A key problem in the application of Archimedean Copulas istheselection oftheparametric formofϕaswellasthelimitations onthe expressiveness of commonly used copula.

A central problem in machine learning and statistics is to model joint densities of random variables from data. Copulas are joint cumulative distribution functions with uniform marginal distributions and are used to capture interdependencies in isolation from marginals. Copulas are widely used within statistics, but have not gained traction in the context of modern deep learning. In this paper, we introduce ACNet, a novel differentiable neural network architecture that enforces structural properties and enables one to learn an important class of copulas--Archimedean Copulas. Unlike Generative Adversarial Networks, Variational Autoencoders, or Normalizing Flow methods, which learn either densities or the generative process directly, ACNet learns a generator of the copula, which implicitly defines the cumulative distribution function of a joint distribution. We give a probabilistic interpretation of the network parameters of ACNet and use this to derive a simple but efficient sampling algorithm for the learned copula. Our experiments show that ACNet is able to both approximate common Archimedean Copulas and generate new copulas which may provide better fits to data.

Kernel Stein discrepancies (KSDs) have become a principal tool for goodness-of-fit testing, but standard KSDs are often insensitive to higher-order dependency structures, such as tail dependence, which are critical in many scientific and financial domains. We address this gap by introducing the Copula-Stein Discrepancy (CSD), a novel class of discrepancies tailored to the geometry of statistical dependence. By defining a Stein operator directly on the copula density, CSD leverages the generative structure of dependence, rather than relying on the joint density's score function. For the broad class of Archimedean copulas, this approach yields a closed-form Stein kernel derived from the scalar generator function. We provide a comprehensive theoretical analysis, proving that CSD (i) metrizes weak convergence of copula distributions, ensuring it detects any mismatch in dependence; (ii) has an empirical estimator that converges at the minimax optimal rate of $O_P(n^{-1/2})$; and (iii) is provably sensitive to differences in tail dependence coefficients. The framework is extended to general non-Archimedean copulas, including elliptical and vine copulas. Computationally, the exact CSD kernel evaluation scales linearly in dimension, while a novel random feature approximation reduces the $n$-dependence from quadratic $O(n^2)$ to near-linear $\tilde{O}(n)$, making CSD a practical and theoretically principled tool for dependence-aware inference.

This is the most common metric in the machine learning community.

Classical estimators, the cornerstones of statistical inference, face insurmountable challenges when applied to important emerging classes of Archimedean copulas. These models exhibit pathological properties, including numerically unstable densities, non-monotonic parameter-to-dependence mappings, and vanishingly small likelihood gradients, rendering methods like Maximum Likelihood (MLE) and Method of Moments (MoM) inconsistent or computationally infeasible. We introduce IGNIS, a unified neural estimation framework that sidesteps these barriers by learning a direct, robust mapping from data-driven dependency measures to the underlying copula parameter theta. IGNIS utilizes a multi-input architecture and a theory-guided output layer (softplus(z) + 1) to automatically enforce the domain constraint theta_hat >= 1. Trained and validated on four families (Gumbel, Joe, and the numerically challenging A1/A2), IGNIS delivers accurate and stable estimates for real-world financial and health datasets, demonstrating its necessity for reliable inference in modern, complex dependence models where traditional methods fail.

Conventional survival metrics, such as Harrell's concordance index and the Brier Score, rely on the independent censoring assumption for valid inference in the presence of right-censored data. However, when instances are censored for reasons related to the event of interest, this assumption no longer holds, as this kind of dependent censoring biases the marginal survival estimates of popular nonparametric estimators. In this paper, we propose three copula-based metrics to evaluate survival models in the presence of dependent censoring, and design a framework to create realistic, semi-synthetic datasets with dependent censoring to facilitate the evaluation of the metrics. Our empirical analyses in synthetic and semi-synthetic datasets show that our metrics can give error estimates that are closer to the true error, mainly in terms of predictive accuracy.

In survival analysis, subjects often face competing risks; for example, individuals with cancer may also suffer from heart disease or other illnesses, which can jointly influence the prognosis of risks and censoring. Traditional survival analysis methods often treat competing risks as independent and fail to accommodate the dependencies between different conditions. In this paper, we introduce HACSurv, a survival analysis method that learns Hierarchical Archimedean Copulas structures and cause-specific survival functions from data with competing risks. HACSurv employs a flexible dependency structure using hierarchical Archimedean copulas to represent the relationships between competing risks and censoring. By capturing the dependencies between risks and censoring, HACSurv achieves better survival predictions and offers insights into risk interactions. Experiments on synthetic datasets demonstrate that our method can accurately identify the complex dependency structure and precisely predict survival distributions, whereas the compared methods exhibit significant deviations between their predictions and the true distributions. Experiments on multiple real-world datasets also demonstrate that our method achieves better survival prediction compared to previous state-of-the-art methods.