# Tests for qualitative features in the random coefficients model

## TL;DR

This paper develops statistical tests to detect qualitative features like modes and density increases in the joint distribution of random coefficients in a linear model, addressing the challenges of ill-posedness and high dimensionality.

## Contribution

It introduces a multiple testing framework for identifying shape features of the joint density in random coefficients models, using Gaussian approximation techniques.

## Key findings

- Effective detection of modes and density increases in simulated data
- Method applied successfully to real data examples
- Provides confidence statements for qualitative features

## Abstract

The random coefficients model is an extension of the linear regression model that allows for unobserved heterogeneity in the population by modeling the regression coefficients as random variables. Given data from this model, the statistical challenge is to recover information about the joint density of the random coefficients which is a multivariate and ill-posed problem. Because of the curse of dimensionality and the ill-posedness, pointwise nonparametric estimation of the joint density is difficult and suffers from slow convergence rates. Larger features, such as an increase of the density along some direction or a well-accentuated mode can, however, be much easier detected from data by means of statistical tests. In this article, we follow this strategy and construct tests and confidence statements for qualitative features of the joint density, such as increases, decreases and modes. We propose a multiple testing approach based on aggregating single tests which are designed to extract shape information on fixed scales and directions. Using recent tools for Gaussian approximations of multivariate empirical processes, we derive expressions for the critical value. We apply our method to simulated and real data.