Statistics and Its Interface

Volume 16 (2023)

Number 1

Special issue on recent developments in complex time series analysis – Part I

Guest editors: Robert T. Krafty (Emory Univ.), Guodong Li (Univ. of Hong Kong), Anatoly Zhigljavsky (Cardiff Univ.)

Testing threshold effect in single-index models

Pages: 43 – 56



Zhaoxing Gao (Center for Data Science, Zhejiang University, China)

Zichuan Mi (Shanxi University of Finance and Economics, China)

Shiqing Ling (Hong Kong University of Science and Technology)


This paper studies the supremum-type score test for the single-index model against a threshold single-index model. It is shown that the test weakly converges a maxima of a Gaussian process under the null hypothesis. The bootstrap method is used to tackle the bias problem and provide the $p$-values of our test statistic. Simulations are carried out to assess the performance of our procedure and real data examples are given for its illustration.


single-index model, threshold, local linear smoother, bootstrap, bias correction

The authors’ work was partially supported by by Hong Kong Research Grants Commission Grants (16500117, 16303118, 16301620 and 16300621) and Australian Research Council and by National Natural Science Foundation of China (11731015, 11571148) (Shiqing Ling).

Received 13 March 2021

Accepted 2 August 2022

Published 28 December 2022