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1. Uniqueness and global optimality of the maximum likelihood estimator for the generalized extreme value distribution

   https://doi.org/10.1093/biomet/asab043  

   Zhang, Likun; Shaby, Benjamin A (August 2021, Biometrika)

   Summary The three-parameter generalized extreme value distribution arises from classical univariate extreme value theory, and is in common use for analysing the far tail of observed phenomena, yet important asymptotic properties of likelihood-based estimation under this standard model have not been established. In this paper we prove that the maximum likelihood estimator is global and unique. An interesting secondary result entails the uniform consistency of a class of limit relations in a tight neighbourhood of the true shape parameter. 

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2. Modeling Spatial Extremes using Non-Gaussian Spatial Autoregressive Models via Convolutional Neural Networks

   Rai, S; Nychka, D; Bandyopadhyay, S (August 2026, Environmetrics)

   Data derived from remote sensing or numerical simulations often have a regular gridded structure and are large in volume, making it challenging to find accurate spatial models that can fill in missing grid cells or simulate the process effectively, especially in the presence of spatial heterogeneity and heavy-tailed marginal distributions. To overcome this issue, we present a spatial autoregressive modeling framework, which maps observations at a location and its neighbors to independent random variables. This is a highly flexible modeling approach and is well-suited for non-Gaussian fields, providing simpler interpretability than hierarchical spatial models. In particular, we consider the spatial autoregressive model with Generalized Extreme Value distribution innovations to combine the observation at a central grid location with its neighbors, capturing extreme spatial behavior based on the heavy-tailed innovations. While these models are fast to simulate by exploiting the sparsity of the key matrices in the computations, the maximum likelihood estimation of the parameters is prohibitive due to the intractability of the likelihood, making optimization challenging. To overcome this, we train a convolutional neural network on a large training set of spatial fields that are generated using parameters covering a useful parameter space, and then use the trained network for fast parameter estimation. We further employ quantile regression for uncertainty quantification, incorporating a novel construction of the training dataset to obtain confidence bounds. Finally, we apply this model to analyze annual maximum precipitation data from ERA-Interim-driven Weather Research and Forecasting (WRF) simulations, allowing us to explore its spatial extreme behavior across North America. 

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3. DeepExtrema: A Deep Learning Approach for Forecasting Block Maxima in Time Series Data

   https://doi.org/10.24963/ijcai.2022/413  

   Galib, Asadullah Hill; McDonald, Andrew; Wilson, Tyler; Luo, Lifeng; Tan, Pang-Ning (July 2022, Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence)

   Accurate forecasting of extreme values in time series is critical due to the significant impact of extreme events on human and natural systems. This paper presents DeepExtrema, a novel framework that combines a deep neural network (DNN) with generalized extreme value (GEV) distribution to forecast the block maximum value of a time series. Implementing such a network is a challenge as the framework must preserve the inter-dependent constraints among the GEV model parameters even when the DNN is initialized. We describe our approach to address this challenge and present an architecture that enables both conditional mean and quantile prediction of the block maxima. The extensive experiments performed on both real-world and synthetic data demonstrated the superiority of DeepExtrema compared to other baseline methods. 

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4. A Computationally Efficient Method for Learning Exponential Family Distributions

   Shah, A; Shah, D; Wornell, G. W. (December 2021, Advances in neural information processing systems)

   We consider the question of learning the natural parameters of a k parameter \textit{minimal} exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the natural parameters are appropriately bounded. While the traditional maximum likelihood estimator for this class of exponential family is consistent, asymptotically normal, and asymptotically efficient, evaluating it is computationally hard. In this work, we propose a computationally efficient estimator that is consistent as well as asymptotically normal under mild conditions. We provide finite sample guarantees to achieve an l2 error of α in the parameter estimation with sample complexity O(poly(k/α)) and computational complexity O(poly(k/α)). To establish these results, we show that, at the population level, our method can be viewed as the maximum likelihood estimation of a re-parameterized distribution belonging to the same class of exponential family. 

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5. Catalytic prior distributions with application to generalized linear models

   https://doi.org/10.1073/pnas.1920913117  

   Huang, Dongming; Stein, Nathan; Rubin, Donald B.; Kou, S. C. (June 2020, Proceedings of the National Academy of Sciences)

   A catalytic prior distribution is designed to stabilize a high-dimensional “working model” by shrinking it toward a “simplified model.” The shrinkage is achieved by supplementing the observed data with a small amount of “synthetic data” generated from a predictive distribution under the simpler model. We apply this framework to generalized linear models, where we propose various strategies for the specification of a tuning parameter governing the degree of shrinkage and study resultant theoretical properties. In simulations, the resulting posterior estimation using such a catalytic prior outperforms maximum likelihood estimation from the working model and is generally comparable with or superior to existing competitive methods in terms of frequentist prediction accuracy of point estimation and coverage accuracy of interval estimation. The catalytic priors have simple interpretations and are easy to formulate. 

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