Kurtosis comparison of the Cauchy and double exponential distributions

We first describe a class of quantile-based kurtosis orderings on symmetric distributions that use density matching to match the scales of distributions before kurtosis comparisons are made. We then use the orderings to give a meaningful comparison of the kurtosis properties of the Cauchy and Double Exponential distributions. Since these distributions are often used as models for heavy-tailed distributions and there appears some confusion about their properties such a comparison should be useful