# Estimation of Heteroskedasticity Semiparametric Regression Curve Using Fourier Series Approach

## Authors

• Rahmawati Pane Department of Mathematics, Universitas Sumatera Utara, Medan, 20155, Indonesia
• Sutarman Department of Mathematics, Universitas Sumatera Utara, Medan, 20155, Indonesia
• Andi Tenri Ampa Department of Statistics, Institut Teknologi Sepuluh Nopember, Surabaya, 60111, Indonesia

## Keywords:

Fourier Series, Heteroskedastic Semiparametric Regression,, Bandwidth parameter, WPLS

## 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

2020-02-24

## How to Cite


R. Pane, Sutarman, and Andi Tenri Ampa, “Estimation of Heteroskedasticity Semiparametric Regression Curve Using Fourier Series Approach”, J. of Research in Math. Trends and Tech., vol. 2, no. 1, pp. 14-20, Feb. 2020.

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