The Estimation of Residual Variance in Nonparametric Regression

Abstract

English

Given a nonparametric regression model Yi = g(xi) + ei, i = 1, 2, …, n, where Y is a dependent variable, x is an independent variable, g is an unknown function and e is an error assumed to be an independent, identical, and is distributed with mean 0 and variance σ2. In this research Rice estimator is used to determine the biased value of a residual variance estimator. The Rice estimator is given as follows:. The biased value of residual variance estimator of the Rice method is:, where and. Using the Rice estimator, the Tong-Wang residual variance estimator is obtained, that is:, Where,,,,, k = 1, 2, …, m. Based upon the data simulation by considering the exponential, arithmetical, and trigonometrical models, it is found that the MSE value of the Tong-Wang estimator tends to be less compared to those of the Rice estimator as well as the GSJ (Gasser, Sroka, and Jennen) estimator.

Indonesian

Diberikan model regresi nonparametrik Yi = g(xi) + ei, i = 1, 2, …, n, dengan Y adalah variabel dependen, x variabel independen, g fungsi yang tidak ditetapkan bentuknya dan e eror yang diasumsikan independen, identik dan berdistribusi dengan rataan 0 dan varians σ2. Dalam penelitian ini digunakan penduga Rice untuk mendapatkan nilai bias penduga varians residual. Penduga Rice diberikan oleh:. Nilai bias penduga varians residual metode Rice adalah:, dimana dan. Dengan menggunakan penduga Rice diperoleh penduga varians residual Tong-Wang yaitu:, dimana,,,,, k = 1, 2, …, m. Berdasarkan data simulasi dengan mempertimbangkan model eksponensial, aritmetika dan trigonometri diperoleh nilai MSE penduga Tong-Wang cenderung lebih kecil dibandingkan dengan penduga Rice dan penduga GSJ (Gasser, Sroka, dan Jennen).

Keywords

• Nonparametric regression
• Rice estimator
• GSJ estimator
• Tong-Wang estimator