs, the asymptotic properties are established. In simulations, the new estimator has the less mean square errors than its main competitors in the presence of platykurtic and heavy-tailed errors.