Search for: All records

Abstract Expected shortfall (ES), also known as superquantile or conditional value-at-risk, is an important measure in risk analysis and stochastic optimisation and has applications beyond these fields. In finance, it refers to the conditional expected return of an asset given that the return is below some quantile of its distribution. In this paper, we consider a joint regression framework recently proposed to model the quantile and ES of a response variable simultaneously, given a set of covariates. The current state-of-the-art approach to this problem involves minimising a non-differentiable and non-convex joint loss function, which poses numerical challenges and limits its applicability to large-scale data. Motivated by the idea of using Neyman-orthogonal scores to reduce sensitivity to nuisance parameters, we propose a statistically robust and computationally efficient two-step procedure for fitting joint quantile and ES regression models that can handle highly skewed and heavy-tailed data. We establish explicit non-asymptotic bounds on estimation and Gaussian approximation errors that lay the foundation for statistical inference, even with increasing covariate dimensions. Finally, through numerical experiments and two data applications, we demonstrate that our approach well balances robustness, statistical, and numerical efficiencies for expected shortfall regression. 

In this paper, we propose differentially private algorithms for robust (multivariate) mean estimation and inference under heavy-tailed distributions, with a focus on Gaussian differential privacy. First, we provide a comprehensive analysis of the Huber mean estimator with increasing dimensions, including non-asymptotic deviation bound, Bahadur representation, and (uniform) Gaussian approximations. Secondly, we privatize the Huber mean estimator via noisy gradient descent, which is proven to achieve near-optimal statistical guarantees. The key is to characterize quantitatively the trade-off between statistical accuracy, degree of robustness and privacy level, governed by a carefully chosen robustification parameter. Finally, we construct private confidence intervals for the proposed estimator by incorporating a private and robust covariance estimator. Our findings are demonstrated by simulation studies. 

The data-driven newsvendor problem with features has recently emerged as a significant area of research, driven by the proliferation of data across various sectors such as retail, supply chains, e-commerce, and healthcare. Given the sensitive nature of customer or organizational data often used in feature-based analysis, it is crucial to ensure individual privacy to uphold trust and confidence. Despite its importance, privacy preservation in the context of inventory planning remains unexplored. A key challenge is the nonsmoothness of the newsvendor loss function, which sets it apart from existing work on privacy-preserving algorithms in other settings. This paper introduces a novel approach to estimating a privacy-preserving optimal inventory policy within the f-differential privacy framework, an extension of the classical [Formula: see text]-differential privacy with several appealing properties. We develop a clipped noisy gradient descent algorithm based on convolution smoothing for optimal inventory estimation to simultaneously address three main challenges: (i) unknown demand distribution and nonsmooth loss function, (ii) provable privacy guarantees for individual-level data, and (iii) desirable statistical precision. We derive finite-sample high-probability bounds for optimal policy parameter estimation and regret analysis. By leveraging the structure of the newsvendor problem, we attain a faster excess population risk bound compared with that obtained from an indiscriminate application of existing results for general nonsmooth convex loss. Our bound aligns with that for strongly convex and smooth loss function. Our numerical experiments demonstrate that the proposed new method can achieve desirable privacy protection with a marginal increase in cost. This paper was accepted by J. George Shanthikumar, data science. Funding: This work was supported by the National Science Foundation [Grants DMS-2113409 and DMS 2401268 to W.-X. Zhou, and FRGMS-1952373 to L. Wang]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01268 .