The functional kNN estimator of the conditional expectile: Uniform consistency in number of neighbors

The main purpose of the present paper is to investigate the problem of the nonparametric estimation of the expectile regression in which the response variable is scalar while the covariate is a random function. More precisely, an estimator is constructed by using the k Nearest Neighbor procedures (kNN). The main contribution of this study is the establishment of the Uniform consistency in Number of Neighbors (UNN) of the constructed estimator. The usefulness of our result for the smoothing parameter automatic selection is discussed. Short simulation results show that the finite sample performance of the proposed estimator is satisfactory in moderate sample sizes. We finally examine the implementation of this model in practice with a real data in financial risk analysis.

A Discrete Density Approach to Bayesian Quantile and Expectile Regression with Discrete Responses

For decades, regression models beyond the mean for continuous responses have attracted great attention in the literature. These models typically include quantile regression and expectile regression. But there is little research on these regression models for discrete responses, particularly from a Bayesian perspective. By forming the likelihood function based on suitable discrete probability mass functions, this paper introduces a discrete density approach for Bayesian inference of these regression models with discrete responses. Bayesian quantile regression for discrete responses is first developed, and then this method is extended to Bayesian expectile regression for discrete responses. The posterior distribution under this approach is shown not only coherent irrespective of the true distribution of the response, but also proper with regarding to improper priors for the unknown model parameters. The performance of the method is evaluated via extensive Monte Carlo simulation studies and one real data analysis.