Unsupervised tail modeling via noisy cross-entropy minimization

Estimation of dynamic mixture distributions is a difficult task, because the density contains an intractable normalizing constant. To overcome this difficulty, we develop an approach that maximizes, by means of the cross-entropy method, a Monte Carlo approximation of the log-likelihood function. The proposed noisy cross-entropy approach is unsupervised, since it does not require the specification of a threshold between the distributions. Moreover, it bypasses the evaluation of the normalizing constant, combining good statistical properties with a modest computational burden. Both simulation-based evidence and empirical applications suggest that noisy cross-entropy estimation is comparable or preferable to existing methods in terms of statistical efficiency, but is less demanding from the computational point of view.