  A new unimodal distribution family indexed by the mode and three other
parameters is derived from a mixture of a Gumbel distribution for the maximum
and a Gumbel distribution for the minimum. Properties of the proposed
distribution are explored, including model identifiability and flexibility in
capturing heavy-tailed data that exhibit different directions of skewness over
a wide range. Both frequentist and Bayesian methods are developed to infer
parameters in the new distribution. Simulation studies are conducted to
demonstrate satisfactory performance of both methods. By fitting the proposed
model to simulated data and data from an application in hydrology, it is shown
that the proposed flexible distribution is especially suitable for data
containing extreme values in either direction, with the mode being a location
parameter of interest. Using the proposed unimodal distribution, one can easily
formulate a regression model concerning the mode of a response given
covariates. We apply this model to data from an application in criminology to
reveal interesting data features that are obscured by outliers. Computer
programs for implementing all considered inference methods in the study are
available at https://github.com/rh8liuqy/flexible_Gumbel.
