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

@article{Holzmann2008ALR, title={A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU}, author={Hajo Holzmann and Sebastian Vollmer}, journal={AStA Advances in Statistical Analysis}, year={2008}, volume={92}, pages={57-69} }

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… Expand

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#### References

SHOWING 1-10 OF 33 REFERENCES

Modified likelihood ratio test in finite mixture models with a structural parameter

- Mathematics
- 2005

Abstract The finite mixture model is an example of a non-regular parametric family, and most classical asymptotic results cannot be directly applied. In particular, the asymptotic properties of… Expand

A modified likelihood ratio test for homogeneity in finite mixture models

- Mathematics
- 2001

Testing for homogeneity in finite mixture models has been investigated by many researchers. The asymptotic null distribution of the likelihood ratio test (LRT) is very complex and difficult to use in… Expand

Fitting Regional Income Distributions in the European Union

- Economics
- 2005

Cross-sectional distribution of per capita (log) GDP across the European Union regions from 1977 to 1996 is analysed. Kernel density estimates reveal a multimodal structure of the distribution during… Expand

Testing for Convergence: Evidence from Non-parametric Multimodality Tests

- Mathematics
- 1995

The convergence hypothesis in growth theory implies that the frequency of the density distribution of GDP in a cross-section of countries tends to approach unimodality as we move forward in time. The… Expand

Empirical evidence of income dynamics across EU regions

- Mathematics
- 2006

This paper analyses the distribution of purchasing power standardized per capita income across EU-12 regions between 1977 and 1996. Dispersion of incomes between regions is measured taking into… Expand

ON THE CALIBRATION OF SILVERMAN'S TEST FOR MULTIMODALITY

- Mathematics
- 2001

It is known that Silverman's bootstrap test for multimodality tends to- wards conservatism, even in large samples, in the sense that the actual level tends to be less than the nominal one. In this… Expand

Some descriptive properties of normal mixtures

- Mathematics
- 1969

Abstract When two single normal densities are mixed together the shape of the resulting distribution will vary in the extent to which it indicates the underlying structure, depending on the values of… Expand

On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems.

- Mathematics, Medicine
- Biometrika
- 2010

This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η), H(0) : γ = γ(0), based on the pseudolikelihood L(θ,… Expand

On the Modes of a Mixture of Two Normal Distributions

- Mathematics
- 1970

This paper presents a procedure for answering the question of whether a mixture of two normal distributions, with five known parameters μ1, μ2, σ1, σ2, p, is unimodal or not. The approach of the… Expand

Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions

- Mathematics
- 1987

Abstract Large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described. Specifically, the asymptotic distribution of… Expand