  Given a random sample of observations, mixtures of normal densities are often
used to estimate the unknown continuous distribution from which the data come.
Here we propose the use of this semiparametric framework for testing symmetry
about an unknown value. More precisely, we show how the null hypothesis of
symmetry may be formulated in terms of normal mixture model, with weights about
the centre of symmetry constrained to be equal one another. The resulting model
is nested in a more general unconstrained one, with same number of mixture
components and free weights. Therefore, after having maximised the constrained
and unconstrained log-likelihoods by means of a suitable algorithm, such as the
Expectation-Maximisation, symmetry is tested against skewness through a
likelihood ratio statistic. The performance of the proposed mixture-based test
is illustrated through a Monte Carlo simulation study, where we compare two
versions of the test, based on different criteria to select the number of
mixture components, with the traditional one based on the third standardised
moment. An illustrative example is also given that focuses on real data.
