Reproducing Kernel Hilbert Space Approach to Multiresponse Smoothing Spline Regression Function

Abstract

In statistical analyses, especially those using a multiresponse regression model approach, a mathematical model that describes a functional relationship between more than one response variables and one or more predictor variables is often involved. The relationship between these variables is expressed by a regression function. In the multiresponse nonparametric regression (MNR) model that is part of the multiresponse regression model, estimating the regression function becomes the main problem, as there is a correlation between the responses such that it is necessary to include a symmetric weight matrix into a penalized weighted least square (PWLS) optimization during the estimation process. This is, of course, very complicated mathematically. In this study, to estimate the regression function of the MNR model, we developed a PWLS optimization method for the MNR model proposed by a previous researcher, and used a reproducing kernel Hilbert space (RKHS) approach based on a smoothing spline to obtain the solution to the developed PWLS optimization. Additionally, we determined the symmetric weight matrix and optimal smoothing parameter, and investigated the consistency of the regression function estimator. We provide an illustration of the effects of the smoothing parameters for the estimation results using simulation data. In the future, the theory generated from this study can be developed within the scope of statistical inference, especially for the purpose of testing hypotheses involving multiresponse nonparametric regression models and multiresponse semiparametric regression models, and can be used to estimate the nonparametric component of a multiresponse semiparametric regression model used to model Indonesian toddlers' standard growth charts.

Estimation of Multiresponse Multipredictor Nonparametric Regression Model Using Mixed Estimator

“…Since L x ∈ H is a bounded linear functional, then according to Yuan and Cai [45] and Lestari et al [35], there exists a representer ξ i ∈ H that meets the following equation:…”

“…where W is a symmetrical weight matrix that is the inverse of a covariance matrix of random errors. Details of the weight matrix can be found in Lestari et al [18,35]. In addition, the PWLS optimization function presented in Equation (31) shows that the goodness of fit component of PWLS optimization is n…”

“…and W is a special symmetric weight matrix, namely, a diagonal matrix, which is an inverse of a covariance matrix of random errors. In this case, based on the assumption of random errors of the MMNR model, the weight matrix W is given as follows [18,35]:…”

“…To investigate the consistency of the regression function estimator of the MMNR model, m(λ,K) (x, t), firstly, we should investigate asymptotic properties of the mixed smoothing spline and Fourier series estimator, m(λ,K) (x, t), based on an integrated mean squared error (IMSE) criterion. In this study, we develop the IMSE from the uniresponse with one predictor case proposed by Wand and Jones [48] and multiresponse with one predictor case proposed by Lestari et al [35] into a multiresponse with multipredictor case. Next, by considering Equation (48), the IMSE can be decomposed into two components, namely, the Biased 2 (δ) component and Var(δ) component, as follows:…”

“…Currently, many researchers are paying attention to developing and applying the smoothing spline estimator in many areas of research. For example, Chamidah et al [17], Lestari et al [18] discussed smoothing splines in MSR models; Lestari et al [35] and Wang et al [28] discussed RKHS in an MNR model and applied an MNR model to determine associations between hormones, respectively. Because of its powerful and flexible properties, the smoothing spline estimator is one of the most popular estimators used for estimating the regression functions of UNR and MNR models.…”