Open Access
August 2019 Adaptive-to-model checking for regressions with diverging number of predictors
Falong Tan, Lixing Zhu
Ann. Statist. 47(4): 1960-1994 (August 2019). DOI: 10.1214/18-AOS1735


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.


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Falong Tan. Lixing Zhu. "Adaptive-to-model checking for regressions with diverging number of predictors." Ann. Statist. 47 (4) 1960 - 1994, August 2019.


Received: 1 January 2018; Revised: 1 May 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082276
MathSciNet: MR3953441
Digital Object Identifier: 10.1214/18-AOS1735

Primary: 62G10
Secondary: 62M07

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

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 4 • August 2019
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