z Test for the significance of the mean of a stable probability distribution with 1<α≤2

Abstract

The analysis of data using a stable probability distribution with tail parameter α<2 (sometimes called a Pareto–Levy distribution) seems to have been avoided in the past in part because of the lack of a significance test for the mean, even though it appears to be the correct distribution to use for describing returns in the financial markets. A z test for the significance of the mean of a stable distribution with tail parameter 1<α≤2 is defined. Tables are calculated and displayed for the 5% and 1% significance levels for a range of tail and skew parameters α and β. Through the use of maximum likelihood estimates, the test becomes a practical tool even when α and β are not that accurately determined. As an example, the z test is applied to the daily closing prices for the Dow Jones Industrial average from 2 January 1940 to 19 March 2010.

Keywords:

• statistic
• Pareto–Levy
• returns
• financial markets

Acknowledgements

The author thanks T.J. Pearson and the California Institute of Technology for making the graphics software pgplot v5.5.2 available for use.