TY - JOUR AU - Pane, Rahmawati AU - Sutarman, AU - Andi Tenri Ampa, PY - 2020/02/24 Y2 - 2024/03/28 TI - Estimation of Heteroskedasticity Semiparametric Regression Curve Using Fourier Series Approach JF - Journal of Research in Mathematics Trends and Technology JA - J. of Research in Math. Trends and Tech. VL - 2 IS - 1 SE - DO - 10.32734/jormtt.v2i1.3744 UR - https://talenta.usu.ac.id/jormtt/article/view/3744 SP - 14-20 AB - <p>A heteroskedastic semiparametric regression model consists of two main <br>components, i.e. parametric component and nonparametric component. The model assumes <br>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 <br>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] . <br>Random error ε i is independent on zero mean and variance<br>σ 2 . Estimation of the <br>heteroskedastic semiparametric regression model was conducted to evaluate the parametric <br>and nonparametric components. The nonparametric component f(t i ) regression was <br>approximated by Fourier series F(t) = bt + 1<br>2 α 0 + ∑ α k 𝑐 𝑜𝑠 kt Kk=1 . The estimation was <br>obtained by means of Weighted Penalized Least Square (WPLS): min f∈C(0,π) {n −1 (y̰− Xβ̰−<br>f̰) ′ W −1 (y̰− Xβ̰− f̰) + λ ∫ 2<br>π [f ′′ (t)] 2 dt π<br>0 } . The WPLS solution provided nonparametric <br>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</p> ER -