Spline Estimator in Multi-Response Nonparametric Regression Model with Unequal Correlation of Errors

Lestari Lestari

doi:10.3844/jmssp.2010.327.332

Abstract

Problem statement: In many applications two or more dependent variables are observed at several values of the independent variables, such as at time points. The statistical problems are to estimate functions that model their dependences on the independent variables and to investigate relationships between these functions. Nonparametric regression model, especially smoothing splines provide powerful tools to model the functions which draw association of these variables. Approach: Penalized weighted least-squares was used to jointly estimate nonparametric functions from contemporaneously correlated data. We apply Generalized Maximum Likelihood (GML), Generalized Cross Validation (GCV) and leaving-out-one-pair Cross Validation (CV) for estimating the smoothing parameters, the weighting parameters and the correlation parameter Results: In this study we formulated the multi-response nonparametric regression model with unequal correlation of errors and give a theoretical method for both obtaining distribution of the response and estimating the nonparametric function in the model. We also estimate the smoothing parameters, the weighting parameters and the correlation parameter simultaneously by applying three methods GML, GCV and CV. Conclusion: Distribution of responses is normal. With multiple correlated responses it is better to estimate these functions jointly using the penalized weighted least-squares.

Highlights

The functions which draw association of two or more dependent variables are observed at several values of the independent variables, such as at multiple time points, can be modeled by using smoothing spline
We extend method as in Wang (1998) to multi-response nonparametric regression model with unequal correlation of errors
In this study we propose the following three methods to estimate the smoothing parameters λk, the weighting parameters rk and γij and the correlation parameter ρi simultaneously, i.e., an extension of the Generalized Maximum Likelihood (GML) method based on a Bayesian model; an extension of the Generalized Cross Validation (GCV) method and leavingout-one-pair cross validation

Summary

Introduction

The functions which draw association of two or more dependent variables are observed at several values of the independent variables, such as at multiple time points, can be modeled by using smoothing spline. Budiantara et al (1997) studied weighted spline estimator in nonparametric regression model with different variance. All these writers studied spline estimators in case of single response nonparametric models only.

Results

Conclusion