Estimation of a Multivariate Non-parametric Regression Function of a Non-normal Distributions by Using Kernel Smoothing

Abstract

Distributions with heavy tails are regarded as one of the most significant mixed probability distributions that have wide applications in various fields of life, particularly in fields related to economics, which is concerned with the issue of environmental pollution due to its economic effects. Therefore, the multivariate nonparametric regression function was estimated in the research when it followed the error limit. The model has random distributions with heavy tails, which are represented by the distribution of the generalized hyperbolic matrix and the generalized basil matrix (generalized hyperbolic symmetric matrix). The multivariate nonparametric regression model was converted to a linear model based on the local polynomial smoother, and through the traditional method, the multivariate Nadaraya-Watson smoother and the multivariate local linear smoother were obtained, presuming that the shape and torsion parameters are known. We concluded that the Nadaraya-Watson smoother and the local linear smoother for (k) of the response variables when the error term follows the distribution of a generalized Bessel matrix are the same as the smoothers when the random error term of the model follows the distribution of the normal matrix. In addition to the application on actual data, which is represented by environmental pollution and air pollution data for the year (2018), and through the results of the application, we also observe the superiority of the multivariate Nadaraya-Watson paver, and when the error distribution of the model is a generalized hyperbolic matrix distribution based on the MSE criterion.