  In many settings it is critical to accurately model the extreme tail
behaviour of a random process. Non-parametric density estimation methods are
commonly implemented as exploratory data analysis techniques for this purpose
as they possess excellent visualisation properties, and can naturally avoid the
model specification biases implied by using parametric estimators. In
particular, kernel-based estimators place minimal assumptions on the data, and
provide improved visualisation over scatterplots and histograms. However kernel
density estimators are known to perform poorly when estimating extreme tail
behaviour, which is important when interest is in process behaviour above some
large threshold, and they can over-emphasise bumps in the density for heavy
tailed data. In this article we develop a transformation kernel density
estimator, and demonstrate that its mean integrated squared error (MISE)
efficiency is equivalent to that of standard, non-tail focused kernel density
estimators. Estimator performance is illustrated in numerical studies, and in
an expanded analysis of the ability of well known global climate models to
reproduce observed temperature extremes in Sydney, Australia.
