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Parsimonious Compactly Supported Covariance Models in the Gauss Hypergeometric Class: Identifiability, Reparameterizations, and Asymptotic Properties

Moreno Bevilacqua, Christian Caamaño-Carrillo, Tarik Faouzi, Xavier Emery

Abstract

We study covariance functions in the Gauss hypergeometric ($\mathcal{GH}$) class, a flexible family that encompasses the Generalized Wendland ($\mathcal{GW}$) and Matérn ($\mathcal{MT}$) models. We derive sharp validity conditions, providing a complete characterization of the admissible parameter space, and show that the model exhibits structural identifiability issues under both increasing- and fixed-domain asymptotics. To resolve this issue, we introduce a parsimonious compactly supported subclass selected via a maximum integral range criterion. The resulting hypergeometric model can be viewed as a structural refinement of the $\mathcal{GW}$ family and admits compact-support reparameterizations that recover the $\mathcal{MT}$ model as a limit case. We further establish strong consistency and asymptotic normality of the maximum likelihood estimator of the associated microergodic parameter under fixed-domain asymptotics. Simulation experiments and a real-data application to climate data illustrate the finite-sample behavior and practical performance of the proposed model.

Parsimonious Compactly Supported Covariance Models in the Gauss Hypergeometric Class: Identifiability, Reparameterizations, and Asymptotic Properties

Abstract

We study covariance functions in the Gauss hypergeometric () class, a flexible family that encompasses the Generalized Wendland () and Matérn () models. We derive sharp validity conditions, providing a complete characterization of the admissible parameter space, and show that the model exhibits structural identifiability issues under both increasing- and fixed-domain asymptotics. To resolve this issue, we introduce a parsimonious compactly supported subclass selected via a maximum integral range criterion. The resulting hypergeometric model can be viewed as a structural refinement of the family and admits compact-support reparameterizations that recover the model as a limit case. We further establish strong consistency and asymptotic normality of the maximum likelihood estimator of the associated microergodic parameter under fixed-domain asymptotics. Simulation experiments and a real-data application to climate data illustrate the finite-sample behavior and practical performance of the proposed model.

Paper Structure

This paper contains 23 sections, 13 theorems, 133 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background: the Matérn and Generalized Wendland correlation models
  3. The Gauss hypergeometric model and the identifiability problem
  4. Fixing the identifiability problem: the Hypergeometric correlation model
  5. Some special cases and bridges to the Matérn model
  6. Asymptotic properties of the maximum likelihood estimator of the Hypergeometric model
  7. A simulation study
  8. A real data application
  9. Concluding Remarks
  10. Proofs
  11. Proof of Theorem \ref{['Wv_vs_Wv']}
  12. Proof of Theorem \ref{['theopp']}
  13. Proof of Theorem \ref{['theopkkp']}
  14. Proof of Theorem \ref{['theopkkp2']}
  15. Proof of Theorem \ref{['theopkkp333']}
  16. ...and 8 more sections

Key Result

Theorem 1

Let $P(\sigma_i^2{\cal GH}_{\delta,\beta_i,\gamma_i,a_i})$, $i=0, 1$, be two zero-mean Gaussian measures and let $\delta>\frac{d}{2}$. If $\frac{d+1}{2}+3\delta<\min(\beta_1+\gamma_1,\beta_0+\gamma_0),$ for any bounded infinite set $D\subset \mathbb{R}^d$, $d=1,2,3$, the Gaussian measures $P(\sigma_ where $L(\beta_i,\gamma_i)=\dfrac{2^{2\delta-d}\Gamma(\beta_i-d/2)\Gamma(\gamma_i-d/2)}{\Gamma(\gam

Figures (6)

  • Figure 1: Left part: ${\cal GW}_{0,\mu,1}$ for $\mu=\mu_1(\frac{1}{2}),3,10,100$ with $\delta=\frac{d+1}{2}+\kappa$ and $\kappa=0$. Right part: the same Figure but with $\kappa=1$. In both cases $d=1$.
  • Figure 2: Left part: $\mathcal{GH}_{\delta,\delta+\frac{\mu(l) }{2},\delta+\frac{\mu(l) }{2}+l,1}$ with $\delta=\frac{d+1}{2}+\kappa$, $\kappa=0$, $\mu(l) =\delta-l+\frac{1}{2}$ and $l=0, \frac{1}{2}, \frac{d}{2}+\kappa$ (black, red and blue lines respectively). Right part: the same Figure but with $\kappa=1$. The case $l=\frac{1}{2}$ (red line) corresponds to the $\mathcal{GW}_{\kappa,\mu,1}$ model. The blue line corresponds to the correlation model that maximizes the integral range. In both cases $d=2$.
  • Figure 3: Two Gaussian RF realizations with ${\cal H}_{\kappa,4,0.2,2}$ correlation model, for $\kappa = 0, 1$ (from left to right).
  • Figure 4: Left part: Boxplots of ML estimates of $\sigma^2$, $a$ and the microergodic parameter $\frac{\sigma^2}{a^{2\kappa+1}}$ (from left to right) when estimating the covariance model $\sigma^2\mathcal{H}_{\kappa,\mu,a,d}$ with $\sigma^2=1$, for $\mu=4,6,8$ and $a=0.1$ (increasing domain scenario). Right part: the same Figure with $a=0.5$ (fixed domain scenario). In both cases $\kappa=0$.
  • Figure 5: Left: spatial map of residuals. Right: empirical semivariogram.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 12 more