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

M M

doi:10.3844/jmssp.2012.377.384

Abstract

Abstract: Problem statement: Assume that data (yki, tki), k = 1,2,..., p; i = 1,2,...,nk where nk represents the number of repeated measurement of k 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.

Highlights

Smoothing spline can be used to estimate functions which represent association of two or more dependent variables are observed at several values of the independent variables, such as at multiple time points
We assume that data, k = 1,2,..., p; i = 1,2,...,nk where nk represents the number of repeated measurement of kth object follows multi-response nonparametric regression model Eq 1: yki = fk + εki Regression curves form ƒi, ƒ2,..., ƒp are unknown and assumed to be smooth which are contained in Sobolev space W2m[ak, bk ]
Estimating of nonparametric regression curve is the main problem in heteroscedastic multi-response nonparametric regression

Summary

INTRODUCTION

Smoothing spline can be used to estimate functions which represent association of two or more dependent variables are observed at several values of the independent variables, such as at multiple time points. Aydin (2007) showed goodness of spline estimator rather than kernel estimator in estimating nonparametric regression model for gross national product data. There are many researchers who have considered spline estimator for estimating regression curve of nonparametric regression model. All these researchers studied spline estimators in case of single response nonparametric regression models only. Math. & Stat., 8 (3): 377-384, 2012 response data. Wegman (1981); Miller and Wegman

RESULTS

MATERIALS AND METHODS

DISCUSSION

CONCLUSION