Modal Regression Models Based On B-splines

Mean regression models use the conditional mean as the center of the conditional distributions,which perform the best estimates while the noises are Gaussian and are widely applied in economies,biologies,humanities and social sciences.Quantile regression models use the conditional quantiles to describe the characteristics of the conditional distributions,particularly,which let the medians as the centers of the conditional distributions.Compared to mean regression models,quantile regression models can describe the conditional distributions more completely and hold better robustness.However,if the noises are multimodal,skewed or heavy-tailed,these two models neither capture the centers of the conditional distributions well,the modal regression models are the more reasonable choice.The parametric modal regression models have been well studied,but the nonparametric ones are not yet.The given local polynomial modal regression models perform well on estimates,but which are pointwise and need large amount of calculating.This paper presents the nonparametric modal regression models based on B-splines.The solutions of models are given via MEM algorithms,the consistency and the asymptotic normality properties of estimations are studied,and the simulations are used to illustrate that which performs as well as local polynomial modal regression models on estimates but with less computational complexities.Furthermore,a novel cross validation rule for selecting hyper-parameters is proposed,which is based on the prediction intervals and outperforms the traditional ones in simulations and applications.In the face of complex data,the mode regression model based on B-spline proposed in this paper is one of the important supplements of robust regression analysis tools.