Bi-responses nonparametric regression model using MARS and its properties

In many cases of regression analysis, we can find unknown relationship pattern between two response variables with many predictor variables (multi-predictor) and both of responses are correlated each other. Consequently such regression problem should be solved using bi-responses nonparametric regression model, which able to overcome high dimensionality case in multi-predictor data. One of the methods is Multivariate Adaptive Regression Spline (MARS). This paper aims to study how MARS method could be utilized on bi-responses nonparametric regression model. The study begins with obtaining model equation form. Thus we estimating regression function and weighted from its model equation. Last step is done by investigating the estimator properties of its model. Results show that the equation form of its model is done by investigating the estimator properties of its model. Results show that the equation form of its model is y = Bα +ε, with regression function and weighted of its equation estimation are f (x) = B (B^TW^-1B)^-1 and W matrix respectively, where the diagonal elements of W as (y^T₁ A₁₁ y₁/y^T₂ A₂₂y₂)^1/2 and (y^T₂ A₂₂ y₂/y^T₁ A₁₁y₁)^1/2 whereas the off-diagonal elements of W matrix as y^T₁A₁₂y₂/(y^T₁A₁₁y₁y^T₂A₂₂y₂)^1/2 and y^T₂A₂₁y₁/(y^T₂A₂₂y₂y^T₁A₁₁y₁)^1/2. The results also show that the properties of this model is a bias estimator and linear in observation y.