# Likelihood ratio tests for many groups in high dimensions

## TL;DR

This paper studies the asymptotic behavior of likelihood ratio tests in high-dimensional models with many groups, deriving central limit theorems for test statistics as both the number of groups and data dimensions grow large.

## Contribution

It introduces new asymptotic results for likelihood ratio tests in high-dimensional, multi-group settings, extending classical theory to many groups.

## Key findings

- Derived CLTs for test statistics in high dimensions
- Compared asymptotic distributions with two-step approximation
- Provided insights into the behavior of likelihood ratio tests with many groups

## Abstract

In this paper we investigate the asymptotic distribution of likelihood ratio tests in models with several groups, when the number of groups converges with the dimension and sample size to infinity. We derive central limit theorems for the logarithm of various test statistics and compare our results with the approximations obtained from a central limit theorem using a two step procedure: first consider the number of groups fixed and assume that the sample size and dimension converge to infinity, secondly investigating the resulting distribution if the number of groups converges to infinity.