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A class of skew-multivariate distributions for spatial data

Pavel Krupskii

TL;DR

This paper develops a new class of skew‑multivariate copula models for spatial data built from Pareto‑mixture constructions, enabling joint modeling of bulk dependence and rich tail behavior, including tail dependence and asymptotic independence as well as permutation asymmetry. The core framework expresses $X_i = Z_i P^{1/ u_i}$, combining a bulk process $oldsymbol{Z}$ with a heavy‑tailed factor $P$, and yields tractable likelihoods in several special cases. The authors derive the stable tail dependence function and extremal copulas, present special cases with closed‑form densities (notably grouped‑$t$ and skew‑multivariate variants), and propose IFM and semiparametric margins for scalable inference. Through extensive simulations and a Oklahoma temperature study, the models demonstrate robust finite‑sample performance and superior tail capturing relative to Gaussian or symmetry‑restricted copulas, with tangible improvements in spatial interpolation on extreme days. The work suggests a practical path for flexible, tail‑aware spatial modeling in environmental and geostatistical applications.

Abstract

This paper introduces a class of copula models for spatial data, based on multivariate Pareto-mixture distributions. We explore the tail properties of these models, demonstrating their ability to capture both tail dependence and asymptotic independence, as well as the tail asymmetry frequently observed in real-world data. The proposed models also offer flexibility in accounting for permutation asymmetry and can effectively represent both the bulk and extreme tails of the distribution. We consider special cases of these models with computationally tractable likelihoods and present an extensive simulation study to assess the finite-sample performance of the maximum likelihood estimators. Finally, we apply our models to analyze a temperature dataset, showcasing their practical utility.

A class of skew-multivariate distributions for spatial data

TL;DR

This paper develops a new class of skew‑multivariate copula models for spatial data built from Pareto‑mixture constructions, enabling joint modeling of bulk dependence and rich tail behavior, including tail dependence and asymptotic independence as well as permutation asymmetry. The core framework expresses , combining a bulk process with a heavy‑tailed factor , and yields tractable likelihoods in several special cases. The authors derive the stable tail dependence function and extremal copulas, present special cases with closed‑form densities (notably grouped‑ and skew‑multivariate variants), and propose IFM and semiparametric margins for scalable inference. Through extensive simulations and a Oklahoma temperature study, the models demonstrate robust finite‑sample performance and superior tail capturing relative to Gaussian or symmetry‑restricted copulas, with tangible improvements in spatial interpolation on extreme days. The work suggests a practical path for flexible, tail‑aware spatial modeling in environmental and geostatistical applications.

Abstract

This paper introduces a class of copula models for spatial data, based on multivariate Pareto-mixture distributions. We explore the tail properties of these models, demonstrating their ability to capture both tail dependence and asymptotic independence, as well as the tail asymmetry frequently observed in real-world data. The proposed models also offer flexibility in accounting for permutation asymmetry and can effectively represent both the bulk and extreme tails of the distribution. We consider special cases of these models with computationally tractable likelihoods and present an extensive simulation study to assess the finite-sample performance of the maximum likelihood estimators. Finally, we apply our models to analyze a temperature dataset, showcasing their practical utility.
Paper Structure (20 sections, 7 theorems, 78 equations, 6 figures, 3 tables)

This paper contains 20 sections, 7 theorems, 78 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Pareto-mixture distributions and their tail properties
  3. The model and its copula
  4. Tail properties
  5. Special cases
  6. Grouped-$t$ distribution
  7. Skew-multivariate distribution with lower and upper tail dependence
  8. Skew-multivariate distribution with flexibility in the upper tail
  9. Inference
  10. Likelihood function
  11. Models with tractable likelihoods
  12. Simulation studies
  13. Real data application
  14. Model fitting
  15. Spatial interpolation
  16. ...and 5 more sections

Key Result

Proposition 1

Consider the random vector $\mathbf{X}$ as given in main-eq. Let $F_{\mathbf{Z}}$ be the joint CDF of $\mathbf{Z}$ and assume there exists $\epsilon > 0$ such that the marginal survival function $\bar{F}_{Z_i}(z) = o(z^{-(1+\epsilon)\nu_i})$ for $i \in \{1,\ldots, d\}$ as $z \to \infty$. Under these where $\zeta_i = \int_0^{\infty}\bar{F}_{Z_i}(y^{1/\nu_i}) \mathrm{d} y.$

Figures (6)

  • Figure 1: Plots of the Pickands dependence function $A(t)$ for $\rho = 0.5$ and $0 < t < 1$ (left), and the upper tail dependence coefficient $\lambda_U(\rho)$ for $0 < \rho <1$ (right), for the grouped-$t$ distribution with parameters $\Omega_{12} = \rho$ and three choice of degrees of freedom: $\nu_1 = \nu_2 = 4$ (black), $\nu_1 = 5, \nu_2 = 10$ (red), and $\nu_1 = 1, \nu_2 = 3$ (green).
  • Figure 2: Top row: Pickands dependence function $A(t)$ for $\rho = 0.5$ and $0 < t < 1$ for the pairs $(W_1,W_2)^{\top}$ (left) and $(-W_1,-W_2)^{\top}$. Bottom row: upper and lower tail dependence coefficients, $\lambda_U(\rho)$ and $\lambda_L(\rho)$ for $0 < \rho < 1$ (left and right, respectively) for model \ref{['eq-skewmodel1']} with parameters $\Omega_{12} = \rho, \tau = 0, \alpha_U = 2, \alpha_U = 0.8$, and $\alpha_1 = \alpha_2 = 0$ (black), $\alpha_1 = 0, \alpha_2 = 3$ (red), and $\alpha_1 = -3, \alpha_2 = 3$ (green).
  • Figure 3: Plots of the Pickands dependence function for $\rho = 0.5$ and $0 < t < 1$ (left) and the upper tail dependence coefficient $\lambda_U(\rho)$ for $0 < \rho < 1$ (right), for model \ref{['eq-skewmodel2']} with parameters $\Omega_{12} = \rho, \ \alpha_1 = \alpha_2 = 1$, and $\beta_1 = \beta_2 = 0.3$ (black), $\beta_1 = 0.3, \beta_2 = 0.9$ (red), and $\beta_1 = 0.1, \beta_2 = 0.95$ (green).
  • Figure 4: Autocorrelation plots for 6 stations with Ljung-Box test p-values below 0.01.
  • Figure 5: Normal scores scatter plots of the estimated residuals for some pairs of stations.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Corollary 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Lemma 1