Orthogonal series method for uncertain nonparametric regression with application to carbon dioxide emissions

Parametric regression is an important branch of statistics that assumes the regression model is known with a mathematical expression, while nonparametric regression has few restrictions on the regression model that is determined by the data. As a tool to explore the relationship between the observations characterized by uncertain variables, uncertain nonparametric regression has not obtained enough attention except for B-spline and local polynomial regression. In order to get a more global and smooth estimate, this paper employs the orthogonal series to approximate the nonparametric regression model. We intend to choose an appropriate number of the orthogonal bases by the leave-one-point-out cross-validation. Then, a new prediction technology is proposed to derive the response variable's forecast value and confidence interval. Finally, a numerical example and a real data example of carbon dioxide emissions are put forward to demonstrate the effectiveness and accuracy of the method.