Rejoinder: Nonparametric inference with generalized likelihood ratio tests

We are very grateful to the Editors, Maria Angeles Gil and Leandro Pardo, for organizing this stimulating discussion. We would like to take this opportunity to thank all discussants for their insightful and constructive comments regarding our paper, opening new avenues for the GLR tests. They have made valuable contributions to the understanding of various testing problems. As stressed in our paper, we reviewed nonparametric inference using the GLR tests, laid down some interesting topics, and stressed the importance of the structured alternatives to maintain reasonable power, but we touched only the surface of this exciting field. We are very happy that the discussants responded part of our topics, offered some complementary views and new insights, and raised some interesting problems for further study. The evolving statistical theory and practice in science and technology leads us to a vast discipline with many challenging statistical problems. The GLR test methods have been developed only for limited models based on limited smoothing platforms. We appreciate all efforts of discussnats that widen the scope of our paper. Before going to technical aspects, let us response to the fundamental question raised by Professors Lafferty and Wasserman: Does hypothesis testing answer the right question? This depends certainly on the context. In the statistical learning or empirical model building that Professors Lafferty and Wasserman alluded to, we are sympathetic with their views. However, in many scientific endeavors, we have hypotheses in mind and we do need to answer the question if the model is supported by the data. For example, in the examples mentioned in section 1.1,