[Paper Review] A semi-parametric estimation for max-mixture spatial processes

Manaf Ahmed, Véronique Maume‐Deschamps+2 more|arXiv (Cornell University)|Oct 23, 2017

Spatial and Panel Data Analysis21 references3 citations

TL;DR

This paper proposes a semi-parametric estimation method for max-mixture spatial processes using the F-madogram to improve parameter estimation, especially in asymptotically independent scenarios. The method minimizes the squared difference between theoretical and empirical F-madograms, demonstrating superior performance over composite likelihood in near-asymptotic-independence settings, with strong empirical validation on East Australian rainfall data.

ABSTRACT

We proposed a semi-parametric estimation procedure in order to estimate the parameters of a max-mixture model and also of a max-stable model (inverse max-stable model) as an alternative to composite likelihood. A good estimation by the proposed estimator required the dependence measure to detect all dependence structures in the model, especially when dealing with the max-mixture model. We overcame this challenge by using the F-madogram. The semi-parametric estimation was then based on a quasi least square method, by minimizing the square difference between the theoretical F-madogram and an empirical one. We evaluated the performance of this estimator through a simulation study. It was shown that on an average, the estimation is performed well, although in some cases, it encountered some difficulties. We apply our estimation procedure to model the daily rainfalls over the East Australia.

Motivation & Objective

• To address the limitations of composite likelihood estimation in max-mixture spatial processes, particularly in estimating asymptotic independence (AI) components.
• To develop a semi-parametric alternative using the F-madogram as a dependence measure that captures both asymptotic dependence (AD) and AI behavior.
• To evaluate the performance of the F-madogram-based least squares estimator through simulation and real data application.
• To provide a computationally less intensive alternative to composite likelihood, especially avoiding the need for Godambe matrix computation.
• To select the best-fitting model using information criteria (MIC and CLIC) under both estimation frameworks.

Proposed method

• The method uses the F-madogram, a spatial dependence measure defined as $ \nu^F(h) = \mathbb{E}\left[ \left| \Lambda(F(X(s+h))) - \Lambda(F(X(s))) \right| \right] $, where $ \Lambda $ is the standard Fréchet distribution function.
• Theoretical F-madogram expressions are derived for max-mixture models combining a max-stable process (e.g., TEG, Brown-Resnick) and an asymptotically independent process (e.g., inverse Brown-Resnick, inverse Smith).
• A least squares estimation procedure minimizes the integrated squared difference between empirical and theoretical F-madograms over a set of spatial lags.
• Model selection is performed using the MIC (for LS-madogram) and CLIC (for composite likelihood) criteria, with smaller values indicating better fit.
• Data are transformed to unit Fréchet margins non-parametrically using empirical cumulative distribution functions to avoid parametric GEV fitting.
• The consistency of the LS-madogram estimator is proven under the condition that parameters are identifiable via the F-madogram.

Experimental results

Research questions

• RQ1 Can the F-madogram serve as a reliable and informative dependence measure for max-mixture spatial processes combining asymptotic dependence and asymptotic independence?
• RQ2 Does the least squares estimation based on the F-madogram outperform composite likelihood estimation in terms of accuracy and robustness, especially when the true process is near-asymptotically independent?
• RQ3 How do the LS-madogram and composite likelihood estimators compare in terms of computational cost and model selection performance on real environmental data?
• RQ4 Is the F-madogram-based estimator consistent and identifiable under the max-mixture model framework?
• RQ5 Which model structure (e.g., TEG + inverse Brown-Resnick) best fits East Australian daily rainfall data, as determined by information criteria under both estimation methods?

Key findings

• The LS-madogram estimator outperforms composite likelihood in simulations when the true model is near-asymptotically independent, particularly in estimating the AI component.
• For the East Australian rainfall data, both LS-madogram and composite likelihood selected the same best model (MM1), indicating robustness across methods.
• The MIC and CLIC criteria selected the same model (MM1) under both estimation procedures, validating model selection consistency.
• The LS-madogram method required significantly less computation than composite likelihood, as it avoids estimating the Godambe matrix.
• The F-madogram successfully captures dependence structure in max-mixture models, enabling joint estimation of AD and AI parameters through a single, interpretable measure.
• The empirical F-madogram showed strong agreement with the theoretical F-madogram under the best-fitting MM1 model, supporting the method's validity.