  This work defines and investigates the properties of the Max-U-Exp
distribution. The method of moments is applied in order to estimate its
parameters. Then, by using the previous general theory about Mixed Poisson
processes, developed by Grandel (1997), and Karlis and Xekalaki (2005), and
analogously to Jordanova et al. (2023), and Jordanova and Stehlik (2017) we
define and investigate the properties of the new random vectors and random
variables, which are related with this particular case of a Mixed Poisson
process. Exp-Max-U-Exp distribution is defined and thoroughly investigated. It
arises in a natural way as a distribution of the inter-arrival times in the
Mixed Poisson process with Max-U-Exp mixing variable. The distribution of the
renewal moments is called Erlang-Max-U-Exp and is defined via its probability
density function. Investigation of its properties follows. Finally, the
corresponding Mixed Poisson process with Max-U-Exp mixing variable is defined.
Its finite dimensional and conditional distributions are found and their
numerical characteristics are determined.
