Log-regularly varying scale mixture of asymmetric Laplaces for robust Bayesian quantile regression

Bayesian Regularized Quantile Beta Regression for Robust Estimation in Skewed Bounded Data

Skewed distributions often contain shape parameters that determine the direction and magnitude of asymmetry. In other cases, skewness arises naturally from the form of the distribution. Ignoring skewness when modeling with symmetric distributions may yield biased or misleading inferences. Bayesian regularized quantile regression has proven effective for skewed responses, yet existing approaches rely on the asymmetric Laplace distribution (ALD), whose unbounded support makes it unsuitable for bounded data. To address this limitation, we propose a Bayesian Regularized Quantile Beta Regression (BRQBR) model for analyzing bounded data supported on (0,1) with inherent skewness. The proposed model estimates conditional quantiles of a Beta-distributed response using a hierarchical Bayesian regularization framework with global–local shrinkage priors. A Gibbs sampler is developed for posterior computation, and the model's performance is evaluated under different skewness levels and contamination scenarios (5% and 10% outliers) using Beta and logit-normal distributions. Application to a real-world seaweed drying dataset demonstrates consistent improvements in predictive accuracy. Across simulation and empirical analysis, BRQBR outperforms or matches maximum likelihood estimation (MLE) while exhibiting strong robustness to outliers. The proposed framework offers a flexible and accurate solution for modeling skewed bounded responses.

Mixing-Law Uncertainty in Multivariate Normal Mean-Variance Mixtures: Semi-parametric Estimation and Robust Cumulative-Prospect Decisions

The distribution of a normal mean-variance mixture depends on the law of its positive mixing variable. We compare six parametric mixing laws with a grid nonparametric maximum likelihood estimator under the same determinant identification constraint. The mixing mean $m=\E(Z)$ is estimated and is not fixed at one. A paired block bootstrap is used to compare multivariate holdout log scores. The models that cannot be distinguished from the model with the largest score define a finite ambiguity set. We then consider a cumulative prospect problem on a common portfolio direction. For each model in the set, the NMVM representation gives a scalar projected return and a corresponding prospect-value function of the exposure. The distributionally robust decision maximizes the lower envelope of these functions. We prove existence of a solution, give the candidate points for the piecewise smooth problem, derive a reference-gap scaling result, and construct an interval branch-and-bound certificate for the finite-scenario optimum. In an application to 30 stock returns, the mixture models give higher holdout density scores than the multivariate Gaussian model. Several parametric and semi-parametric models, however, remain in the ambiguity set. The worst-case model is therefore determined at the portfolio optimization stage rather than selected in advance from a point estimate of the holdout score.

Causal Inference for Heterogeneous Extreme Quantiles with Heavy-Tailed Outcomes

We propose a framework for estimating conditional extreme quantile treatment effects (CEQTEs) in observational studies with heavy-tailed outcomes. Our procedure first estimates intermediate conditional quantiles using inverse-probability-weighted (IPW) quantile regression and then extrapolates them to extreme levels using extreme value theory. Under a linear conditional quantile model, we show that the conditional and marginal distributions of each potential outcome share a common extreme value index (EVI), motivating two complementary Hill-type EVI estimators based on conditional and marginal information, respectively. On the theoretical front, we introduce an IPW tail quantile score process that bridges regression quantile score processes and uniform tail empirical processes while accounting for treatment assignment. We establish its functional weak convergence under mild regularity conditions, without requiring a max-domain-of-attraction condition. This result provides the probabilistic foundation for the asymptotic analysis of the proposed CEQTE estimators. Simulation studies demonstrate favorable finite-sample performance, and an application to NLSY79 data reveals substantial heterogeneity in the effect of college education on extremely high hourly wages across confounder-defined subpopulations.

Generalized Linear Models for Extremes: Estimation and Inference in High Dimensions

We propose a regression model for the extreme tail of a response variable, in which covariates rescale the tail without changing its shape. A single covariate-dependent function then characterizes the entire conditional tail, in contrast to extreme quantile regression, which targets a quantile at a pre-specified level. The tail shape itself is left unrestricted: heavy-, light- and short-tailed responses are covered by the same framework. We specify the function through a link function and a linear combination of the covariates, which is in the spirit of a generalized linear model. In estimation, we match the parametric specification to the underlying tail function under a Bregman divergence, over a region localized at the largest observations. The resulting loss is convex, and an $\ell_1$-penalty allows the number of covariates to exceed the effective sample size. The tail localization makes the asymptotic theory deviate from that for classical penalized generalized linear models. Only the tail observations selected by a random threshold are used in the statistical analysis, making them dependent. We derive the convergence rate of the penalized estimator and propose a debiased estimator that is asymptotically normal, yielding confidence intervals for individual coefficients. Its asymptotic variance is determined by the covariance of the score, which under tail localization differs from the Hessian and must be estimated separately. We apply the method to automobile insurance claims data.

Laplace Factor Models in High-Dimensional Data

Laplace factor models (LFMs) provide a heavy-tailed alternative to Gaussian factor models by representing high-dimensional observations through a low-rank common component and Laplace-distributed idiosyncratic errors. This paper develops an assumption-consistent finite-sample analysis of matrix concentration, covariance estimation, and Monte Carlo integration under this model. We first formulate the model with explicit dimensional, independence, covariance, and identifiability conditions. Standard matrix Laplace-transform and matrix Bernstein inequalities are then recalled with their precise applicability conditions. Because untruncated Laplace variables are neither almost surely bounded nor strongly log-concave, these standard results cannot be applied directly in the forms commonly used for bounded or Gaussian-like observations. To address this issue, we analyze a coordinatewise truncated covariance estimator and derive an operator-norm bound that separates the stochastic estimation error from the truncation bias. The resulting rate depends on the effective rank and the logarithm of the ambient dimension and is therefore not dimension-free. For Monte Carlo integration, we replace strong-log-concavity arguments by a sub-exponential concentration analysis that is compatible with independent Laplace errors and yields non-asymptotic absolute- and relative-error bounds. Simulation studies compare empirical tails with the classical matrix Bernstein bound, evaluate ordinary, truncated, winsorized, PCA, POET-type, and Huberized covariance estimators, and we compare Laplace-based and Studentized confidence intervals. The results show that the classical Bernstein bound can be conservative, and truncation involves a substantial bias–variance trade-off. In a Wine chemical-analysis application, three factors explain 66.53% of the standardized variance, and POET-type covariance estimation attains a cross-validated balanced accuracy of 0.9901. These findings clarify both the scope and the limitations of finite-sample analysis for LFMs.

A Unified Inferential Framework for the Log-Exponential Fréchet Distribution Under Balanced Ranked Set Sampling