componentwise probability integral transform

This study addresses goodness-of-fit testing for multivariate distributions, focusing on uniformity, normality, spherical and elliptical symmetry, and independence. It introduces a novel approach based on the decomposition of Brownian sheets. Under the null hypothesis, the problem is transformed via probability integral transforms into testing uniformity over the unit hypercube, and interactions among coordinates are effectively disentangled using a zero-margin measure decomposition. The method uniquely integrates the Gaussian process decomposition of Brownian sheets with empirical distribution functions, substantially enhancing sensitivity to joint dependence structures. Simulation studies demonstrate that the proposed test achieves power comparable to or exceeding that of current state-of-the-art methods, particularly excelling in detecting complex dependency patterns.

Conventional copula models often lack sufficient flexibility in capturing complex dependence structures. Method: This paper introduces a class of univariate W-transforms that preserve uniformity—constructed via distribution functions and piecewise strictly monotonic functions on [0,1]—ensuring transformed margins remain standard uniform. These transforms naturally induce copula-to-copula mappings, yielding W-transformed copulas. Contribution/Results: Theoretically, we derive closed-form expressions, establish conditions for density existence, and characterize rank correlations (Spearman’s ρ, Kendall’s τ), tail dependence coefficients, and symmetry properties. Methodologically, we propose an interpretable parametric family enabling independent control over central and tail dependence. Empirical results demonstrate that W-transformed copulas significantly improve fit to intricate real-world dependence patterns, particularly asymmetric tail dependence.

This paper addresses inference on general parameter transformations of cumulative distribution functions (CDFs) in the presence of nuisance parameters. We propose a unified, nonparametric, asymptotically size-controlled testing framework applicable to joint inference on one-, two-, and multi-sample CDFs. The method constructs test statistics via numerical bootstrap, obviating analytical critical value derivation and ensuring implementation simplicity. We establish theoretical guarantees of asymptotic size control and consistency. Monte Carlo simulations and empirical analyses demonstrate strong finite-sample robustness and high statistical power. Our key contribution is the first unified, nonparametric, asymptotically valid, and structure-free test for transformations of CDFs with nuisance parameters—overcoming the restrictive functional-form and parametric-structure assumptions inherent in conventional approaches.

This paper addresses joint testing of the mean vector and covariance matrix under a uniform block structure in high-dimensional data with missing observations. Method: We develop the first unified statistical inference framework accommodating missing data, introducing a novel block-wise Hadamard product representation for uniformly structured block matrices. This enables closed-form expressions for the likelihood ratio and information statistics, along with their exact null distributions. We further propose an FDP-controlled simultaneous marginal mean testing procedure. Contribution/Results: Theoretical analysis and extensive simulations demonstrate accurate distributional characterization of test statistics, robust and reliable FDP control, and strong robustness against perturbations in the covariance structure and arbitrary missingness mechanisms. The method is successfully applied to hypothesis testing in high-dimensional neuroimaging data, substantially broadening the practical applicability of block-structured covariance models in high-dimensional inference.

This work addresses the longstanding limitation in conditional density estimation—namely, the absence of closed-form solutions for multivariate conditional densities under non-Gaussian assumptions. We propose a generative conditional density estimation framework grounded in copula modeling and analytic conditionalization in latent space. Methodologically, we first establish the inheritability of “conditional stability” under mixture and transformation operations, thereby extending analytically tractable conditional families to non-Gaussian, nonlinear, and cross-dimensional settings. The core components include a Gaussian Mixture Copula Model (GMCM), an explicit latent-space conditionalization mechanism, and joint copula modeling. Experiments on synthetic and real-world datasets demonstrate substantial improvements in conditional density estimation accuracy and robustness to missing data imputation. Crucially, our approach enables efficient, differentiable, and sampling-free deterministic conditional inference.

This study addresses the limited power of traditional goodness-of-fit tests in high-dimensional settings by proposing a novel approach based on the joint distribution of multiple samples. The method employs principal component analysis for dimensionality reduction and constructs confidence sets using k-nearest neighbor estimates of high-density regions, effectively integrating information from order statistics, empirical distribution function values, and moments. It is further extended to the two-sample case. To enhance test power, the approach incorporates non-uniform transformations and probability integral transforms, with inference carried out via permutation testing. Simulation studies demonstrate that the proposed method outperforms or matches existing classical and graphical tests across a range of alternative hypotheses, substantially improving both power and applicability in high-dimensional scenarios.

This study addresses the limitation of conventional global goodness-of-fit tests in multivariate settings, which often fail to pinpoint localized model misspecifications. To overcome this, the authors propose a local calibration test based on adaptive partitioning via Beta-trees. Departing from single-statistic global frameworks, the method evaluates whether predicted probabilities fall within finite-sample confidence intervals across data-driven subregions, enabling precise identification and visualization of model inadequacies. By leveraging k-means clustering to generate null distributions and constructing rigorous confidence intervals, the approach effectively detects local deviations in both simulated and real-world datasets, demonstrating superior performance in tasks such as selecting the number of components in mixture models.

This study addresses the lack of theoretically grounded goodness-of-fit tests for high-dimensional independent component analysis, particularly when the data dimension grows proportionally with the sample size. The authors propose a novel goodness-of-fit test that eliminates the need for pre-whitening—a longstanding requirement in conventional approaches—and establish, for the first time, a theoretically valid testing procedure within a high-dimensional asymptotic framework. The test statistic is constructed based on high-dimensional asymptotic theory, and extensive numerical simulations demonstrate its reliable size control and strong power across various settings. Empirical analysis further highlights the method’s practical diagnostic potential in real-world applications, such as gene expression data analysis.

This study addresses the challenges of low statistical power and difficulty in level control arising from subtle element-wise differences and sample dependence in testing the equality of high-dimensional covariance matrices. To overcome these issues, this work proposes a studentized L2-type statistic combined with a zero-preserving transformation, introducing a nonsingular linear transformation to incorporate structural information without dimension reduction. Furthermore, it employs a Gaussian multiplier bootstrap to approximate the null distribution, thereby avoiding the construction of the full covariance matrix, and establishes the asymptotic invariance when replacing population transformations with their sample estimates. The contributions include deriving finite-sample Gaussian approximation bounds, demonstrating substantially improved test size accuracy and power under weak dependence, and validating the method's effectiveness through breast cancer gene expression data.

Existing methods for testing (conditional) independence among non-Euclidean random objects struggle to balance geometric flexibility with theoretical tractability and cannot accommodate object-valued conditioning variables. This work proposes Distance Profile Embedding (DPE), a novel approach that maps random objects in general metric spaces into a Hilbert space of square-integrable functions, thereby establishing a unified framework for both marginal and conditional independence testing. DPE is the first method capable of handling object-valued conditioning variables without requiring isometric embeddings or bijective correspondence assumptions. It further provides a closed-form asymptotic null distribution, enabling analytical p-value computation. Both theoretical analysis and empirical evaluations demonstrate that DPE achieves strong performance and practical utility across synthetic data as well as real-world applications, including gut microbiome compositions and global human mortality distributions.