Estimation in quantile regression models with jump discontinuities

In quantile regression, the quantile function is often discontinuous so that it is needful to detect the jump points. In the existing literature, it is generally assumed the jump number to be known. In most applications, however, it is not uncommon that the jump number is often unknown so that the existing methods of detecting jump discontinuities are no longer applicable. In this paper, we first propose a new procedure of detecting jump by using the local polynomial techniques for the case of the unknown jump number. We then propose the estimators of jump points, jump sizes and jump number, and study their asymptotic properties. Finally, we conduct simulation studies to assess the finite sample performance of our proposed estimators and present their biases and standard errors to demonstrate that our proposed method performs well in a wide range of practical settings.