  Mixed modeling of extreme values and random effects is relatively unexplored
topic. Computational difficulties in using the maximum likelihood method for
mixed models and the fact that maximum likelihood method uses available data
and does not use the prior information motivate us to use Bayesian method. Our
simulation studies indicate that random effects modeling produces more reliable
estimates when heterogeneity is present. The application of the proposed model
to the climate data and return values of some economic indexes reveals the same
pattern as the simulation results and confirms the usefulness of mixed modeling
of random effects and extremes. As the nature of climate and economic data are
massive and there is always a possibility of missing a considerable part of
data, saving the information included in past data is useful. Our simulation
studies and applications show the benefit of Bayesian method to save the
information from the past data into the posterior distributions of the
parameters to be used as informative prior distributions to fit the future
data. We show that informative prior distributions obtained from the past data
help to estimate the return level in Block Maxima method and Value-at-Risk and
Expected Shortfall in Peak Over Threshold method with less bias than using
uninformative prior distributions.
