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We propose a robust method for constructing conditionally valid prediction intervals based on models for conditional distributions such as quantile and distribution regression. Our approach can be applied to important prediction problems, including cross-sectional prediction, k–step-ahead forecasts, synthetic controls and counterfactual prediction, and individual treatment effects prediction. Our method exploits the probability integral transform and relies on permuting estimated ranks. Unlike regression residuals, ranks are independent of the predictors, allowing us to construct conditionally valid prediction intervals under heteroskedasticity. We establish approximate conditional validity under consistent estimation and provide approximate unconditional validity under model misspecification, under overfitting, and with time series data. We also propose a simple “shape” adjustment of our baseline method that yields optimal prediction intervals.

Counterfactual distributions are important ingredients for policy analysis and decomposition analysis in empirical economics. In this article, we develop modeling and inference tools for counterfactual distributions based on regression methods. The counterfactual scenarios that we consider consist of ceteris paribus changes in either the distribution of covariates related to the outcome of interest or the conditional distribution of the outcome given covariates. For either of these scenarios, we derive joint functional central limit theorems and bootstrap validity results for regression-based estimators of the status quo and counterfactual outcome distributions. These results allow us to construct simultaneous confidence sets for function-valued effects of the counterfactual changes, including the effects on the entire distribution and quantile functions of the outcome as well as on related functionals. These confidence sets can be used to test functional hypotheses such as no-effect, positive effect, or stochastic dominance. Our theory applies to general counterfactual changes and covers the main regression methods including classical, quantile, duration, and distribution regressions. We illustrate the results with an empirical application to wage decompositions using data for the United States. As a part of developing the main results, we introduce distribution regression as a comprehensive and flexible tool for modeling and estimating the entire conditional distribution. We show that distribution regression encompasses the Cox duration regression and represents a useful alternative to quantile regression. We establish functional central limit theorems and bootstrap validity results for the empirical distribution regression process and various related functionals.

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterization of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterization thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood-based estimation for conditional transformation models that allows the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.

Quantile and quantile effect (QE) functions are important tools for descriptive and causal analysis due to their natural and intuitive interpretation. Existing inference methods for these functions do not apply to discrete random variables. This article offers a simple, practical construction of simultaneous confidence bands for quantile and QE functions of possibly discrete random variables. It is based on a natural transformation of simultaneous confidence bands for distribution functions, which are readily available for many problems. The construction is generic and does not depend on the nature of the underlying problem. It works in conjunction with parametric, semiparametric, and nonparametric modeling methods for observed and counterfactual distributions, and does not depend on the sampling scheme. We apply our method to characterize the distributional impact of insurance coverage on health care utilization and obtain the distributional decomposition of the racial test score gap. We find that universal insurance coverage increases the number of doctor visits across the entire distribution, and that the racial test score gap is small at early ages but grows with age due to socio-economic factors especially at the top of the distribution. Supplementary materials (additional results, R package, replication files) for this article are available online.

Visual object tracking using Siamese networks has proven effective by matching a reference target with candidate regions. However, their performance is limited by static templates, insufficient context modeling, and weak multi-level feature integration, especially under occlusion, background clutter, and appearance variation. To address these limitations, we propose TSDTrack, a transformer-augmented Siamese tracker designed for quality-aware and robust tracking. Our framework employs a ResNet backbone to extract multi-scale hierarchical features, which are fused using a transformer-based module that applies global attention to enhance semantic and spatial consistency. The prediction head consists of two branches: a confidence aware branch (CAB) that assesses the confidence of classification responses, and a regression distribution learning (RDL) branch that models bounding box localization as discrete probability distributions, improving precision under uncertainty. Furthermore, we introduce a confidence-gated template update strategy that selectively refreshes the target representation based on the CAB score, enabling adaptive appearance modeling while avoiding drift. Experiments on LaSOT, GOT-10k, OTB100, and UAV123 demonstrate that TSDTrack achieves state-of-the-art performance in both accuracy and robustness, achieving 55.5% success on LaSOT, 67.5% AO on GOT-10k, 71.6% AUC on OTB100, and 66.4% success on UAV123, outperforming recent transformer-based and Siamese trackers.

