Mixed Estimator of Kernel and Fourier Series in Semiparametric Regression

Given paired observation (x_i, v_1i, v_2i, ⋯, v_pi, t_1i, t_2i, ⋯, t_qi, y_i), i = 1, 2, ⋯, n, follow the additive semiparametric regression model y_i = μ(x_i, v_i, t_i) +_i, where μ(xi,vt,ti)=f(xi)+∑j=1pgj(νji)+∑s=1qhs(tsi) v_i = (v_1i, v_2i, ⋯, v_pi)^′, and t_i = (t_1i, t_2i, ⋯, t_qi)^′. Random errors_i is a normal distribution with mean 0 and variance σ ². To obtain a mixed estimator μ(x_i, v_i, t_i), the regression curve f(x_i) is approached by linier parametric, g_j(v_ji) is kernel with bandwidths Φ = (φ₁, φ₂, ⋯, φ_p)^′ and the regression curve component fourier series h_s (t_si) is approached by with oscillation paremeter N. The estimator is where . Penalized Least Squares (PLS) method give Minc,β{ L(c)+L(β)+∑s=1qθsS(Hs(tsi)) } with smoothing parameter θ = (θ₁, θ₂, ⋯, θ_q)^′, the estimator f(x) is and is , where and . So that, μΦ,θ,N(vi,ti)=Z(Φ,θ,N)y is the mixed estimator of μ(v_i, t_i) where Z(Φ, θ, N) = C(Φ, θ, N) + V(Φ) + E(Φ, θ, N) Matrix C(Φ, θ, N), V(Φ) and E(Φ, θ, N) are depended on Φ, θ and N. Optimal Φ, θ and N can be obtained by the smallest Generalized Cross Validation (GCV).