arXiv subjects

Ruodu Wang

Publications and source records attributed to Ruodu Wang.

At least 55 records · Page 3Linked to original sources

arXiv2025

Optimal allocations with distortion risk measures and mixed risk attitudes

We study Pareto-optimal risk sharing in economies with heterogeneous attitudes toward risk, where agents' preferences are modeled by distortion risk measures. Building on comonotonic and counter-monotonic improvement results, we show that agents with similar attitudes optimally share risks comonotonically (risk-averse) or counter-monotonically (risk-seeking). We show how the general $n$-agent problem can be reduced to a two-agent formulation between representative risk-averse and risk-seeking agents, characterized by the infimal convolution of their distortion risk measures. Within this two-agent framework, we establish necessary and sufficient conditions for the existence of optimal allocations, and we identify when the infimal convolution yields an unbounded value. When existence fails, we analyze the problem under nonnegative allocation constraints, and we characterize optima explicitly, under piecewise-linear distortion functions and Bernoulli-type risks. Our findings suggest that the optimal allocation structure is governed by the relative strength of risk aversion versus risk seeking behavior, as intuition would suggest.

econ.TH↗

arXiv2025

Elicitability and identifiability of tail risk measures

Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value-at-Risk, Expected Shortfall and Range Value-at-Risk being prime examples. They are induced by law-based risk measures, called their generators, evaluated on the tail distribution. This paper establishes joint identifiability and elicitability results of tail risk measures together with the corresponding quantile, provided that their generators are identifiable and elicitable, respectively. As an example, we establish the joint identifiability and elicitability of the tail expectile together with the quantile. The corresponding consistent scores constitute a novel class of weighted scores, nesting the known class of scores of Fissler and Ziegel for the Expected Shortfall together with the quantile. For statistical purposes, our results pave the way to easier model fitting for tail risk measures via regression and the generalized method of moments, but also model comparison and model validation in terms of established backtesting procedures.

q-fin.ST↗

arXiv2025

Invariant Modeling for Joint Distributions

A common theme underlying many problems in statistics and economics involves the determination of a systematic method of selecting a joint distribution consistent with a specified list of categorical marginals, some of which have an ordinal structure. We propose guidance in narrowing down the set of possible methods by introducing Invariant Aggregation (IA), a natural property that requires merging adjacent categories in one marginal not to alter the joint distribution over unaffected values. We prove that a model satisfies IA if and only if it is a copula model. This characterization ensures i) robustness against data manipulation and survey design, and ii) allows seamless incorporation of new variables. Our results provide both theoretical clarity and practical safeguards for inference under marginal constraints.

econ.TH↗

arXiv2025

Improved thresholds for e-values

The rejection threshold used for e-values and e-processes is by default set to $1/α$ for a guaranteed type-I error control at $α$, based on Markov's and Ville's inequalities. This threshold can be wasteful in practical applications. We discuss how this threshold can be improved under additional distributional assumptions on the e-values; some of these assumptions are naturally plausible and empirically observable, without knowing explicitly the form or model of the e-values. For small values of $α$, the threshold can roughly be improved (divided) by a factor of $2$ for decreasing or unimodal densities, and by a factor of $e$ for decreasing or unimodal-symmetric densities of log-transformed e-values. Moreover, we propose to use the supremum of comonotonic e-values, which is shown to preserve the type-I error guarantee. We also propose some preliminary methods to boost e-values in the e-BH procedure under some distributional assumptions while controlling the false discovery rate. Through a series of simulation studies, we demonstrate the effectiveness of our proposed methods in various testing scenarios, showing enhanced power.

math.ST↗

arXiv2025

Prudence and higher-order risk attitudes in the rank-dependent utility model

We obtain a full characterization of consistency with respect to higher-order stochastic dominance within the rank-dependent utility model. Different from the results in the literature, we do not assume any condition on the utility functions and the probability weighting functions, such as differentiability or continuity. It turns out that the level of generality that we offer leads to models that do not have a continuous probability weighting function and yet they satisfy prudence. In particular, the corresponding probability weighting function can only have a jump at 1, and must be linear on [0,1).

econ.TH↗

arXiv2025

Higher-order Gini indices: An axiomatic approach

Via an axiomatic approach, we characterize the family of n-th order Gini deviation, defined as the expected range over n independent draws from a distribution, to quantify joint dispersion across multiple observations. This family extends the classical Gini deviation, which relies solely on pairwise comparisons. The normalized version is called a high-order Gini coefficient. The generalized indices grow increasingly sensitive to tail inequality as n increases, offering a more nuanced view of distributional extremes. The higher-order Gini deviations admit a Choquet integral representation, inheriting the desirable properties of coherent deviation measures. Furthermore, we show that both the n-th order Gini deviation and the n-th order Gini coefficient are statistically n-observation elicitable, allowing for direct computation through empirical risk minimization. Data analysis using World Inequality Database data reveals that higher-order Gini coefficients capture disparities that the classical Gini coefficient may fail to reflect, particularly in cases of extreme income or wealth concentration.

