  We study the problem of selecting features associated with extreme values in
high dimensional linear regression. Normally, in linear modeling problems, the
presence of abnormal extreme values or outliers is considered an anomaly which
should either be removed from the data or remedied using robust regression
methods. In many situations, however, the extreme values in regression modeling
are not outliers but rather the signals of interest; consider traces from
spiking neurons, volatility in finance, or extreme events in climate science,
for example. In this paper, we propose a new method for sparse high-dimensional
linear regression for extreme values which is motivated by the Subbotin, or
generalized normal distribution, which we call the extreme value linear
regression model. For our method, we utilize an $\ell_p$ norm loss where $p$ is
an even integer greater than two; we demonstrate that this loss increases the
weight on extreme values. We prove consistency and variable selection
consistency for the extreme value linear regression with a Lasso penalty, which
we term the Extreme Lasso, and we also analyze the theoretical impact of
extreme value observations on the model parameter estimates using the concept
of influence functions. Through simulation studies and a real-world data
example, we show that the Extreme Lasso outperforms other methods currently
used in the literature for selecting features of interest associated with
extreme values in high-dimensional regression.
