TY - JOUR
T1 - Adaptive-to-model checking for regressions with diverging number of predictors
AU - Tan, Falong
AU - Zhu, Lixing
N1 - Funding Information:
Supported by the Fundamental Research Funds for the Central Universities, a grant from the University Grants Council of Hong Kong and National Natural Science Foundation of China Grant #NSFC11671042.
Publisher copyright:
© Institute of Mathematical Statistics, 2019
PY - 2019/8
Y1 - 2019/8
N2 - 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(n1/3/ 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.
AB - 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(n1/3/ 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.
KW - Adaptive-to-model test
KW - Diverging number of predictors
KW - Empirical process
KW - Martingale transformation
KW - Parametric single-index models
KW - Sufficient dimension reduction
UR - http://www.scopus.com/inward/record.url?scp=85072298949&partnerID=8YFLogxK
U2 - 10.1214/18-AOS1735
DO - 10.1214/18-AOS1735
M3 - Journal article
AN - SCOPUS:85072298949
SN - 0090-5364
VL - 47
SP - 1960
EP - 1994
JO - Annals of Statistics
JF - Annals of Statistics
IS - 4
ER -