A Confidence Corridor for Expectile Functions

Abstract

Let (X1; Y1), …, (Xn; Yn) be i.i.d. rvs and denote v(x) be the unknown τ - expectile regression curve of Y conditional on X. An expectile-smoother vn(x) is a localized, nonlinear estimator of v(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {vn(x) – v(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |vn(x) – v(x)|. The derived result helps in the construction of a uniform confidence band for the expectile curve v(x). This paper considers fitting a simultaneous confidence corridor (SCC) around the estimated expectile function of the conditional distribution of Y given x based on the observational data generated according to a nonparametric regression model. Moreover, we construct the simultaneous confidence corridors around the expectiles of the residuals from the temperature models to investigate the temperature risk drivers.