[Paper Review] On the asymptotic approximation to the probability distribution of extremal precipitation

V. Yu. Korolev, Andrey Gorshenin|arXiv (Cornell University)|May 31, 2017

Hydrology and Drought Analysis16 references3 citations

TL;DR

This paper proposes an asymptotic approximation for the distribution of maximum daily precipitation within wet periods, modeling wet duration via a negative binomial distribution with shape parameter <1. The limit distribution is a scale mixture of Fréchet laws with gamma mixing, equivalent to a power of a Snedecor–Fisher distributed random variable, enabling accurate modeling and simulation of extreme precipitation events through parameter estimation and mixture representations.

ABSTRACT

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.

Motivation & Objective

• To develop a statistically sound asymptotic approximation for the distribution of maximum daily precipitation within wet periods, addressing the inadequacy of traditional geometric models.
• To model wet period durations using the negative binomial distribution with shape parameter <1, which better fits real meteorological data than geometric distributions.
• To establish a limit distribution for extreme precipitation as a scale mixture of Fréchet laws with gamma mixing, derived from mixed Poisson and extreme order statistics theory.
• To provide computationally feasible parameter estimation methods using least squares and probability plotting for real-world data fitting.
• To validate the model through empirical fitting to long-term daily precipitation data from Potsdam and Elista, demonstrating high model adequacy.

Proposed method

• Represents the negative binomial distribution as a mixed geometric and mixed Poisson distribution, enabling application of limit theorems for extreme order statistics in random sample sizes.
• Applies limit theorems for extreme order statistics in mixed Poisson samples to derive the asymptotic distribution of maximum daily precipitation.
• Derives the limit distribution as a scale mixture of Fréchet distributions with gamma-distributed mixing, equivalent to the positive power of a Snedecor–Fisher distributed random variable.
• Proposes two parameter estimation methods: a 'rough' method based on probability plotting and a least squares method using order statistics and transformed quantiles.
• Utilizes product representations of the limit law for simulation, including stable, Weibull, Pareto, and folded normal laws under specific conditions.
• Employs censoring techniques to assess the asymptotic behavior of the model across varying wet period thresholds (1 to 15 days), evaluating model accuracy.

Experimental results

Research questions

• RQ1 Can a negative binomial model with shape parameter <1 better describe the duration of wet periods than the traditional geometric distribution in real precipitation data?
• RQ2 What is the asymptotic distribution of the maximum daily precipitation within a wet period when wet durations follow a negative binomial distribution with r < 1?
• RQ3 How can the resulting limit distribution be represented as a scale mixture of Fréchet laws, and what are its analytic properties such as infinite divisibility?
• RQ4 What are effective and accurate methods for estimating the parameters of the asymptotic model from real precipitation data?
• RQ5 How well does the proposed asymptotic model approximate empirical extreme precipitation distributions across different wet period length thresholds?

Key findings

• The asymptotic distribution of maximum daily precipitation is a scale mixture of Fréchet distributions with gamma mixing, equivalent to the positive power of a Snedecor–Fisher distributed random variable.
• Under certain conditions, the limit distribution is infinitely divisible and can be represented as a scale mixture of stable, Weibull, Pareto, or folded normal laws.
• The least squares parameter estimation method (equations 22 and 23) provides more accurate estimates than the 'rough' probability plotting method.
• The model provides excellent fit to empirical data from Potsdam (r = 0.847) and Elista (r = 0.876), even with censoring thresholds up to 15 days and sample sizes above 150.
• The approximation remains accurate for censoring thresholds ≥3 days, with the threshold having a more pronounced effect on accuracy than sample size.
• The model’s mixture representations enable efficient computer simulation of extreme precipitation events using product representations of the limit random variable.