Tests for qualitative features in the random coefficients model

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.

Nonparametric comparison of epidemic time trends: the case of COVID-19

“…A general approach to construct scale-dependent critical values was pioneered by Dümbgen and Spokoiny (2001) and has been used in many other studies since then; cp. for example Rohde (2008), Dümbgen and Walther (2008), Rufibach andWalther (2010), Schmidt-Hieber et al (2013), Eckle et al (2017) and Dunker et al (2019). In our context, the approach of Dümbgen and Spokoiny (2001) leads to the critical values…”

Nonparametric comparison of epidemic time trends: The case of COVID-19

“…A general approach to construct scale-dependent critical values was pioneered by Dümbgen and Spokoiny (2001) and has been used in many other studies since then; see e.g. Rohde, 2008 , Dümbgen and Walther, 2008 , Rufibach and Walther, 2010 , Schmidt-Hieber et al, 2013 , Eckle et al, 2017 and Dunker et al (2019) . In our context, the approach of Dümbgen and Spokoiny (2001) leads to the critical values where and are scale-dependent constants and the quantity is determined by the following consideration: Since we need to choose the quantity as the -quantile of the statistic in order to ensure control of the FWER at level .…”

Nonparametric Estimation of the Random Coefficients Model in Python

“…0 7 5 ] ] p l o t r m l e ( r e s u l t ) p l o t r m l e ( r e s u l t s h i f t , p l t t y p e= ' s u r f a c e ' ) The final example demonstrates how to use the module on a real dataset, and shows the user how to load the data into Python from a comma-separated value (CSV) format and how to pre-process it, if necessary, into a usable form. The data used here are from the British Family Expenditure Surey that was used in Dunker et al (2019Dunker et al ( , 2021.…”