Pendugaan Varians Residual Dalam Regresi Nonparametrik

Wahab, Abdul (2006) Pendugaan Varians Residual Dalam Regresi Nonparametrik. Masters thesis, Institut Teknologi Sepuluh November.

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Abstract

Diberikan model regresi nonparametrik
Yᵢ = g(xᵢ) + εᵢ, i = 1, 2, ..., n, dengan Y adalah variabel dependenvariabel independen, g fungsi yang tidak ditetapkan bentuknya dan & eror yang diasumsikan independen, identik dan berdistribusi dengan rataan 0 dan varians . Dalam penelitian ini digunakan penduga Rice untuk mendapatkan nilai bias penduga varians residual. Penduga Rice diberikan oleh:
σ̂ᵣ² = [ 1 / (2(n - 1)) ] × ∑ᵢ₌₂ⁿ (Yᵢ - Yᵢ₋₁)²
Nilai bias penduga varians residual metode Rice adalah:
θ = J ⋅ dₖ + O( k³ / n³ ) dimana J = (1/2) ∫₀¹ (g'(x))² dx dan dₖ = k² / n²
Dengan menggunakan penduga Rice diperoleh penduga varians residual Tong-Wang yaitu:
σ̂ₜₚ² = s_w - [ ∑ₖ₌₁ᵐ wₖ ⋅ sₖ ⋅ (dₖ - d_w) / ∑ₖ₌₁ᵐ wₖ ⋅ (dₖ - d_w)² ] × d_w
dimana wₖ = (n - k) / N, N = ∑ₖ₌₁ᵐ (n - k), sₖ = [ 1 / (2(n - k)) ] × ∑ᵢ₌ₖ₊₁ⁿ (Yᵢ - Yᵢ₋ₖ)²
s_w = ∑ₖ₌₁ᵐ wₖ ⋅ sₖ, d_w = ∑ₖ₌₁ᵐ wₖ ⋅ dₖ, 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).
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Given a nonparametric regression model
Yᵢ = g(xᵢ) + εᵢ, i = 1, 2, ..., n, with Yis the dependent variable-independent variable, gis a function whose form is not specified and \varepsilonis an error assumed to be independent, identically distributed with mean 0 and variance . In this study, the Rice estimator is used to obtain the bias value of the residual variance estimator. The Rice estimator is given by:
σ̂ᵣ² = [ 1 / (2(n - 1)) ] × ∑ᵢ₌₂ⁿ (Yᵢ - Yᵢ₋₁)²
The bias value of the residual variance estimator using the Rice method is:
θ = J ⋅ dₖ + O( k³ / n³ ) where J = (1/2) ∫₀¹ (g'(x))² dx dan dₖ = k² / n²
Using the Rice estimator, the Tong-Wang residual variance estimator is obtained as:
σ̂ₜₚ² = s_w - [ ∑ₖ₌₁ᵐ wₖ ⋅ sₖ ⋅ (dₖ - d_w) / ∑ₖ₌₁ᵐ wₖ ⋅ (dₖ - d_w)² ] × d_w
where wₖ = (n - k) / N, N = ∑ₖ₌₁ᵐ (n - k), sₖ = [ 1 / (2(n - k)) ] × ∑ᵢ₌ₖ₊₁ⁿ (Yᵢ - Yᵢ₋ₖ)²
s_w = ∑ₖ₌₁ᵐ wₖ ⋅ sₖ, d_w = ∑ₖ₌₁ᵐ wₖ ⋅ dₖ, k = 1, 2, ..., m
Based on simulation data by considering exponential, arithmetic, and trigonometric models, the MSE value of the Tong-Wang estimator tends to be smaller compared to the Rice estimator and the GSJ (Gasser, Sroka and Jennen) estimator.

Item Type:	Thesis (Masters)
Additional Information:	RTSt 519.536 Wah p
Uncontrolled Keywords:	Regresi nonparametrik, Penduga Rice, penduga GSJ, penduga Tong-Wang, Nonparametric regression, Rice estimator, GSJ estimator, Tong-Wang estimator.
Subjects:	Q Science > QA Mathematics > QA278.2 Regression Analysis. Logistic regression
Divisions:	Faculty of Mathematics and Science > Statistics > 49101-(S2) Master Thesis
Depositing User:	magang .
Date Deposited:	30 Sep 2026 08:15
Last Modified:	30 Sep 2026 08:15
URI:	http://repository.its.ac.id/id/eprint/145101

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