  A novel statistical method is proposed and investigated for estimating a
heavy tailed density under mild smoothness assumptions. Statistical analyses of
heavy-tailed distributions are susceptible to the problem of sparse information
in the tail of the distribution getting washed away by unrelated features of a
hefty bulk. The proposed Bayesian method avoids this problem by incorporating
smoothness and tail regularization through a carefully specified semiparametric
prior distribution, and is able to consistently estimate both the density
function and its tail index at near minimax optimal rates of contraction. A
joint, likelihood driven estimation of the bulk and the tail is shown to help
improve uncertainty assessment in estimating the tail index parameter and offer
more accurate and reliable estimates of the high tail quantiles compared to
thresholding methods.
