Data-driven goodness-of-fit tests

TL;DR

This paper introduces a versatile, data-driven framework for constructing consistent statistical goodness-of-fit tests that incorporate model selection and penalization, applicable to various complex testing scenarios.

Contribution

It develops a general method unifying multiple test types with model selection, providing theoretical guarantees and broad applicability to inverse, multi-sample, and nonparametric testing.

Findings

The tests are consistent under null and alternative hypotheses.

Explicit detectability rules for alternative hypotheses are derived.

The method applies to inverse problems, multi-sample, and nonparametric tests.

Abstract

We propose and study a general method for construction of consistent statistical tests on the basis of possibly indirect, corrupted, or partially available observations. The class of tests devised in the paper contains Neyman's smooth tests, data-driven score tests, and some types of multi-sample tests as basic examples. Our tests are data-driven and are additionally incorporated with model selection rules. The method allows to use a wide class of model selection rules that are based on the penalization idea. In particular, many of the optimal penalties, derived in statistical literature, can be used in our tests. We establish the behavior of model selection rules and data-driven tests under both the null hypothesis and the alternative hypothesis, derive an explicit detectability rule for alternative hypotheses, and prove a master consistency theorem for the tests from the class. The paper shows that the tests are applicable to a wide range of problems, including hypothesis testing in statistical inverse problems, multi-sample problems, and nonparametric hypothesis testing.