Electronic Journal of Statistics

Tests for qualitative features in the random coefficients model

Fabian Dunker, Konstantin Eckle, Katharina Proksch, and Johannes Schmidt-Hieber

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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, 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.

Article information

Electron. J. Statist., Volume 13, Number 2 (2019), 2257-2306.

Received: July 2018
First available in Project Euclid: 3 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G15: Tolerance and confidence regions
Secondary: 62G20: Asymptotic properties

Gaussian approximation mode detection monotonicity multiscale statistics shape constraints Radon transform ill-posed problems

Creative Commons Attribution 4.0 International License.


Dunker, Fabian; Eckle, Konstantin; Proksch, Katharina; Schmidt-Hieber, Johannes. Tests for qualitative features in the random coefficients model. Electron. J. Statist. 13 (2019), no. 2, 2257--2306. doi:10.1214/19-EJS1570. https://projecteuclid.org/euclid.ejs/1562140822

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