A nonparametric least-squares test for checking a polynomial relationship

In this paper the interest is in testing whether a regression function is a polynomial of a certain degree. One possible approach to this testing problem is to do a parametric polynomial fit and a nonparametric fit and to reject the null hypothesis of a polynomial function if the distance between the two fits is too large. Another approach consists of looking at the residuals from the parametric fit. In this paper we propose an entirely new approach to deal with the testing problem. When testing whether a regression function is a polynomial of degree smaller than or equal to p, the key idea is to first obtain a nonparametric local polynomial estimate of the pth derivative of the unknown regression function, and then to proceed with a classical least-squares test for a general linear model for testing whether this derivative is constant. This is a quite appealing approach since it just relies on ordinary least-squares tests, and hence is simple to use. The performance of the method is illustrated via a simulation study.