Generalized Gumbel model for r-largest order statistics, with an application to peak streamflow

Area of Science:

• Extreme value analysis
• Statistical modeling
• Probability distributions

Background:

• Extreme values are rare, necessitating efficient analysis methods.
• The Gumbel distribution for r-largest order statistics (rGD) is limited by its two-parameter structure.
• Block maxima methods can be less efficient than using r-largest order statistics (rLOS).

Purpose of the Study:

• To extend the Gumbel distribution for rLOS to a more flexible three-parameter model.
• To introduce the generalized Gumbel distribution for rLOS (rGGD).
• To provide robust statistical inference methods for the new rGGD model.

Main Methods:

• Derivation of probability functions for the rGGD.
• Application of maximum likelihood estimation and the delta method for parameter estimation.
• Utilizing entropy difference tests and cross-validated likelihood for model selection and inference.

Main Results:

• The proposed rGGD model demonstrates greater flexibility in capturing the variability of r-largest data.
• Monte Carlo simulations confirm the model's usefulness and effectiveness.
• Application to peak streamflow data shows practical applicability.

Conclusions:

• The generalized Gumbel distribution for rLOS (rGGD) offers a more adaptable approach to extreme value analysis.
• The developed inference methods provide a solid framework for applying the rGGD.
• This model can aid in the design of engineering structures to mitigate risks from extreme events.