Estimation of Heteroskedasticity Semiparametric Regression Curve Using Fourier Series Approach - Pane, Rahmawati; Sutarman; Andi Tenri Ampa (Journal of Research in Mathematics Trends and Technology, 2020)
A heteroskedastic semiparametric regression model consists of two main components, i.e. parametric component and nonparametric component. The model assumes that any data (xÌ° i′ , t i , y i ) follows y i = xÌ° i′ Î²Ì°+ f(t i ) + Ïƒ i Îµ i , where i = 1,2, … , n , xÌ° i′ = (1, x i1 , x i2 , … , x ir ) and t i is the predictor variable. Parameter vector Î²Ì° = (Î² 1 , Î² 2 , … , Î² r ) ′ âˆˆ â„œ r is unknown and f(t i ) is also unknown and is assumed to be in interval of C[0,Ï€] . Random error Îµ i is independent on zero mean and variance Ïƒ 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 Î± 0 + âˆ‘ Î± k ð‘ ð‘œð‘ kt Kk=1 . The estimation was obtained by means of Weighted Penalized Least Square (WPLS): min fâˆˆC(0,Ï€) n âˆ’1 (yÌ°âˆ’ XÎ²Ì°âˆ’ fÌ°) ′ W âˆ’1 (yÌ°âˆ’ XÎ²Ì°âˆ’ fÌ°) + Î» âˆ« 2 Ï€ [f ′′ (t)] 2 dt Ï€ 0 . The WPLS solution provided nonparametric component fÌ‚Ì° Î» (t) = M(Î»)yÌ° âˆ— for a matrix M(Î») and parametric component Î²Ì°Ì‚ = [X ′ T(Î»)X] âˆ’1 X ′ T(Î»)yÌ°