Testing for polynomial regression using nonparametric regression techniques

Abstract

In regression analysis, it is always important to test the validity of the assumed model prior to making inferences regarding the population of interest. In this investigaton, we utilize nonparametric regression techniques to test the validity of a k th order polynomial model. The departures from the polynomial model are assumed to belong to a smooth class of functions; a parametric form is not assumed. Two tests based on nonparametric regression fits to the residuals from k th order polynomial regression are proposed. The first utilizes a polynomial regression fit of order (m + k - 1) to the residuals from k th order polynomial regression. Then m is allowed to grow with n, the sample size, as n tends to infinity. A test statistic based on this estimator is formulated and its asymptotic distribution under alternatives converging to the null at a rate of m^[1/4]/[square root(n)] is derived. The second test proposed is based on a statistic utilizing a 2k th order smoothing spline fit to the residuals from k th order polynomial regression. Its asymptotic distributon under alternatives converging to the null at a rate of (nλ^[1/4k])^[-1/2] where λ is the smoothing parameter, is derived. We note that these rates of convergence are slower than the parametric rate of n^[-1/2]. Large sample comparisons of the two tests are conducted via Pitman asymptotic relative efficiency and the smoothing spline test is seen to be more efficient than the polynomial regression based test. A small-scale simulation study conducted in order to compare the two tests in finite samples did not produce a clear winner in terms of power.