An adaptive-to-model test for partially parametric single-index models

TL;DR

This paper introduces an adaptive-to-model testing procedure for partially parametric single-index models that effectively addresses high-dimensional data challenges by combining dimension reduction, eigenvalue ratio estimation, and Monte-Carlo null distribution approximation.

Contribution

It proposes a novel adaptive testing method that is robust against general alternatives and automatically determines the number of covariate combinations, overcoming high-dimensional limitations.

Findings

The new test performs well in simulations compared to existing methods.

It effectively detects a wide range of alternative models.

Real data analysis confirms its practical utility.

Abstract

Residual marked empirical process-based tests are commonly used in regression models. However, they suffer from data sparseness in high-dimensional space when there are many covariates. This paper has three purposes. First, we suggest a partial dimension reduction adaptive-to-model testing procedure that can be omnibus against general global alternative models although it fully use the dimension reduction structure under the null hypothesis. This feature is because that the procedure can automatically adapt to the null and alternative models, and thus greatly overcomes the dimensionality problem. Second, to achieve the above goal, we propose a ridge-type eigenvalue ratio estimate to automatically determine the number of linear combinations of the covariates under the null and alternatives. Third, a Monte-Carlo approximation to the sampling null distribution is suggested. Unlike existing bootstrap approximation methods, this gives an approximation as close to the sampling null distribution as possible by fully utilising the dimension reduction model structure under the null. Simulation studies and real data analysis are then conducted to illustrate the performance of the new test and compare it with existing tests.