Testing linearity of regression models with dependent errors by kernel based methods

In a recent paper Gonzalez Manteiga and Vilar Fernandez (1995) considered the problem of testing linearity of a regression under MA(infinity) structure of the errors using a weighted L-2-distance between a parametric and a nonparametric fit. They established asymptotic normality of the corresponding test statistic under the hypothesis and under local alternatives. In the present paper we extend these results and establish asymptotic normality of the statistic under fixed alternatives. This result is then used to prove that the optimal (with respect to uniform maximization of power) weight function in the test of Gonzalez Manteiga and Vilar Fernandez (1995) is given by the Lebesgue measure independently of the design density. The paper also discusses several extensions of tests proposed by Azzalini and Bowman (1993), Zheng (1996) and Dette (1999) to the case of non-independent errors and compares these methods with the method of Gonzalez Manteiga and Vilar Fernandez (1995). It is demonstrated that among the kernel based methods the approach of the latter authors is the most efficient from an asymptotic point of view.