Non-parametric estimation of copula density functions presents significant challenges. One issue is the unboundedness of certain copula density functions and their derivatives at the corners of the unit square. Another is the boundary bias inherent in kernel density estimation. This paper presents a kernel-based method for estimating bounded copula density functions, addressing boundary bias through the mirror reflection technique. Optimal smoothing parameters are derived via Asymptotic Mean Integrated Squared Error (AMISE) minimization and cross-validation, with theoretical guarantees of consistency and asymptotic normality. Two kernel smoothing strategies are proposed: the rule-of-thumb approach and least squares cross-validation (LSCV). Simulation studies highlight the efficacy of the rule-of-thumb method in bandwidth selection for copulas with unbounded marginal supports. The methodology is further validated through an application to the Wisconsin Breast Cancer Diagnostic Dataset (WBCDD), where LSCV is used for bandwidth selection.

We aim to establish estimation bounds for various divergences, including total variation, Kullback-Leibler (KL) divergence, Hellinger divergence, and Pearson χ ² divergence, within the GAN estimator. We derive an inequality based on empirical and population objective functions of the GAN model, achieving almost surely convergence rates. Subsequently, this inequality was employed to derive estimation bounds for total variation, Kullback-Leibler (KL) divergence, Hellinger divergence, and Pearson χ ² divergence, leading to almost surely convergence rates and differences between the expected outputs of the discriminator on real data and generated data. Our study demonstrates better results compared to some existing ones, which are a specific case of the general objective function.

The Information Maximizing Generative Adversarial Network (InfoGAN) can be formulated as a minimax problem involving a generator and a discriminator, augmented by a mutual information regularization term. Despite strong empirical performance, rigorous generalization guarantees for InfoGAN-type objectives remain limited, particularly when additional structural components are introduced. In this paper, we study an InfoGAN-inspired adversarial framework obtained by removing the latent code component and introducing an explicit regularization term on the generator, yielding an analytically tractable generator-regularized adversarial objective. We establish generalization error bounds by analyzing the gap between empirical and population objective functions using Rademacher complexity arguments for the discriminator, the generator, and their composition. The resulting bounds reveal explicit n ^−1/2 and m ^−1/2 decay rates with respect to the discriminator and generator sample sizes and clarify the role of the generator regularization parameter. The theory is further specialized to two-layer neural networks with Lipschitz continuous and non-decreasing activation functions, where explicit entropy-based complexity bounds are derived. Experiments on the CIFAR-10 dataset validate the predicted scaling behavior and demonstrate that the generalization gap decreases systematically as sample size increases, highlighting the stabilizing effect of generator regularization. Overall, this work provides one of the first rigorous generalization analyses for an InfoGAN-inspired adversarial objective with explicit generator regularization.

Nonparametric estimation of copula density functions using kernel estimators presents significant challenges. One issue is the potential unboundedness of certain copula density functions at the corners of the unit square. Another is the boundary bias inherent in kernel density estimation. This paper presents a kernel-based method for estimating bounded copula density functions, addressing boundary bias through the mirror-reflection technique. Optimal smoothing parameters are derived via Asymptotic Mean Integrated Squared Error (AMISE) minimization and cross-validation, with theoretical guarantees of consistency and asymptotic normality. Two kernel smoothing strategies are proposed: the rule-of-thumb approach and least squares cross-validation (LSCV). Simulation studies highlight the efficacy of the rule-of-thumb method in bandwidth selection for copulas with unbounded marginal supports. The methodology is further validated through an application to the Wisconsin Breast Cancer Diagnostic Dataset (WBCDD), where LSCV is used for bandwidth selection.