# Adaptive-to-model hybrid of tests for regressions

## TL;DR

This paper introduces an adaptive hybrid testing method for regressions that combines the strengths of nonparametric and empirical process-based tests, improving sensitivity and distribution tractability.

## Contribution

It proposes a novel adaptive-to-model hybrid test that inherits the advantages of both test types and enhances nonparametric tests with empirical process properties.

## Key findings

- The hybrid test effectively detects various alternative models.
- Numerical studies confirm the test's superior power.
- The method is adaptable to different data types.

## Abstract

In model checking for regressions, nonparametric estimation-based tests usually have tractable limiting null distributions and are sensitive to oscillating alternative models, but suffer from the curse of dimensionality. In contrast, empirical process-based tests can, at the fastest possible rate, detect local alternatives distinct from the null model, but is less sensitive to oscillating alternative models and with intractable limiting null distributions. It has long been an issue on how to construct a test that can fully inherit the merits of these two types of tests and avoid the shortcomings. We in this paper propose a generic adaptive-to-model hybrid of moment and conditional moment-based test to achieve this goal. Further, a significant feature of the method is to make nonparametric estimation-based tests, under the alternatives, also share the merits of existing empirical process-based tests. This methodology can be readily applied to other kinds of data and constructing other hybrids. As a by-product in sufficient dimension reduction field, the estimation of residual-related central subspace is used to indicate the underlying models for model adaptation. A systematic study is devoted to showing when alternative models can be indicated and when cannot. This estimation is of its own interest and can be applied to the problems with other kinds of data. Numerical studies are conducted to verify the powerfulness of the proposed test.