  We propose a semiparametric method for fitting the tail of a heavy-tailed
population given a relatively small sample from that population and a larger
sample from a related background population. We model the tail of the small
sample as an exponential tilt of the better-observed large-sample tail, using a
robust sufficient statistic motivated by extreme value theory. In particular,
our method induces an estimator of the small-population mean, and we give
theoretical and empirical evidence that this estimator outperforms methods that
do not use the background sample. We demonstrate substantial efficiency gains
over competing methods in simulation and on data from a large controlled
experiment conducted by Facebook.
