Testing the Isotropic Cauchy Hypothesis

TL;DR

This paper investigates the effectiveness of likelihood ratio tests in distinguishing isotropic Cauchy distributions from Gaussian ones, revealing that error probabilities decay logarithmically rather than exponentially, with differing Bayesian error behaviors.

Contribution

It characterizes the asymptotic error decay rates of likelihood ratio tests between isotropic Cauchy and Gaussian distributions, highlighting non-exponential decay and contrasting Bayesian error behaviors.

Findings

Error probability decays logarithmically with sample size

Constants in the decay rate are explicitly determined

Bayesian error probabilities exhibit different asymptotic behaviors

Abstract

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.