# A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU

A parametric test for bimodality based on the likelihood principle by using two-component mixtures. For the given two normal distributions, the test checks if their mixture is bimodal or unimodal.

## Abstract

We propose a parametric test for bimodality based on the likelihood principle by using two-component mixtures. The test uses explicit characterizations of the modal structure of such mixtures in terms of their parameters. Examples include the univariate and multivariate normal distributions and the von Mises distribution. We present the asymptotic distribution of the proposed test and analyze its finite sample performance in a simulation study. To illustrate our method, we use mixtures to investigate the modal structure of the cross-sectional distribution of per capita log GDP across EU regions from 1977 to 1993. Although these mixtures clearly have two components over the whole time period, the resulting distributions evolve from bimodality toward unimodality at the end of the 1970s.

## Reference

Hajo Holzmann, Sebastian Vollmer

“A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU”(2008) DOI: 10.1007/s10182-008-0057-2

```
@Article{holzmann2008,
title = {A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU},
abstract = {We propose a parametric test for bimodality based on the likelihood principle by using two-component mixtures. The test uses explicit characterizations of the modal structure of such mixtures in terms of their parameters. Examples include the univariate and multivariate normal distributions and the von Mises distribution. We present the asymptotic distribution of the proposed test and analyze its finite sample performance in a simulation study. To illustrate our method, we use mixtures to investigate the modal structure of the cross-sectional distribution of per capita log GDP across EU regions from 1977 to 1993. Although these mixtures clearly have two components over the whole time period, the resulting distributions evolve from bimodality toward unimodality at the end of the 1970s.},
volume = {92},
issn = {1863-818X},
doi = {10.1007/s10182-008-0057-2},
number = {1},
journal = {AStA Advances in Statistical Analysis},
publisher = {Springer Science and Business Media LLC},
author = {Holzmann, Hajo and Vollmer, Sebastian},
year = {2008},
month = {feb},
pages = {57–69}
}
```