q-fin.MF↗

arXiv2025

Risk sharing, measuring variability, and distortion riskmetrics

We address the problem of sharing risk among agents with preferences modelled by a general class of comonotonic additive and law-based functionals that need not be either monotone or convex. Such functionals are called distortion riskmetrics, which include many statistical measures of risk and variability used in portfolio optimization and insurance. The set of Pareto-optimal allocations is characterized under various settings of general or comonotonic risk sharing problems. We solve explicitly Pareto-optimal allocations among agents using the Gini deviation, the mean-median deviation, or the inter-quantile difference as the relevant variability measures. The latter is of particular interest, as optimal allocations are not comonotonic in the presence of inter-quantile difference agents; instead, the optimal allocation features a mixture of pairwise counter-monotonic structures, showing some patterns of extremal negative dependence.

q-fin.RM↗

arXiv2025

Hypothesis testing with e-values

This book is written to offer a humble, but unified, treatment of e-values in hypothesis testing. It is organized into three parts: Fundamental Concepts, Core Ideas, and Advanced Topics. The first part includes four chapters that introduce the basic concepts. The second part includes five chapters of core ideas such as universal inference, log-optimality, e-processes, operations on e-values, and e-values in multiple testing. The third part contains seven chapters of advanced topics. The book collates important results from a variety of modern papers on e-values and related concepts, and also contains many results not published elsewhere. It offers a coherent and comprehensive picture on a fast-growing research area, and is ready to use as the basis of a graduate course in statistics and related fields.

math.ST↗

arXiv2025

Max- and min-stability under first-order stochastic dominance

Max-stability is the property that taking a maximum between two inputs results in a maximum between two outputs. We study max-stability with respect to first-order stochastic dominance, the most fundamental notion of stochastic dominance in decision theory. Under two additional standard axioms of nondegeneracy and lower semicontinuity, we establish a representation theorem for functionals satisfying max-stability, which turns out to be represented by the supremum of a bivariate function. A parallel characterization result for min-stability, that is, with the maximum replaced by the minimum in max-stability, is also established. By combining both max-stability and min-stability, we obtain a new characterization for a class of functionals, called the Lambda-quantiles, that appear in finance and political science.

q-fin.MF↗

arXiv2025

Asymptotic and compound e-values: multiple testing and empirical Bayes

We explicitly define the notions of (bona fide, approximate or asymptotic) compound p-values and e-values, which have been implicitly presented and used in the recent multiple testing literature. While it is known that the e-BH procedure with compound e-values controls the FDR, we show the converse: every FDR controlling procedure can be recovered by instantiating the e-BH procedure with certain compound e-values. Since compound e-values are closed under averaging, this allows for combination and derandomization of arbitrary FDR procedures. We then connect compound e-values to empirical Bayes. In particular, we use the fundamental theorem of compound decision theory to derive the log-optimal simple separable compound e-value for testing a set of point nulls against point alternatives: it is a ratio of mixture likelihoods. As one example, we construct asymptotic compound e-values for multiple t-tests, where the (nuisance) variances may be different across hypotheses. Our construction may be interpreted as a data-driven instantiation of the optimal discovery procedure, and our results provide the first type-I error guarantees for the same, along with significant power gains.

stat.ME↗

arXiv2025

Elementary proofs of several results on false discovery rate

We collect self-contained elementary proofs of four results in the literature on the false discovery rate of the Benjamini-Hochberg (BH) procedure for independent or positive-regression dependent p-values, the Benjamini-Yekutieli correction for arbitrarily dependent p-values, and the e-BH procedure for arbitrarily dependent e-values. As a corollary, the above proofs also lead to some inequalities of Simes and Hommel.

math.ST↗

arXiv2025

Risk exchange under infinite-mean Pareto models

We study the optimal decisions and equilibria of agents who aim to minimize their risks by allocating their positions over extremely heavy-tailed (i.e., infinite-mean) and possibly dependent losses. The loss distributions of our focus are super-Pareto distributions, which include the class of extremely heavy-tailed Pareto distributions. Using a recent result on stochastic dominance, we show that for a portfolio of super-Pareto losses, non-diversification is preferred by decision makers equipped with well-defined and monotone risk measures. The phenomenon that diversification is not beneficial in the presence of super-Pareto losses is further illustrated by an equilibrium analysis in a risk exchange market. First, agents with super-Pareto losses will not share risks in a market equilibrium. Second, transferring losses from agents bearing super-Pareto losses to external parties without any losses may arrive at an equilibrium which benefits every party involved.

