Minimax Risk in Estimating Kink Threshold and Testing Continuity
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We 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, we 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.
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