A Monte Carlo Experiment to Evaluate the Efficiency of the Mandelbrot-Levy Characteristic Exponent as a Test Statistic for Cauchy Distributed Random Variables

A refinement of Mandelbrot's technique for estimating the Mandelbrot-Levy characteristic exponent is applied to Cauchy distributed pseudorandom samples. Tables of percentage points of the test statistic are calculated under the null of a Cauchy-distributed population for various sample sizes. An iterative least squares procedure is used to determine how much of the sample is needed to characterize the linear part of the upper tail. For various sample sizes, the characteristic exponent is estimated for 10,000 sets of Cauchy distributed samples, providing the distribution under the null hypothesis. With the distribution under the Cauchy null now established, the characteristic exponent can be used as a test statistic for the Cauchy distribution, and evaluated using the tables of probability values calculated here