Abstract
In the real cases, we are frequently faced the problem in which two or more dependent variables are observed at several values of the independent variables, such as at multiple time points. Multi-response nonparametric regression model, especially smoothing spline model, provides powerful tools to model the function which represents association of among the variables. The problem is how to estimate nonparametric regression curve of the multi-response nonparametric regression model. The nonparametric regression curve can be estimated using spline estimator approach, that is by carrying out penalized weighted least-squares optimation. Therefore, we need a covariance matrix which will be used as a weight of the optimation. In this paper, we determine the construction of covariance matrix for both equal and unequal of correlations cases. The results show that the covariance matrices have quite similar construction of diagonal elements but the elements outside the diagonal have very different construction that depend on the construction of the Jordan matrix.
Highlights
In the real cases, we are frequently faced the problem in which two or more dependent variables are observed at several values of the independent variables, such as at multiple time points
Lestari et al (2010) membahas estimator spline dalam regresi nonparametrik multirespon untuk kasus korelasi tidak sama
J. (1981) Vector splines and estimation of filter function, Technometrics, 23, 83-89
Summary
Fungsi yang menggambarkan hubungan antara dua atau lebih variabel terikat yang diobservasi pada beberapa nilai variabel bebas dapat dimodelkan dengan menggunakan pendekatan spline dalam regresi nonparametrik multirespon. Wang et al (2000) membahas penghalusan spline untuk mengestimasi fungsi nonparametrik untuk data bivariat. Lestari et al (2010) membahas estimator spline dalam regresi nonparametrik multirespon untuk kasus korelasi tidak sama. Jika matriks kovariansi tidak diketahui maka harus diestimasi dari data dan hal ini dapat mempengaruhi estimasi parameter penghalus (Wang, 1998). Masalah estimasi fungsi atau kurva regresi nonparametrik dengan menggunakan pendekatan penghalusan spline yang diperoleh dengan penyelesaian optimasi penalized weighted least-squares diperlukan suatu bobot yang ditentukan oleh matriks kovariansinya. Dalam paper ini akan dibahas bagaimana mendapatkan konstruksi matriks kovariansi pada kasus korelasi sama maupun pada kasus korelasi tidak sama
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