# Estimation and adaptive-to-model testing for regressions with diverging number of predictors

## TL;DR

This paper develops a new test for parametric single-index models with a diverging number of predictors, analyzing estimator properties and constructing an adaptive test statistic suitable for high-dimensional settings.

## Contribution

It introduces an adaptive-to-model residual empirical process and a martingale transformation for model checking in high-dimensional regressions with diverging predictors.

## Key findings

- The test maintains good size and power in simulations.
- Asymptotic properties are established under null and alternative hypotheses.
- Estimator properties are characterized for diverging dimensions.

## Abstract

The research described in this paper is motivated by model checking for parametric single-index models with diverging number of predictors. To construct a test statistic, we first study the asymptotic property of the estimators of involved parameters of interest under the null and alternative hypothesis when the dimension is divergent to infinity as the sample size goes to infinity. For the testing problem, we study an adaptive-to-model residual-marked empirical process as the basis for constructing a test statistic. By modifying the approach in the literature to suit the diverging dimension settings, we construct a martingale transformation. Under the null, local and global alternative hypothesis, the weak limits of the empirical process are derived and then the asymptotic properties of the test statistic are investigated. Simulation studies are carried out to examine the performance of the test.