Asymmetric cusp estimation in regression models

Abstract

We consider the problem of estimating the location of an asymmetric cusp ${\theta }_0$ in a regression model. That means, we focus on regression functions, which are continuous at ${\theta }_0$, but the degree of smoothness from the left $p_0$ could be different to the degree of smoothness from the right $q_0$. 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 ${\hat{\theta }}_n$ depend on the minimum b of $p_0$ and $q_0$ and that our estimator converges to a maximizer of a Gaussian process. In the regular case, that is, for b greater than $\frac{1}{2}$, we have a rate of $\sqrt{n}$ and the asymptotic normality property. In the non-regular case, we have a representation of the limit distribution of ${\hat{\theta }}_n$ as maximizer of a fractional Brownian motion with drift.

Keywords:

• cusp
• change point
• nonlinear regression
• m-estimates
• asymptotic

AMS Subject Classification:

• 62J02

Acknowledgements

The author would like to thank the Editor, the Associate Editor and the Referees for their careful reading and comments. These comments and suggestions have been very helpful for revising and improving the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.