Distribution regression is the regression case where the input objects are distributions. Many machine learning problems can be analyzed in this framework, such as multi-instance learning and learning from noisy data. This paper attempts to build a conformal predictive system (CPS) for distribution regression, where the prediction of the system for a test input is a cumulative distribution function (CDF) of the corresponding test label. The CDF output by a CPS provides useful information about the test label, as it can estimate the probability of any event related to the label and be transformed to prediction interval and prediction point with the help of the corresponding quantiles. Furthermore, a CPS has the property of validity as the prediction CDFs and the prediction intervals are statistically compatible with the realizations. This property is desired for many risk-sensitive applications, such as weather forecast. To the best of our knowledge, this is the first work to extend the learning framework of CPS to distribution regression problems. We first embed the input distributions to a reproducing kernel Hilbert space using kernel mean embedding approximated by random Fourier features, and then build a fast CPS on the top of the embeddings. While inheriting the property of validity from the learning framework of CPS, our algorithm is simple, easy to implement and fast. The proposed approach is tested on synthetic data sets and can be used to tackle the problem of statistical postprocessing of ensemble forecasts, which demonstrates the effectiveness of our algorithm for distribution regression problems.

In this paper, we investigated the coefficient-based regularized distribution regression for data generated by unbounded sampling processes. The algorithm adopts a two-stage sampling framework: the first-stage sample consists of probability distributions, from which the second-stage sample is drawn. A rigorous capacity-dependent convergence analysis was conducted under more general conditions, and its performance was comparable to that of one-stage sampling learning. Regularization was imposed on the coefficients and the kernel Kwas permitted to be indefinite. The important feature of this algorithm is that it can improve the saturation effect suffered by classical kernel ridge regression (KRR). Notably, the output sample values were assumed to satisfy a moment condition (rather than the stricter uniform boundedness constraint common in related works). We derived the convergence error bounds via the novel integral operator techniques, and further established the mini-max optimal learning rates of the algorithm, which were comparable to those achieved under bounded sampling settings.

Computer vision researchers prefer to estimate age from face images because facial features provide useful information. However, estimating age from face images becomes challenging when people are distant from the camera or occluded. A person’s gait is a unique biometric feature that can be perceived efficiently even at a distance. Thus, gait can be used to predict age when face images are not available. However, existing gait-based classification or regression methods ignore the ordinal relationship of different ages, which is an important clue for age estimation. This paper proposes an ordinal distribution regression with a global and local convolutional neural network for gait-based age estimation. Specifically, we decompose gait-based age regression into a series of binary classifications to incorporate the ordinal age information. Then, an ordinal distribution loss is proposed to consider the inner relationships among these classifications by penalizing the distribution discrepancy between the estimated value and the ground truth. In addition, our neural network comprises a global and three local sub-networks, and thus, is capable of learning the global structure and local details from the head, body, and feet. Experimental results indicate that the proposed approach outperforms state-of-the-art gait-based age estimation methods on the OULP-Age dataset.

We establish a deep learning theory for distribution regression with deep convolutional neural networks (DCNNs). Deep learning based on structured deep neural networks has been powerful in practical applications. Generalization analysis for regression with DCNNs has been carried out very recently. However, for the distribution regression problem in which the input variables are probability measures, there is no mathematical model or theoretical analysis of DCNN-based learning theory. One of the difficulties is that the classical neural network structure requires the input variable to be a Euclidean vector. When the input samples are probability distributions, the traditional neural network structure cannot be directly used. A well-defined DCNN framework for distribution regression is desirable. In this paper, we overcome the difficulty and establish a novel DCNN-based learning theory for a two-stage distribution regression model. Firstly, we realize an approximation theory for functionals defined on the set of Borel probability measures with the proposed DCNN framework. Then, we show that the hypothesis space is well-defined by rigorously proving its compactness. Furthermore, in the hypothesis space induced by the general DCNN framework with distribution inputs, by using a two-stage error decomposition technique, we derive a novel DCNN-based two-stage oracle inequality and optimal learning rates (up to a logarithmic factor) for the proposed algorithm for distribution regression.

Triangular systems with nonadditively separable unobserved heterogeneity provide a theoretically appealing framework for the modeling of complex structural relationships. However, they are not commonly used in practice due to the need for exogenous variables with large support for identification, the curse of dimensionality in estimation, and the lack of inferential tools. This paper introduces two classes of semiparametric nonseparable triangular models that address these limitations. They are based on distribution and quantile regression modeling of the reduced form conditional distributions of the endogenous variables. We show that average, distribution, and quantile structural functions are identified in these systems through a control function approach that does not require a large support condition. We propose a computationally attractive three-stage procedure to estimate the structural functions where the first two stages consist of quantile or distribution regressions. We provide asymptotic theory and uniform inference methods for each stage. In particular, we derive functional central limit theorems and bootstrap functional central limit theorems for the distribution regression estimators of the structural functions. These results establish the validity of the bootstrap for three-stage estimators of structural functions, and lead to simple inference algorithms. We illustrate the implementation and applicability of all our methods with numerical simulations and an empirical application to demand analysis.