Asymmetric cusp estimation in regression models

Abstract

We consider the problem of estimating the location of an asymmetric cusp in a regression model. That means, we focus on regression functions, which are continuous at , but the degree of smoothness from the left could be different to the degree of smoothness from the right . The degrees of smoothness have to be estimated as well. We investigate the consistency with increasing sample size n of the least-squares estimates. It turns out that the rates of convergence of depend on the minimum b of and and that our estimator converges to a maximizer of a Gaussian process. In the regular case, that is, for b greater than , we have a rate of and the asymptotic normality property. In the non-regular case, we have a representation of the limit distribution of as maximizer of a fractional Brownian motion with drift.