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The statistical method for the financial returns plays a key role in measuring the goodness-of-fit of a given distribution to real data. As is well known, the Normal Inverse Gaussian (NIG) and Generalized Hyperbolic Skew-t (GHST) distributions have been found to successfully describe the data of the returns from the financial market. In this paper, we mainly consider the discrimination between these distributions. It is observed that the maximum likelihood estimators cannot be obtained in closed form. We propose to use the EM algorithm to compute the maximum likelihood estimators. The observed Fisher information matrix, as well as the standard deviation of the MLEs are derived. We then perform a number of goodness-of-fit to compare the NIG and GHST distributions for the stock exchange data. Moreover, the Vuong type test based on their Kullback-Leibler distances has been considered to select the best candidate models. An important implication of the present study is that the GHST distribution function, in contrast to NIG distribution, may describe more appropriate for the proposed data.

Discriminating between the normal inverse Gaussian and generalized hyperbolic skew-t distributions with a follow-up the stock exchange data