Stochastic Dominance Research Articles

Tests for almost stochastic dominance

We introduce a 2-dimensional stochastic dominance (2DSD) index to characterize both strict and almost stochastic dominance. Based on this index, we derive an estimator for the minimum violation ratio (MVR), also known as the critical parameter, of the almost stochastic ordering condition between two variables. We determine the asymptotic properties of the empirical 2DSD index and MVR for the most frequently used stochastic orders. We also provide conditions under which the bootstrap estimators of these quantities are strongly consistent. As an application, we develop consistent bootstrap testing procedures for almost stochastic dominance. The performance of the tests is checked via simulations and the analysis of real data.

Correlation aversion and bivariate stochastic dominance with respect to reference functions

This paper introduces an extension of stochastic dominance, moving from univariate to bivariate analysis by incorporating a reference function. Our approach offers flexibility in reference function selection, improving upon previous studies cohesively. Bivariate orderings are invaluable tools in actuarial sciences, facilitating the assessment and management of dependencies between risks and lifelengths within multiple insurance contracts. These advancements hold promising practical implications, particularly within the actuarial sciences domain.

Tests for the first‐order stochastic dominance

AbstractWe study the first‐order stochastic dominance (SD) test in the context of two independent random samples. We introduce several test statistics that effectively capture violations of the dominance relationship, particularly in the tail regions. Additionally, we develop a resampling procedure to compute the ‐values or critical values for these tests. The proposed tests have asymptotic type I error rates for frontal configurations equal to the nominal level . Furthermore, their powers approach 1 for any fixed alternatives. Through simulation experiments, we demonstrate that our SD tests outperform the recentring test proposed by Donald and Hsu (2016) as well as the integral‐type test presented by Linton et al. (2010) in various scenarios discussed in existing literature. We also employ the proposed tests to analyze changes in the distribution of household income in the United Kingdom over time. The proposed tests offer some insights into potential dominance relationships within this context.

Stochastic optimization problems with nonlinear dependence on a probability measure via the Wasserstein metric

Nonlinear dependence on a probability measure has recently been encountered with increasing intensity in stochastic optimization. This type of dependence corresponds to many situations in applications; it can appear in problems static (one-stage), dynamic with finite (multi-stage) or infinite horizon, and single- and multi-objective ones. Moreover, the nonlinear dependence can appear not only in the objective functions but also in the constraint sets. In this paper, we will consider static one-objective problems in which the nonlinear dependence appears in the objective function and may also appear in the constraint sets. In detail, we consider “deterministic” constraint sets, whose dependence on the probability measure is nonlinear, constraint sets determined by second-order stochastic dominance, and sets given by mean-risk problems. The last mentioned instance means that the constraint set corresponds to solutions which guarantee acceptable values of both criteria. To obtain relevant assertions, we employ the stability results given by the Wasserstein metric, based on the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {{\\mathcal {L}}}_{1} $$\\end{document} norm. We mainly focus on the case in which a solution has to be obtained on the basis of the data and of investigating a relationship between the original problem and its empirical version.

Estimation of the Local Conditional Tail Average Treatment Effect

The conditional tail average treatment effect (CTATE) is defined as a difference between the conditional tail expectations of potential outcomes, which can capture heterogeneity and deliver aggregated local information on treatment effects over different quantile levels and is closely related to the notion of second-order stochastic dominance and the Lorenz curve. These properties render it a valuable tool for policy evaluation. In this article, we study estimation of the CTATE locally for a group of compliers (local CTATE or LCTATE) under the two-sided noncompliance framework. We consider a semiparametric treatment effect framework under endogeneity for the LCTATE estimation using a newly introduced class of consistent loss functions jointly for the CTE and quantile. We establish the asymptotic theory of our proposed LCTATE estimator and provide an efficient algorithm for its implementation. We then apply the method to evaluate the effects of participating in programs under the Job Training Partnership Act in the United States.

On the desire to make a difference

True benevolence is, most fundamentally, a desire that the world be better. It is natural and common, however, to frame thinking about benevolence indirectly, in terms of a desire to make a difference to how good the world is. This would be an innocuous shift if desires to make a difference were extensionally equivalent to desires that the world be better. This paper shows that at least on some common ways of making a “desire to make a difference” precise, this extensional equivalence fails. Where it fails, “difference-making preferences” run counter to the ideals of benevolence. In particular, in the context of decision making under uncertainty, coupling a “difference-making” framing in a natural way with risk aversion leads to preferences that violate stochastic dominance, and that lead to a strong form of collective defeat, from the point of view of betterness. Difference-making framings and true benevolence are not strictly mutually inconsistent, but agents seeking to implement true benevolence must take care to avoid the various pitfalls that we outline.

