Transformed estimates of densities of heavy-tailed distributions and classification

Nonparametric estimation of the density of a heavy-tailed probability distribution is investigated. The initial data are transformed to a bounded interval and the distribution density is determined by an inverse transformation of the distribution density estimate of transformed data. An adaptive data transformation is studied, in which the order of decay of the tail of the true distribution density is preserved and stable estimation of the deviation in tail index estimates is guaranteed. In classification, the empirical risk of erroneous classification by the Bayes empirical classifier is used as a measure for the quality of distribution density estimates.