Simple tests for peakedness, fat tails and leptokurtosis based on quantiles

Schmid Friedrich, Trede Mark

Abstract
The conventional test for leptokurtosis is based on the fourth moment of the standardized sample. This test suffers from various weaknesses: It cannot account for peakedness and fat tails separately and it is extremely sensitive with respect to outliers resulting in a lack of robustness concerning the error of the first kind. We suggest alternative tests for fat tails, peakedness and leptokurtosis. They are based on selector statistics as introduced by Hogg. Quantiles of the test statistics under normality for sample sizes of up to 2000 are derived by Monte-Carlo simulation. The asymptotic distribution is derived analytically as well. We apply the tests to daily, weekly and monthly asset returns of three German stocks. One can conclude that the new tests give a much more subtle picture of the structure of leptokurtosis in the data than the conventional test. (C) 2003 Elsevier Science B.V. All rights reserved.

Keywords
alpha-robustness validity robustness selector statistics