A Characteristic Exponent Test for the Cauchy Distribution
[JEL: C12, C15, C40]

Atlantic Economic Journal 28:4, December 2000

Western Carolina University and the State University of New York at Binghamton

The Mandelbrot-Levy distributions are a family of infinite-variance distributions without explicit analytical expressions, except for special cases.  Limiting distributions include the normal, with finite variance, and the Cauchy, with the most extreme platykurtosis or fat tails. Paul Levy developed the theory of these distributions [Calcul des Probibilites, Paris: Gauthier Villars, 1925].  The Hurst exponent H introduced in the hydrological study of the Nile valley is the reciprocal of the characteristic exponent [Transactions of the American Society of Civil Engineers, 1951, v. 116, pp. 770-799].

The characteristic function of a Mandelbrot-Levy random variable is:

log f(t) = i(delta)t - (gamma)|t|alpha[1 + i(gamma)(sign(t)tan((alpha)(pi)/2))]
where delta is the expectation or mean of t if alpha > 1; gamma is a scale parameter; alpha is the characteristic exponent; and i is the square root of -1.  Gnedenko and Kolmolgorov showed the sum of n iid Mandelbrot-Levy variables is:
n log f(t) = i(delta)t - n(gamma)|t|alpha[1 + i(gamma)(sign(t)tan((alpha)(pi)/2))]
and thus the distributions exhibit stability under addition [Limit Distributions for Sums of Random Variables, Reading MA: Addison-Wesley, 1954].  Many applications of the central limit theorem only demonstrate Mandelbrot-Levy character.  The result of normality generally depends on an unjustified assumption of finite variance.

Mandelbrot introduced a technique for estimating alpha by regression [Annals of Economic and Social Measurement, 1972, 1:3, pp. 259-290], further refined by Lo [Econometrica, 1991, 59:3, pp. 1279-1313].  Various statistics are available to test the null hypothesis of normality, but not for the Cauchy distribution, the other extreme.

Tables of percentages of (alpha) were generated by Monte Carlo experiments with 1,000 iterations for different sample sizes.  Here follow the sample sizes, followed by 1%, 5%, 10%, 90%, 95% and 99% critical values:

0.254; 0.464; 0.550; 1.330; 1.521; 1.965],
0.410; 0.556; 0.631; 1.256; 1.354; 1.634],
0.526; 0.691; 0.749; 1.157; 1.215; 1.324],
0.659; 0.765; 0.802; 1.108; 1.153; 1.227],
0.699; 0.825; 0.854; 1.085; 1.108; 1.154],
0.807; 0.876; 0.899; 1.045; 1.066; 1.096],
0.864; 0.906; 0.926; 1.032; 1.048; 1.071],
0.885; 0.935; 0.946; 1.023; 1.034; 1.049]. 
These critical values can be used to evaluate estimated alphas for the Cauchy null; the null should be rejected if the estimated characteristic exponent lies outside the critical bounds.  Dispersion of alpha around the theoretical value of 1 varies greatly with the sample size.