Estimation and hypothesis test for single-index multiplicative models

• Jun Zhang ^[1] ; Junpeng Zhu ^[1] ; Zhenghui Feng ^[2]
  1. [1] Shenzhen University

     Shenzhen University

     China

     
  2. [2] Xiamen University

     Xiamen University

     China

     
• Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 28, Nº. 1, 2019, págs. 242-268
• Idioma: inglés
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• Resumen

  • Estimation and hypothesis tests for 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. For the hypothesis testing of parametric components, a Wald-type test statistic is proposed. The asymptotic properties of the estimators and test statistics are established, and a smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. A score-type test statistic is then proposed for checking the validity of single-index multiplicative models. The quadratic form of the scaled test statistic has an asymptotic chi-squared distribution under the null hypothesis and follows a noncentral chi-squared distribution under local alternatives, converging to the null hypothesis at a parametric convergence rate. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.