Centre of Statistics and its Applications

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Publications

Fitting heavy Tail distributions with mixture models

Publication . Basílio, Jorge; Oliveira, Amilcar

The normal probability distribution as assumption for financial returns have been recognized as inappropriate, and a source of inaccurate estimation of Value at Risk. Empirical evidence also have been shown that financial returns shows a more accentuated leptokurtic distribution when compared with a Normal distribution and also skewed. This is usually a cause of underestimated values of VaR, specially when the quantiles are very low. Therefore it is necessary to focus on the tail of the distribution and identify models to fit that behavior. We will highlight the differences between the quality of fitting in the tails of the distribution and the fitting for all the distribution. This work compares and interprets the results obtained by applying mixture models as a method to estimate the behavior on the extremes for heavy tail data distributions. This results will be then used to describe an analytical solution of VaR under mixture models.

2020Conference object

Open access

The skewness and kurtosis of the product of two normally distributed random variables

Publication . Seijas-Macias, J. Antonio; Oliveira, Amilcar; Oliveira, Teresa

The analysis of the product of two normally distributed variables does not seem to follow any known distribution. Fortunately, the moment-generating function is available and we can calculate the statistics of the product distribution: mean, variance, the skewness and kurtosis (excess of kurtosis). In this work, we have considered the role played by the parameters of the two normal distributions’ factors (mean and variance) on the values of the skewness and kurtosis of the product. Ranges of variation are defined for kurtosis and the skewness. The determination of the evolution of the skewness and kurtosis values of the product can be used to establish the normality of the product and how to modelize its distribution. Finally, the Pearson Inequality is proved for the skewness and kurtosis of the product of two normal random variables.

2021-04-14Journal article

Restricted access