q-fin.RM↗

arXiv2025

Partial Law Invariance and Risk Measures

We introduce the concept of partial law invariance, generalizing the concepts of law invariance and probabilistic sophistication widely used in decision theory, as well as statistical and financial applications. This new concept is motivated by practical considerations of decision making under uncertainty, thus connecting the literature on decision theory and that on financial risk management. We fully characterize partially law-invariant coherent risk measures via a novel representation formula. Strong partial law invariance is defined to bridge the gap between the above characterization and the classic representation formula of Kusuoka. We propose a few classes of new risk measures, including partially law-invariant versions of the Expected Shortfall and the entropic risk measures, and illustrate their applications in risk assessment under different types of uncertainty. We provide a tractable optimization formula for computing a class of partially law-invariant coherent risk measures and give a numerical example.

q-fin.RM↗

arXiv2025

Composite Sorting

This paper introduces an assignment model with concave costs of skill gaps, which arise generally when firms mitigate costs of mismatch as in Stigler (1939) and Laffont and Tirole (1986, 1991). Concave costs of skill gaps imply that the output function is neither supermodular nor submodular. We thus introduce a tractable model that interpolates between the polar canonical cases of supermodularity and submodularity. We characterize sorting, wages, and comparative statics and show these substantively differ from traditional assignment models. Under composite sorting: (1) distinct worker types work in the same occupation, and (2) worker types are simultaneously part of both positive and negative sorting. Quantitatively, our model can generate and help explain earnings dispersion between and within occupations.

econ.GN↗

arXiv2025

Distorted optimal transport

Classic optimal transport theory is formulated through minimizing the expected transport cost between two given distributions. We propose the framework of distorted optimal transport by minimizing a distorted expected cost, which is the cost under a non-linear expectation. This new formulation is motivated by concrete problems in decision theory, robust optimization, and risk management, and it has many distinct features compared to the classic theory. We choose simple cost functions and study different distortion functions and their implications on the optimal transport plan. We show that on the real line, the comonotonic coupling is optimal for the distorted optimal transport problem when the distortion function is convex and the cost function is submodular and monotone. Some forms of duality and uniqueness results are provided. For inverse-S-shaped distortion functions and linear cost, we obtain the unique form of optimal coupling for all marginal distributions, which turns out to have an interesting ``first comonotonic, then counter-monotonic" dependence structure; for S-shaped distortion functions a similar structure is obtained. Our results highlight several challenges and features in distorted optimal transport, offering a new mathematical bridge between the fields of probability, decision theory, and risk management.

math.OC↗

arXiv2025

Sub-uniformity of harmonic mean p-values

We obtain several inequalities on the generalized means of dependent p-values. In particular, the weighted harmonic mean of p-values is strictly sub-uniform under several dependence assumptions of p-values, including independence, negative upper orthant dependence, the class of extremal mixture copulas, and some Clayton copulas. Sub-uniformity of the harmonic mean of p-values has an important implication in multiple hypothesis testing: It is statistically invalid (anti-conservative) to merge p-values using the harmonic mean unless a proper threshold or multiplier adjustment is used, and this applies across all significance levels. The required multiplier adjustment on the harmonic mean p-value grows sub-linearly to infinity as the number of p-values increases, and hence there does not exist a constant multiplier that works for any number of p-values, even under independence.

math.ST↗

arXiv2025

Coherent risk measures and uniform integrability

We establish a profound connection between coherent risk measures, a prominent object in quantitative finance, and uniform integrability, a fundamental concept in probability theory. Instead of working with absolute values of random variables, which is convenient in studying integrability, we work directly with random loses and gains, which have clear financial interpretation. We introduce a technical tool called the folding score of distortion risk measures. The analysis of the folding score allows us to convert some conditions on absolute values to those on gains and losses. As our main results, we obtain three sets of equivalent conditions for uniform integrability. In particular, a set is uniformly integrable if and only if one can find a coherent distortion risk measure that is bounded on the set, but not finite on $L^1$.

q-fin.RM↗

arXiv2025

Combining exchangeable p-values

The problem of combining p-values is an old and fundamental one, and the classic assumption of independence is often violated or unverifiable in many applications. There are many well-known rules that can combine a set of arbitrarily dependent p-values (for the same hypothesis) into a single p-value. We show that essentially all these existing rules can be strictly improved when the p-values are exchangeable, or when external randomization is allowed (or both). For example, we derive randomized and/or exchangeable improvements of well known rules like ``twice the median'' and ``twice the average'', as well as geometric and harmonic means. Exchangeable p-values are often produced one at a time (for example, under repeated tests involving data splitting), and our rules can combine them sequentially as they are produced, stopping when the combined p-values stabilize. Our work also improves rules for combining arbitrarily dependent p-values, since the latter becomes exchangeable if they are presented to the analyst in a random order. The main technical advance is to show that all existing combination rules can be obtained by calibrating the p-values to e-values (using an $α$-dependent calibrator), averaging those e-values, converting to a level-$α$ test using Markov's inequality, and finally obtaining p-values by combining this family of tests; the improvements are delivered via recent randomized and exchangeable variants of Markov's inequality.

math.ST↗