M-type smoothing splines in nonparametric and semiparametric regression models

Consider the regression model Y _i = g(t _i) + e _i for i = 1,...,n. Here -∞ < Y _i, e _i < ∞, t _i ∈ T ⊂ R ^d, g ∈ H, and H is a specified class of continuous functions from T to R. Based on a finite series expansion g̃ _n of g, an M-estimate ĝ _n of g is constructed, and the asymptotic normality of the estimate is investigated. Meanwhile, a test statistic for testing H ₀ : g(·) = g ₀(·) (a known function) is discussed. We also consider M-estimates for semiparametric regression models and show that they are consistent and asymptotically normal.