  The Peaks-Over Threshold is a fundamental method in the estimation of rare
events such as small exceedance probabilities, extreme quantiles and return
periods. The main problem with the Peaks-Over Threshold method relates to the
selection of threshold above and below which the asymptotic results are valid
for large and small observations respectively. In addition, the main assumption
leading to the asymptotic results is that the observations are independent and
identically distributed. However, in practice, many real life processes yield
data that are non-stationary and/or related to some covariate variables. As a
result, threshold selection gets complicated as it may depend on the
covariates. Strong arguments have been made against the use of constant
threshold as observation that is considered extreme at some covariate level may
not qualify as an extreme observation at another covariate level. Some authors
have attempted to obtain covariate dependent thresholds in different ways: the
most appealing one relies on quantile regression. In this paper, we propose a
covariate dependent threshold based on expectiles. We compare this threshold
with the constant and the quantile regression in a simulation study for
estimating the tail index of the Generalised Pareto distribution. As may be
expected, no threshold is universally the best. However, certain general
observations can be made for the exponential growth data considered. Firstly,
we find that the expectile threshold outperforms the others when the response
variable has smaller to medium values. Secondly, for larger values of the
response variable, the constant threshold is generally the best method. The
threshold selection methods are illustrated in the estimation of the tail index
of an insurance claims data.
