Kolmogorov and Wasserstein Distances between Max-Stable Distributions

Technical Summary: Kolmogorov and Wasserstein Distances Between Max-Stable Distributions

Problem Statement
The paper addresses the fundamental problem of quantifying the distance between two $d$-dimensional max-stable distribution functions, $F_1$ and $F_2$, with unit-$\alpha$-Fréchet margins. This problem is central to both approximation theory and the statistical modeling of extremes. The authors seek to derive explicit comparison bounds for the Kolmogorov distance ($d_K$) and, where finite moments exist, the Wasserstein distance ($W_p$) between these laws. The bounds are expressed in terms of the underlying spectral or parametric representations, specifically focusing on de Haan representers, angular measures, and stable tail dependence functions.

Methodology
The analysis relies on the spectral representation of max-stable laws and the homogeneity of their exponent functions. The core methodology involves three distinct approaches:

1. Reduction to Homogeneous Cross-Sections: The Kolmogorov distance is reduced to a uniform comparison of exponent functions $V_k(x) = -\log F_k(x)$ on a homogeneous cross-section (specifically, the set where $\min_j x_j = 1$). The authors utilize a scalar function $\omega(t)$, defined via the supremum of $|e^{-q} - e^{-(1+t)q}|$, to bound the distance between the distributions based on the difference in their exponent functions.
2. Coupling and Transport Costs:
   • For Kolmogorov bounds, the authors relate the distance between distributions to the Wasserstein distance between powered de Haan representers $(Z^{(k)})^\alpha$ and the total variation distance between their angular measures $H_k$.
   • For Wasserstein bounds between the max-stable laws themselves, the paper employs a synchronous de Haan–LePage coupling. This involves coupling the underlying de Haan representers $Z^{(k)}$ and utilizing the properties of the LePage series representation to bound the transport cost between the resulting max-stable laws.
3. Inf–Argmax Decomposition: The exponent functions are decomposed using a maximiser-weight map $\Psi_j(x)$, allowing the distance bounds to be expressed directly in terms of model parameters for specific parametric families.

Key Contributions and Results

• Kolmogorov Bounds via Representers and Angular Measures:
  The paper establishes that the Kolmogorov distance is bounded by the function $\omega$ applied to the minimum of two quantities: the Wasserstein distance between the powered de Haan representers and the total variation distance between the angular measures.
  $$d_K(F_1, F_2) \leq \omega(W_\alpha \wedge M_\infty \|H_1 - H_2\|_{TV}) \leq \frac{1}{e}(W_\alpha \wedge M_\infty \|H_1 - H_2\|_{TV})$$
  A significant technical contribution is the coefficient $M_\infty = \sup_{s \in S} \|s\|_\infty^\alpha$. When using the positive $\ell_\alpha$-sphere (where $\sum s_j^\alpha = 1$), $M_\infty = 1$. Consequently, for the unnormalised angular measures used in this framework, the coefficient multiplying the setwise angular total-variation distance contains no explicit dimension factor, contrasting with previous results that included dimension-dependent constants.

• Wasserstein Bounds via Synchronous Coupling:
  For $1 \leq p < \alpha$, the paper derives bounds for the $p$-Wasserstein distance between the max-stable laws in terms of the $\alpha$-Wasserstein transport cost between their unpowered de Haan representers:
  $$W_{p, \|\cdot\|_\infty}(F_1, F_2) \leq \Gamma\left(1 - \frac{p}{\alpha}\right)^{1/p} W_{\alpha, \infty}(F_1, F_2)$$
  This result complements existing bounds by utilizing a synchronous coupling mechanism that transports the raw representers rather than their powered versions.

• Fixed Copula and Varying Fréchet Index:
  The authors analyze the case where the extreme-value copula (and thus the stable tail dependence function) is fixed, but the Fréchet index $\alpha$ varies.

  • They provide two-sided Kolmogorov bounds depending only on the one-dimensional Fréchet comparison.
  • They derive an exact formula for the $\ell_1$-Wasserstein distance when $p=1$ and $\alpha_1, \alpha_2 > 1$:
    $$W_{1, \|\cdot\|_1}(F_{\alpha_1}, F_{\alpha_2}) = d \int_0^\infty |t^{-1/\alpha_1} - t^{-1/\alpha_2}| e^{-t} dt$$
    This demonstrates that in the fixed-copula setting, the Wasserstein distance is controlled entirely by the marginal transformation.

• Applications to Specific Models:
  The general bounds are specialized to:

  • Archimax and Clustered Archimax Copulas: Bounds are derived in terms of the generator $\psi$ and the discrepancy of the stable tail dependence functions.
  • Brown–Resnick and Hüsler–Reiss Models: The paper provides explicit bounds based on the covariance matrices of the underlying Gaussian processes. Specifically, for Hüsler–Reiss models, the Kolmogorov distance is bounded by the discrepancy of the $\Psi$-functions derived from the variogram parameters, offering a representation-invariant comparison.

Significance and Claims
The paper claims to provide a unified framework for comparing max-stable distributions that avoids explicit dimension factors in the angular measure bounds when using the $\ell_\alpha$-sphere. By linking the Kolmogorov distance to the Wasserstein distance of representers and the total variation of angular measures, the work offers a "common mechanism" for deriving Kolmogorov estimates. The derivation of an exact $\ell_1$-Wasserstein formula for varying Fréchet indices and the application to complex structures like clustered Archimax copulas and Hüsler–Reiss models extend the utility of these bounds to practical parametric families used in extreme value theory. The results are presented as intrinsic bounds that facilitate the comparison of laws with varying stability indices and angular structures without relying on ad-hoc dimension-dependent constants.