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In nonparametric regression with jump discontinuities, one major challenge is to determine the number of jumps in a regression curve. Most existing methods to solve that problem are based on either a sequence of hypothesis tests or model selection, by introducing some extra tuning parameters that may not be easy to determine in practice. This article aims to develop a data-driven new methodology for determining the number of jumps, using an order-preserved sample-splitting strategy together with a cross-validation-based criterion. Statistical consistency of the determined number of jumps by our proposed method is established. More interestingly, the proposed method allows us to move beyond just point estimation, and it can quantify uncertainty of the proposed estimate. The key idea behind our method is the construction of a series of statistics with marginal symmetry property and this property can be used for choosing a data-driven threshold to control the false discovery rate of our method. The proposed method is computationally efficient. Numerical experiments indicate that it has a reliable performance in finite-sample cases. An R packageis developed to implement the proposed method.

Data-Driven Determination of the Number of Jumps in Regression Curves