X2 goodness-of-fit tests for polynomial regression

In the context of nonparametric regression models, we propose a very general procedure to test the goodness-of-fit of the regression function underlying in a set of (z(i), y(i)) data, to a polynomial type family of regression curves. Test statistics are based on residual sums of squares obtained by a comparison of a nonparametric fit, HY, versus a parametric fit, PY, via RSS (HY, PY) = (HY - PY)(T)(HY - PY) or versus a smoothed parametric fit, via RSS (HY, HPY) = (HY - HPY)(T)(HY - HPY). A chi(2) distribution with degrees of freedom determined by the hat matrixes H and P is used to approximate the distribution of test statistics. The proposed procedure generalizes classical least squares theory and involves a variety of different nonparametric smoothing techniques. A comparison among chi(2) tests with different smoothing techniques and with previous procedures based on a normal distribution and bootstrap is made by means of a simulation study.