Testing the Isotropic Cauchy Hypothesis

Isotropic \(\alpha\)-stable distributions are central in the theory of heavy-tailed distributions and play a role similar to that of the Gaussian density among finite second-moment laws. Given a sequence of \(n\) observations, we are interested in characterizing the performance of Likelihood Ratio T...

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Bibliographic Details
Published in:arXiv.org 2024-08
Main Authors: Fahs, Jihad, Abou-Faycal, Ibrahim, Issa, Ibrahim
Format: Article
Language:English
Subjects:
Error detection
Hypotheses
Likelihood ratio
Online Access:Get full text
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Summary:Isotropic \(\alpha\)-stable distributions are central in the theory of heavy-tailed distributions and play a role similar to that of the Gaussian density among finite second-moment laws. Given a sequence of \(n\) observations, we are interested in characterizing the performance of Likelihood Ratio Tests where two hypotheses are plausible for the observed quantities: either isotropic Cauchy or isotropic Gaussian. Under various setups, we show that the probability of error of such detectors is not always exponentially decaying with \(n\) with the leading term in the exponent shown to be logarithmic instead and we determine the constants in that leading term. Perhaps surprisingly, the optimal Bayesian probabilities of error are found to exhibit different asymptotic behaviors.
ISSN:2331-8422
DOI:10.48550/arxiv.2408.06269