quantile mapping

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 work investigates the propagation of perturbations induced by additive noise in one-dimensional translated density signals within the cumulative distribution transform (CDT) domain, along with the associated displacement recovery problem. Leveraging the linearization property of CDT with respect to translations, the authors derive for the first time a first-order expansion of noise-induced perturbations in the CDT domain, revealing an amplification effect in low-density regions and establishing an explicit covariance structure. Building on this analysis, they propose a unified framework for displacement recovery applicable to both density and signed signals, which jointly estimates the template and displacements via projection or alternating alignment strategies. Theoretical analysis and numerical experiments demonstrate that the proposed method achieves accurate and stable displacement recovery even in the presence of noise.

This work addresses the limited robustness of conventional mean–standard deviation or min–max normalization in financial time-series modeling. We propose the first integration of cumulative distribution function (CDF)-based quantile normalization—i.e., (x mapsto F(x))—into Kolmogorov–Arnold Networks (KANs), leveraging its well-established use in finance. Building upon Legendre-KAN within the Hybrid Coordinate Representation (HCR) framework, we incorporate empirical CDF normalization to map inputs robustly onto the ([0,1]) uniform interval. This yields probabilistically interpretable neuron weights—e.g., quantile sensitivity—and inherently supports distributional propagation and directional control of backpropagation. Empirical evaluation demonstrates that merely substituting the normalization layer significantly improves forecasting accuracy, reduces overfitting, and enhances modeling of local joint distributions and mixed moment estimation. The approach establishes a novel, interpretable, and distribution-aware paradigm for KANs in financial machine learning.

To address feature instability in few-shot image recognition caused by measurement-induced affine transformations, this paper proposes the Generalized Normalized Radon Cumulative Distribution Transform (GN-RCDT). GN-RCDT extends the R-CDT to multidimensional and non-Euclidean spaces via a family of generalized normalization strategies, achieving invariance to arbitrary affine transformations; theoretical analysis establishes its geometric invariance and linear separability. Integrating Wasserstein transport theory, generalized Radon transforms, and sliced Wasserstein distance, the framework unifies representation learning for 2D images, 3D shapes, and 3D rotation matrices. Experiments demonstrate that GN-RCDT achieves near-perfect classification accuracy and clustering performance under limited data, significantly enhancing generalization and robustness. It is the first work to systematically realize invariant feature learning with R-CDT in non-Euclidean and high-dimensional settings.

This paper addresses the quantile spectrum and cross-spectrum—nonstationary frequency-domain features defined over the two-dimensional domain of frequency and quantile level—originally proposed by Li (2012, 2014), and introduces the first nonparametric estimation framework for them. Methodologically, it constructs the quantile discrete Fourier transform (QDFT) and quantile spectral sequences (QSER) via trigonometric quantile regression; spectral estimators are then built from the autocovariance function of QSER using windowing techniques, augmented by novel inter-quantile smoothing to enhance estimation stability. The main contributions are: (i) establishing the first rigorous theoretical framework for QDFT–QSER, enabling fully nonparametric modeling of quantile spectra; and (ii) delivering an estimator that, in simulations, achieves superior statistical accuracy and robustness compared to the classical L-W estimator—particularly through substantial variance reduction—thereby providing a generalizable tool for quantile-based spectral analysis.

This study addresses the deviation of empirical probability integral transforms (PIT) from the theoretical uniform distribution under finite samples, a phenomenon induced by the two-stage sampling structure that invalidates conventional one-sample uniformity tests. The work systematically demonstrates that this non-uniformity arises from dependence structures and variance distortions introduced either by a reference sample or a rolling window: the former converges to a two-sample Kolmogorov–Smirnov distribution, while the latter exhibits temporal autocorrelation. Building on probability integral transform theory, empirical quantile estimation, and two-sample KS asymptotics, this paper establishes—for the first time—that empirical PIT values cannot be treated as independent uniform random variables. Leveraging these insights, the authors develop a corrected statistical inference framework specifically tailored for backtesting forecast calibration.

This study addresses nonparametric distribution estimation from limited quantile observations corrupted by noise, under unknown distributional structure. The authors propose a two-step estimator that incorporates shape constraints on the hazard rate—such as increasing failure rate—by first solving a finite-dimensional convex optimization problem over transformation knots and then reconstructing the cumulative distribution function via shape-preserving interpolation. This work is the first to integrate hazard-rate-based shape constraints with noisy quantile data, establishing a computationally tractable nonparametric framework extendable to various hazard-related properties. Finite-sample error bounds and convergence rates are rigorously derived. Empirical results demonstrate substantial improvements in estimation accuracy and downstream decision quality in revenue management and reliability analysis, offering practical guidance for offline data collection.

This study addresses the unification of calibration concepts across classification and regression tasks, aiming to ensure consistency between predicted distributions and observed outcomes for diverse data types—continuous, discrete, nominal, and binary. The work introduces modal calibration for nominal outcomes and establishes a hierarchical framework distinguishing full, partial, and average calibration. It proposes a generalized definition of calibration based on predictive distribution functionals—such as means, quantiles, and event probabilities—and leverages probability integral transforms alongside constructive algorithms for analysis. Key contributions include demonstrating the logical independence between dual probability integral transform (PIT) calibration and existing discrete calibration notions, clarifying implication and independence relationships among various calibration types, and providing reproducible methods for generating illustrative examples and counterexamples.

This work addresses the challenge of constructing predictive intervals that simultaneously achieve finite-sample marginal coverage, conditional validity, and length efficiency under complex settings such as heteroscedasticity, response skewness, or model misspecification. The authors propose a novel paradigm that estimates the conditional cumulative distribution function (CDF) using neural networks, applies conformal calibration to the probability integral transform (PIT) values of this estimate, and constructs the shortest percentile interval in PIT space. The resulting method provides guaranteed finite-sample marginal coverage, asymptotically valid conditional coverage, and robustness to estimation errors in the conditional CDF. Empirical evaluations on diverse synthetic and real-world datasets demonstrate that the proposed approach significantly shortens prediction intervals while achieving superior conditional calibration compared to existing methods.

This work addresses the stability of sample quantiles under heavy-tailed distributions, where traditional approaches struggle to disentangle the coupled effects of perturbations in projection direction estimation and quantile thresholding. The paper introduces a Q–Q orthogonal decomposition framework that, for the first time, explicitly decomposes the estimation error into three distinct components: directional perturbation, empirical fluctuation along a fixed direction, and a Bahadur remainder term. By integrating Bahadur representation, empirical process theory, halfspace symmetric difference analysis, and Glivenko–Cantelli uniform convergence techniques, the proposed method achieves a refined characterization of local quantile stability without requiring global uniform convergence assumptions, thereby substantially sharpening the precision of stability bounds.