Nonparametric estimation for quadratic regression

The method of least squares provides the most widely used algorithm for fitting a linear model. A variety of nonparametric procedures have been developed that are designed to be robust against model violations and resistant against aberrant points. One such method introduced by Theil [1950. A rank-invariant method of linear and polynomial regression analysis. I, II, III. Proc. Ned. Akad. Wet. 53, 386–392, 521–525, 1397–1412] is based on pairwise estimates. There are many examples in which the data are nonlinear, and in particular, where a quadratic fit may be more appropriate. We here propose a nonparametric method for fitting a quadratic regression.