Spline Smoothing for Multi-Response Nonparametric Regression Model in Case of Heteroscedasticity of Variance

Abstract

Problem statement: Assume that data (y ki , t ki ), k = 1,2,…, p; i = 1,2,…,n k where n k represents the number of repeated measurement of k th object follows multi-response nonparametric regression model with variances of errors are heteroscedastic. Nonparametric regression curves are unknown and assumed to be smooth which are contained in Sobolev space. Random Errors are independent and normally distributed with zero means and unequal of variances. Approach: Smoothing spline can be used to estimate the nonparametric regression curve by carrying out the penalized weighted least-squares optimation. Therefore, reproducing kernel Hilbert space approach is applied to carry out the penalized weighted least-squares optimation. Results: In this study we consider the heteroscedastic multi-response nonparametric regression model and give a mathematical statistics method for obtaining the weighted spline estimator to estimate heteroscedastic multi-response nonparametric regression curves. Conclusion: The reproducing kernel Hilbert space approach gives solution of penalized weighted least-squares optimation for estimating heteroscedastic multi-response nonparametric regression curve which gives the weighted spline estimator. The estimator obtained is a biased estimator for nonparametric regression curve. However, the estimator is linear in observation.

The combination of spline and kernel estimator for nonparametric regression and its properties

“…Nonparametric regression is a regression model approached used if the pattern of relation between predictor variable and response isn't known, or if there is no complete past information on the shape of data pattern [1], [2], [3]. The nonparametric regression models which receive a lot of attention from re-I Nyoman Budiantara et al searchers are Kernel [4], [5], [6], Spline smoothing [2], [7], [8], [9], [10], Fourier Series [11], [12], [13] and Local Polynomial [14], [15]. Nonparametric regression approach has high flexibility because data is expected to look for its shape of regression curve estimation without being influenced by researchers subjective factors [1], [3], [6].…”

“…Among nonparametric and semiparametric models above, spline truncated is one of the models with very specific and good statistical interpretation and visual interpretation [3], [9]. Estimator spline truncated has high flexibility [1].…”

“…Estimator spline truncated has high flexibility [1]. Spline truncated also has very good ability to handle data with changeable behaviors on certain sub-intervals [1], [3], [8], [9], [10]. Kernel estimator in nonparametric and semiparametric regression lately also receive a lot of attention from researchers.…”

Combined estimator Fourier series and spline truncated in multivariable nonparametric regression

“…Moreover, the advantage of spline is able to describe the change of the function pattern in the sub-specified interval and can handle well the data pattern which is dramatically change by using knots [5]. Some recently researches about the application of spline in nonparametric regression could be found in Lestari, Budiantara, Sunaryo and Mashuri [6], Wibowo, Haryatmi and Budiantara [7], and Fernandes, Budiantara, Otok and Suhartono [8].…”