Distinguishing Stable Probability Measures Part I: Discrete Time

Abstract

A sequence of N, independent, identically distributed, random variables is observed from one of two stable distributions with known parameters. The likelihood-ratio test for discriminating between these two distributions is found explicitly and performance limitations are determined. When the two distributions differ only in location, the likelihood-ratio test is sensitive to whether the distribution is nongaussian stable (0 < α < 2) when nonlinear soft limiting of large deviations is used, or gaussian stable (α = 2) when linear processing is used. When the two distributions differ only in scale, the likelihood-ratio test is sensitive to whether 0 < α < 2 when nonlinear soft limiting of large deviations is used, or gaussian (α = 2) when a chi-squared test is used. The analysis of the two remaining cases, distinguishing between one of two characteristic indices, and between one of two skewness parameters, parallels the analysis of distinguishing between one of two scale parameters and is only touched upon briefly.