Abstract
A multiresponse multipredictor semiparametric regression (MMSR) model is a combination of parametric and nonparametric regressions models with more than one predictor and response variables where there is correlation between responses. Due to this correlation we need to construct a symmetric weight matrix. This is one of the things that distinguishes it from the classical method, which uses a parametric regression approach. In this study, we theoretically developed a method of determining a confidence interval for parameters in a MMSR model based on a truncated spline, and investigating asymptotic properties of estimator for parameters in a MMSR model, especially consistency and asymptotic normality. The weighted least squares method was used to estimate the MMSR model. Next, we applied a pivotal quantity method, a Cramer–Wold theorem, and a Slutsky theorem to determine the confidence interval, investigate consistency, and asymptotic normality properties of estimator for parameters in a MMSR model. The obtained results were that the estimated regression function is linear to observation. We also obtained a 1001−α% confidence interval for parameters in the MMSR model, and the estimator for parameters in MMSR model was consistent and asymptotically normally distributed. In the future, these obtained results can be used as a theoretical basis in designing a standard toddlers growth chart to assess nutritional status.
Highlights
A regression model which is used to analyze the functional relationship between response variable and predictor variable in various fields is widely used for both prediction and interpretation purposes
The multiresponse multipredictor semiparametric regression (MMSR) model presented by Equation (1) can be written as follows: yki = hk xk1i, xk2i, . . . , xkqi, tk1i, tk2i, . . . , tkri + εki where hk xk1i, xk2i, . . . , xkqi, tk1i, tk2i, . . . , tkri = fk xk1i, xk2i, . . . , xkqi + gk(tk1i, tk2i . . . , tkri) is an unknown regression function of the MMSR model presented by Equation (1)
To estimate the regression function of the MMSR model presented by Equations (1) and (2) based on truncated spline estimator, we need to develop the truncated spline proposed by [12]
Summary
A regression model which is used to analyze the functional relationship between response variable and predictor variable in various fields is widely used for both prediction and interpretation purposes. The estimators only consider the goodness of fit and do not consider smoothness These estimators are not good to use for estimating models of fluctuating data in the sub intervals, because the estimation results will provide a large mean square error (MSE) value. This is different from the spline estimator which considers goodness of fit and smoothness factors as has been discussed by several researchers. This means that those researchers mentioned above discussed estimators in NR models only
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