On fitting the Pareto-Levy distribution to stock market index data: selecting a suitable cutoff value

The so-called Pareto-Levy or power-law distribution has been successfully used as a model to describe probabilities associated to extreme variations of worldwide stock markets indexes data and it has the form $Pr(X>x) ~ x**(-alpha) for gamma< x <infinity. The selection of the threshold parameter gamma$ from empirical data and consequently, the determination of the exponent alpha, is often is done by using a simple graphical method based on a log-log scale, where a power-law probability plot shows a straight line with slope equal to the exponent of the power-law distribution. This procedure can be considered subjective, particularly with regard to the choice of the threshold or cutoff parameter gamma. In this work is presented a more objective procedure, based on a statistical measure of discrepancy between the empirical and the Pareto-Levy distribution. The technique is illustrated for data sets from the New York Stock Exchange Index and the Mexican Stock Market Index (IPC).