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Motivated by finance applications, we assessed the performance of several univariate density estimation methods, focusing on their ability to deal with heavy-tailed target densities. Four approaches, a fixed bandwidth kernel estimator, an adaptive bandwidth kernel estimator, the Hermite series (SNP) estimator of Gallant and Nychka, and the logspline estimator of Kooperberg and Stone, are compared. We conclude that the logspline and adaptive kernel methods provide superior performance, and the convergence rate of the SNP estimator is remarkably slow compared with the other methods. The Hellinger convergence rate of the SNP estimator is derived as a function of tail heaviness. These findings are confirmed in Monte Carlo experiments. Qualitative assessment reveals the possibility that side lobes in the tails of the fixed kernel and SNP estimates are artefacts of the fitting method. Copyright The Author(s). Journal compilation Royal Economic Society 2008

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The behavior of events that occur infrequently but have a large impact tends to differ from that of the central tendency, and identifying the tail dependence structure among key factors is critical for controlling risks. However, due to technical difficulties, conventional analyses of dependence have focused on the global average dependence. This article proposes a novel approach for analyzing the entire structure of nonlinear dependence between two data sets on the basis of accurate pointwise mutual information estimation. The emphasis is on fat-tailed distributions that tend to appear in events involving sudden large changes. The proposed pointwise mutual information estimator is sufficiently robust and efficient for exploring tail dependence, and its good performance was confirmed in an experimental study. The significance of the identified dependence structure was assessed using the proposed bootstrap procedure. New facts were discovered from its application to daily returns and volume on the New York stock Exchange (NYSE) Composite Index.