Estimation of Heteroskedasticity Semiparametric Regression Curve Using Fourier Series Approach

Abstract

A heteroskedastic semiparametric regression model consists of two main
components, i.e. parametric component and nonparametric component. The model assumes
that any data (x̰ i′ , t i , y i ) follows y i = x̰ i′ β̰+ f(t i ) + σ i ε i , where i = 1,2, … , n , x̰ i′ = (1, x i1 , x i2 , … , x ir ) and t i
is the predictor variable. Parameter vector β̰ = (β 1 , β 2 , … , β r ) ′ ∈ ℜ r is unknown and f(t i ) is also unknown and is assumed to be in interval of C[0,π] .
Random error ε i is independent on zero mean and variance
σ 2 . Estimation of the
heteroskedastic semiparametric regression model was conducted to evaluate the parametric
and nonparametric components. The nonparametric component f(t i ) regression was
approximated by Fourier series F(t) = bt + 1
2 α 0 + ∑ α k 𝑐 𝑜𝑠 kt Kk=1 . The estimation was
obtained by means of Weighted Penalized Least Square (WPLS): min f∈C(0,π) {n −1 (y̰− Xβ̰−
f̰) ′ W −1 (y̰− Xβ̰− f̰) + λ ∫ 2
π [f ′′ (t)] 2 dt π
0 } . The WPLS solution provided nonparametric
component f̰̂ λ (t) = M(λ)y̰ ∗ for a matrix M(λ) and parametric component β̰̂ = [X ′ T(λ)X] −1 X ′ T(λ)y̰