Minimax risk in estimating kink threshold and testing continuity

The authors derive a risk lower bound in estimating the threshold parameter without knowing whether the threshold regression model is continuous or not. The bound goes to zero as the sample size n grows only at the cube-root rate. Motivated by this finding, the authors develop a continuity test for the threshold regression model and a bootstrap to compute its p -values. The validity of the bootstrap is established, and its finite-sample property is explored through Monte Carlo simulations.