  Based on the negative binomial model for the duration of wet periods measured
in days, an asymptotic approximation is proposed for the distribution of the
maximum daily precipitation volume within a wet period. This approximation has
the form of a scale mixture of the Frechet distribution with the gamma mixing
distribution and coincides with the distribution of a positive power of a
random variable having the Snedecor-Fisher distribution. The proof of this
result is based on the representation of the negative binomial distribution as
a mixed geometric (and hence, mixed Poisson) distribution and limit theorems
for extreme order statistics in samples with random sizes having mixed Poisson
distributions. Some analytic properties of the obtained limit distribution are
described. In particular, it is demonstrated that under certain conditions the
limit distribution is mixed exponential and hence, is infinitely divisible. It
is shown that under the same conditions the limit distribution can be
represented as a scale mixture of stable or Weibull or Pareto or folded normal
laws. The corresponding product representations for the limit random variable
can be used for its computer simulation. Several methods are proposed for the
estimation of the parameters of the distribution of the maximum daily
precipitation volume. The results of fitting this distribution to real data are
presented illustrating high adequacy of the proposed model. The obtained
mixture representations for the limit laws and the corresponding asymptotic
approximations provide better insight into the nature of mixed probability
("Bayesian") models.
