Academic journal article Atlantic Economic Journal

A Characteristic Exponent Test for the Cauchy Distribution

Academic journal article Atlantic Economic Journal

A Characteristic Exponent Test for the Cauchy Distribution

Article excerpt

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. Levy [Calcul des Probibilites, 1925] developed the theory of these distributions. The Hurst [Transactions of the American Society of Civil Engineers, 1951] exponent, H, introduced in the hydrological study of the Nile valley, is the reciprocal of the characteristic exponent.

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

log f(t) = i[delta]t - [gamma][[absolute val. of t].sup.[alpha]][1 + i[beta](sign(t)tan([alpha][pi]/2))],

where [delta] is the expectation or mean of t if [alpha] [greater than] 1; [gamma] is a scale parameter; [alpha] is the characteristic exponent; and i is the square root of -1. Gnedenko and Kolmolgorov [Limit Distributions for Sums of Random Variables, 1954] showed the sum of n iid Mandelbrot-Levy variables is:

n log f(t) = in [delta]t - n[gamma][[absolute val. of t].sup.[alpha]][1 + i[beta](sign(t)tan([alpha][pi]/2))].

Thus, the distributions exhibit stability under addition. Many applications of the central limit theorem only demonstrate the Mandelbrot-Levy character. The result of normality generally depends on an unjustified assumption of a finite variance. …

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