Estimation and hypothesis test for partial linear single-index multiplicative models

  • Jun ZhangEmail author
  • Xia Cui
  • Heng Peng


Estimation and hypothesis test for partial linear single-index multiplicative models are considered in this paper. To estimate unknown single-index parameter, we propose a profile least product relative error estimator coupled with a leave-one-component-out method. To test a hypothesis on the parametric components, a Wald-type test statistic is proposed. We employ the smoothly clipped absolute deviation penalty to select relevant variables. To study model checking problem, we propose a variant of the integrated conditional moment test statistic by using linear projection weighting function, and we also suggest a bootstrap procedure for calculating critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for illustration.


Local linear smoothing Model checking Profile least product relative error estimator Single-index Variable selection 



The authors thank the editor, the associate editor and two referees for their constructive suggestions that helped us to improve the early manuscript. Xia Cui is a College Talent Cultivated by Thousand-Hundred-Ten Program of Guangdong Province, and her research was supported by grants from National Natural Science Foundation of China (NSFC) (No. 11471086), Humans and Social Science Research Team of Guangzhou University (No. 201503XSTD) and the Training Program for Excellent Young College Teachers of Guangdong Province (No. Yq201404). Heng Peng’s research was supported in part by CEGR grant of the Research Grants Council of Hong Kong (Nos. HKBU202012 and HKBU 12302615), FRG grants from Hong Kong Baptist University (Nos. FRG2 14-15/064 and FRG2 /16-17/042).

Supplementary material

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Supplementary material 1 (pdf 150 KB)


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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2019

Authors and Affiliations

  1. 1.College of Mathematics and Statistics, Shenzhen-Hong Kong Joint Research Center for Applied Statistical Sciences, Institute of Statistical SciencesShenzhen UniversityShenzhenChina
  2. 2.School of Economics and StatisticsGuangzhou UniversityGuangzhouChina
  3. 3.Department of MathematicsThe Hong Kong Baptist UniversityKowloon TongChina

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