Testing the hypothesis of a general linear model using nonparametric regression estimation

Summary Given the modelY _i =m(χ _i )+ɛi,whereE(ɛ _i) =0,X _i ≠Ci=1, ...,n, andC is ap-dimensional compact set, we have designed a new method for testing the hypothesis that the regression function follows a general linear model,m(·) ∈ {m _θ(·) =A ^t (·)θ}_{θ∈Θ⊂ℛq} , withA a function fromℜ ^p toℜ ^q. The statistic, denoted ΔASE, used fortesting the given hypothesis is defined to be the difference between the average squared errors (ASE) associated with the non-parametric estimator ofm and the minimum distance parametric estimator ofm. The asymptotic normality of both ΔASE and the minimum distance estimators is proved under general conditions. Alternative bootstrap versions of ΔASE are also considered.