Should Normal Distribution be Normal? The Student's T Alternative

Abstract

In the paper we try to answer, whether the Gaussian distribution - called widely the 'normal' distribution - is really basic, natural and normal. In particular, we investigate how the above statement conforms with the distribution of real data, namely daily returns of some stock indexes. It was the authors former experience that, when looking at the distributions of real data, it was very difficult to find there a 'normal', i.e. Gaussian distribution. The data, by their nature, are heterogeneous. If so, then the data should be modelled taking into account their possible heterogeneity. This can be done using mixture models - with mixtures composed from finite or infinite number of components. Students' T (univariate or multivariate) is one prominent example of distributions which may be obtained as a mixture of infinitesimal number of Gaussian distributions. The considerations are illustrated by an example of application to financial time series, namely daily returns of the indexes WIG20 and S&P500. We show, why the normality (i.e. 'Gaussianity') should be rejected and why the 't' distribution is plausible.