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.
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Barro, R., Sala-i-Martin, X.: Convergence across states and regions. Brookings Pap. Econ. Act. 1991(1), 107–182 (1991)
Basu, S., Jammalamadaka, S.R.: Unimodality in circular data: a Bayes test. In: Balakrishnan, N., et al. (eds.) Advances on Methodological and Applied Aspects of Probability and Statistics, pp. 141–153. Taylor and Francis, New York (2002)
Behboodian, J.: On the modes of a mixture of two normal distributions. Technometrics 12, 131–139 (1970)
Bianchi, M.: Testing for convergence: Evidence from non-parametric multimodality tests. J. Appl. Econom. 12, 393–409 (1997)
Chen, J., Kalbfleisch, J.D.: Modified likelihood ratio test in finite mixture models with a structural parameter. J. Stat. Plan. Inference 129, 93–107 (2005)
Chen, J., Chen, H., Kalbfleisch, J.: A modified likelihood ratio test for homogeneity in finite mixture models. J. R. Stat. Soc. Ser. B 63, 19–29 (2001)
Chernoff, H.: On the distribution of the likelihood ratio. Ann. Math. Stat. 25, 573–578 (1954)
Fisher, N.I., Marron, J.S.: Mode testing via the excess mass estimate. Biometrika 88, 499–517 (2001)
Fisher, N.I., Mammen, E., Marron, J.S.: Testing for multimodality. Comput. Stat. Data Anal. 18, 499–512 (1994)
Goffinet, B., Loisel, P., Laurent, B.: Testing in normal mixture models when the proportions are known. Biometrika 79, 842–846 (1992)
Hall, P., York, M.: On the calibration of Silverman’s test for multimodality. Stat. Sin. 11, 515–536 (2001)
Hartigan, J.A., Hartigan, P.M.: The dip test of unimodality. Ann. Stat. 13, 70–84 (1985)
Le Gallo, J.: Space-time analysis of GDP disparities among European regions: a Markov chain approach. Int. Reg. Sci. Rev. 27, 138–163 (2004)
Mammen, E., Marron, J.S., Fisher, N.I.: Some asymptotics for multimodality tests based on kernel density estimates. Probab. Theory Relat. Fields 91, 115–132 (1992)
Mardia, K.V., Sutton, T.W.: On the modes of a mixture of two von Mises distributions. Biometrika 62, 699–701 (1975)
Müller, D.W., Sawitzki, G.: Excess mass estimates and tests for multimodality. J. Am. Stat. Assoc. 86, 738–746 (1991)
Pittau, M.G., Zelli, R.: Fitting regional income distributions in the European Union. Oxf. Bull. Econ. Stat. 67, 135–161 (2005)
Pittau, M.G., Zelli, R.: Empirical evidence of income dynamics across EU regions. J. Appl. Econom. 21, 605–628 (2006)
Quah, D.: Regional convergence clusters across Europe. Eur. Econ. Rev. 40, 1353–1375 (1996)
Quah, D.: Empirics for growth and distribution: polarization, stratification, and convergence clubs. J. Econ. Growth 2, 25–59 (1997)
Ray, S., Lindsay, B.G.: The topography of multivariate normal mixtures. Ann. Stat. 33, 2042–2065 (2005)
Robertson, C.A., Fryer, J.G.: Some descriptive properties of normal mixtures. Skand. Aktuarietidskr. 69, 137–146 (1969)
Sala-i-Martin, X.: The classical approach to convergence analysis. Econ. J. 106, 1019–1036 (1996)
Self, S.G., Liang, K.-Y.: Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Am. Stat. Assoc. 82, 605–610 (1987)
Silverman, B.W.: Using kernel density estimates to investigate multimodality. J. R. Stat. Soc. Ser. B 43, 97–99 (1981)
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Holzmann, H., Vollmer, S. A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU. AStA 92, 57–69 (2008). https://doi.org/10.1007/s10182-008-0057-2
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DOI: https://doi.org/10.1007/s10182-008-0057-2