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Mixtures of equispaced normal distributions and their use for testing symmetry with univariate data


  • Bacci, Silvia
  • Bartolucci, Francesco


Given a random sample of observations, mixtures of normal densities are often used to estimate the unknown continuous distribution from which the data come. The use of this semi-parametric framework is proposed for testing symmetry about an unknown value. More precisely, it is shown how the null hypothesis of symmetry may be formulated in terms of a normal mixture model, with weights about the center of symmetry constrained to be equal one another. The resulting model is nested in a more general unconstrained one, with the same number of mixture components and free weights. Therefore, after having maximized the constrained and unconstrained log-likelihoods, by means of the Expectation–Maximization algorithm, symmetry is tested against skewness through a likelihood ratio statistic with p-value computed by using a parametric bootstrap method. The behavior of this mixture-based test is studied through a Monte Carlo simulation, where the proposed test is compared with the traditional one, based on the third standardized moment, and with the non-parametric triples test. An illustrative example is also given which is based on real data.

Suggested Citation

  • Bacci, Silvia & Bartolucci, Francesco, 2014. "Mixtures of equispaced normal distributions and their use for testing symmetry with univariate data," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 262-272.
  • Handle: RePEc:eee:csdana:v:71:y:2014:i:c:p:262-272
    DOI: 10.1016/j.csda.2013.01.015

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    References listed on IDEAS

    1. Racine, Jeffrey S. & Maasoumi, Esfandiar, 2007. "A versatile and robust metric entropy test of time-reversibility, and other hypotheses," Journal of Econometrics, Elsevier, vol. 138(2), pages 547-567, June.
    2. Modarres, Reza & Gastwirth, Joseph L., 1996. "A modified runs test for symmetry," Statistics & Probability Letters, Elsevier, vol. 31(2), pages 107-112, December.
    3. Francesco Bartolucci, 2005. "Clustering Univariate Observations via Mixtures of Unimodal Normal Mixtures," Journal of Classification, Springer;The Classification Society, vol. 22(2), pages 203-219, September.
    4. Antonietta Mira, 1999. "Distribution-free test for symmetry based on Bonferroni's measure," Journal of Applied Statistics, Taylor & Francis Journals, vol. 26(8), pages 959-972.
    5. repec:taf:gnstxx:v:21:y:2009:i:8:p:943-967 is not listed on IDEAS
    6. Ehab F. Abd-Elfattah & Ronald W. Butler, 2011. "Tests for symmetry with right censoring," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(4), pages 683-693, December.
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    Cited by:

    1. Lyubchich, Vyacheslav & Wang, Xingyu & Heyes, Andrew & Gel, Yulia R., 2016. "A distribution-free m-out-of-n bootstrap approach to testing symmetry about an unknown median," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 1-9.


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