The Annals of Statistics

Adaptive-to-model checking for regressions with diverging number of predictors

Falong Tan and Lixing Zhu

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In this paper, we construct an adaptive-to-model residual-marked empirical process as the base of constructing a goodness-of-fit test for parametric single-index models with diverging number of predictors. To study the relevant asymptotic properties, we first investigate, under the null and alternative hypothesis, the estimation consistency and asymptotically linear representation of the nonlinear least squares estimator for the parameters of interest and then the convergence of the empirical process to a Gaussian process. We prove that under the null hypothesis the convergence of the process holds when the number of predictors diverges to infinity at a certain rate that can be of order, in some cases, $o(n^{1/3}/\log n)$ where $n$ is the sample size. The convergence is also studied under the local and global alternative hypothesis. These results are readily applied to other model checking problems. Further, by modifying the approach in the literature to suit the diverging dimension settings, we construct a martingale transformation and then the asymptotic properties of the test statistic are investigated. Numerical studies are conducted to examine the performance of the test.

Article information

Ann. Statist., Volume 47, Number 4 (2019), 1960-1994.

Received: January 2018
Revised: May 2018
First available in Project Euclid: 21 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing
Secondary: 62M07: Non-Markovian processes: hypothesis testing

Adaptive-to-model test diverging number of predictors empirical process martingale transformation parametric single-index models sufficient dimension reduction


Tan, Falong; Zhu, Lixing. Adaptive-to-model checking for regressions with diverging number of predictors. Ann. Statist. 47 (2019), no. 4, 1960--1994. doi:10.1214/18-AOS1735.

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Supplemental materials

  • Supplementary Material to “Adaptive-to-model checking for regressions with diverging number of predictors.”. This Supplementary Material contains three parts with the regularity conditions, technical lemmas and proofs of the main results.