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Based on the negative binomial model for the duration of wet periods measuredin days, an asymptotic approximation is proposed for the distribution of themaximum daily precipitation volume within a wet period. This approximation hasthe form of a scale mixture of the Frechet distribution with the gamma mixingdistribution and coincides with the distribution of a positive power of arandom variable having the Snedecor-Fisher distribution. The proof of thisresult is based on the representation of the negative binomial distribution asa mixed geometric (and hence, mixed Poisson) distribution and limit theoremsfor extreme order statistics in samples with random sizes having mixed Poissondistributions. Some analytic properties of the obtained limit distribution aredescribed. In particular, it is demonstrated that under certain conditions thelimit distribution is mixed exponential and hence, is infinitely divisible. Itis shown that under the same conditions the limit distribution can berepresented as a scale mixture of stable or Weibull or Pareto or folded normallaws. The corresponding product representations for the limit random variablecan be used for its computer simulation. Several methods are proposed for theestimation of the parameters of the distribution of the maximum dailyprecipitation volume. The results of fitting this distribution to real data arepresented illustrating high adequacy of the proposed model. The obtainedmixture representations for the limit laws and the corresponding asymptoticapproximations provide better insight into the nature of mixed probability("Bayesian") models.

On the asymptotic approximation to the probability distribution of extremal precipitation