Pendekatan Kernel Dalam Regresi Semiparametrik Dan Pemilihan Banwith Optimal

Mulianah, Mulianah (2006) Pendekatan Kernel Dalam Regresi Semiparametrik Dan Pemilihan Banwith Optimal. Masters thesis, Institut Teknologi Sepuluh November.

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Abstract

Dalam analisis regresi tidak semua kasus dapat diselesaikan dengan regresi parametrik, karena tidak adanya informasi yang jelas tentang bentuk hubungan antara variabel prediktor dengan variabel respon sehingga harus menggunakan pendekatan regresi nonparametrik. Dengan menggabungkan dua pendekatan tersebut dalam suatu model regresi maka diperoleh model regresi semiparametrik
Y_i = X_i' β + f(m_1i, m_2i, ..., m_qi) + ε_i, i = 1, 2, ..., n
dimana ε_i adalah variabel random yang i.i.d dengan mean 0 dan varians σ². Tujuan penelitian ini adalah menentukan estimator parameter β dan kurva regresi f. Selanjutnya mengkaji sifat asimtotik dari ĥ_G dengan metode Rice(T). Juga Memperlihatkan sifat optimal asimtotik metode AIC, FPE, S, T, dan CV berdasarkan data simulasi serta menerapkan pemilihan bandwidth optimal dalam regresi semiparametrik dengan metode Rice (T) pada data real. Dengan menggunakan pendekatan kernel diperoleh estimator parametrik :
β̂ = (X̃'X̃)⁻¹ X̃'Ỹ
dan estimator nonparametrik :
f̂(m_1, m_2, ..., m_q) = [ ∑(i=1)^n [K_h1(m_1 - m_1i) ... K_hq(m_q - m_qi)] (Y_i - X_i' β̂) ] / [ ∑(j=1)^n [K_h1(m_1 - m_1j) ... K_hq(m_q - m_qj)] ] Sifat asimtotik metode Rice mengikuti :
lim_(n→∞) E(T̂_(R,n)(h)) = σ² + O(1) Berdasarkan studi simulasi dengan menggunakan metode AIC, FPE, S, T, dan CV diperoleh hasil optimal asimtotik dengan L(h)/G(h) → 1. Dalam penerapan model regresi semiparametrik pada data ASAT (Aspartate Aminotransferase) yaitu fungsi hati sebagai variabel respon, ASAM (Asnine Aminotranforase) sebagai komponen parametrik dan kolesterol sebagai komponen nonparametrik diperoleh model estimasi semiparametrik :
Ŷ = 0,8861 X + [ ∑(i=1)^n (1/30,72) K( (m - m_i)/30,72 ) (Y_i - 0,8861 X_i) ] / [ ∑(j=1)^n (1/30,72) K( (m - m_j)/30,72 ) ]
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Not all cases in the regression analysis can be estimated using a parametric approach because the lack of clear information about the relationship between the
predictor and response variables. For such problem, we can use a nonparametric approach. By combining those two approaches, we will obtain a semiparametric regression model.
Y_i = X_i' β + f(m_1i, m_2i, ..., m_qi) + ε_i, i = 1, 2, ..., n
where ε_i are random error i.i.d with a mean of 0 and variance of σ². The objectives of this research were to determine the estimator of parameter β and the regression curve f . Further, the asymptotic behavior of ĥ_G was examined using the Rice (T) method. In addition, the asymptotic optimal behavior of the method of AIC, FPE, S, T, and CV was shown based upon simulation data. This study also discussed the application of the optimal bandwidth choice in semiparametric regression using the Rice (1) method on real data. Using the kernel approach, we obtained a parametric estimator:
β̂ = (X̃'X̃)⁻¹ X̃'Ỹ
and a nonparametric estimator:
f̂(m_1, m_2, ..., m_q) = [ ∑(i=1)^n [K_h1(m_1 - m_1i) ... K_hq(m_q - m_qi)] (Y_i - X_i' β̂) ] / [ ∑(j=1)^n [K_h1(m_1 - m_1j) ... K_hq(m_q - m_qj)] ]
he asymptotic nature ofthe Rice method satisfied the following: lim_(n→∞) E(T̂_(R,n)(h)) = σ² + O(1)
Based upon a simulation study using the method of AIC, FPE, S, T, and CV the optimal asymptotic with L(h)/G(h) → 1. In applying the semiparametric G(h) regression model at ASAT (Aspartate Aminotransforase) data, that is, the liver function as the response variable, ASAM (Asnine Aminotransforase) as the
parametric component, and cholesterol as the nonparametric component, the semiparametric estimation model was obtained as follows:
Ŷ = 0,8861 X + [ ∑(i=1)^n (1/30,72) K( (m - m_i)/30,72 ) (Y_i - 0,8861 X_i) ] / [ ∑(j=1)^n (1/30,72) K( (m - m_j)/30,72 ) ]

Item Type:	Thesis (Masters)
Additional Information:	RTSt 519.536 Mul p
Uncontrolled Keywords:	Bandwidth, Ke'mel, Regresi Semiparametrik , Bandwidth, Kernel, Semiparametric Regression.
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:	01 Oct 2026 01:45
Last Modified:	01 Oct 2026 01:45
URI:	http://repository.its.ac.id/id/eprint/145111

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