Integrating Uncertainties in a Multi-Criteria Decision Analysis with the Entscheidungsnavi

The Entscheidungsnavi is an open-source decision support system based on multi-attribute utility theory, that offers various methods for dealing with uncertainties. To model decisions with uncertainties, decision-makers can use two categories: Forecast and Parameter Uncertainties. Forecast Uncertainty is modeled with (combined) influence factors using discrete, user-defined probability distributions or predefined ‘worst-median-best’ distributions. Parameter Uncertainty allows imprecision for utilities, objective weights, and probability distributions. To analyze these uncertainties, the Entscheidungsnavi offers several methods and tools, like a robustness check, based on (Monte Carlo) simulations and a sensitivity analysis. The objective weight analysis provides insights into the effects of different objective weight combinations. Indicator impacts, tornado diagrams, and risk profiles visualize the impact of uncertainties in a decision under risk. Risk profiles also enable a check for stochastic and simulation dominance. This article presents the complete range of methods for dealing with uncertainties in the Entscheidungsnavi using a hypothetical case study.

Unifying Star Ratings and Text Reviews in Linguistic Terms for Product Competitiveness Analysis Based on Stochastic Dominance

Unifying Star Ratings and Text Reviews in Linguistic Terms for Product Competitiveness Analysis Based on Stochastic Dominance

ПРОКСИ ДЛЯ РАНЖИРОВАНИЯ КОМПАНИЙ ПО КРИТЕРИЯМ ИННОВАЦИОННОГО МЕНЕДЖМЕНТА

A study of the correspondence of the ranking series of innovative companies ac-cording to the criterion of stochastic domi-nance and using the hierarchy analysis method was conducted. Within the framework of the latter, an analysis of the factors of the innovative potential of companies and their assessment were carried out, quantitative values of the influence of factors were determined. A significant rank correlation was revealed between the results obtained using two methods.

Revisiting development strategy under climate uncertainty: case study of Malawi

This paper analyzes the effectiveness of agriculture-led versus non-agriculture-led development strategies under climate-induced economic uncertainty. Utilizing Malawi as a case study, we introduce the application of Stochastic Dominance (SD) analysis, a tool from decision analysis theory, and compare the two strategies in the context of weather/climate-associated economic uncertainty. Our findings suggest that an agriculture-led development strategy consistently surpasses its non-agriculture-led antagonist in poverty and undernourishment outcomes across almost all possible weather/climate scenarios. This underscores that, despite increasing exposure of the entire economy to weather/climate uncertainty, agriculture-led development remains the optimal strategy for Malawi to reduce poverty and undernourishment. The study also endorses the broader use of SD analysis in policy planning studies, promoting its potential to integrate risk and uncertainty into policymaking.

Too-systemic-to-fail: Empirical comparison of systemic risk measures in the Eurozone financial system

This paper quantifies the Too-Systemic-To-Fail (TSTF) paradigm in the Eurozone since the introduction of the Euro through three primary dimensions: Too-Big-To-Fail (TBTF), Too-Interconnected-To-Fail (TITF), and Too-Many-To-Fail (TMTF). We apply prominent systemic risk measures based on public data, including the Granger-causality network (GCN), Delta Conditional Value-at-Risk (ΔCoVaR), Marginal Expected Shortfall (MES), and Systemic Risk Index (SRISK). Financial interconnectedness and systemic risk exposure within the 17-member states of the Eurozone are measured on two levels: (i) identifying which financial sectors (banking, diversified financials, insurance, and real estate) are most exposed to systemic risk in the Eurozone at the union level; and (ii) identifying which member state is most exposed to systemic risk within each financial sector at the country level. We extend the original ΔCoVaR, MES and SRISK models by incorporating the bootstrap Kolmogorov-Smirnov stochastic dominance test to rank institutions based on their exposure to systemic risk formally.

A neural network framework for portfolio optimization under second-order stochastic dominance

Traditional portfolio optimization strategies often rely on statistical methods and linear programming tools to achieve a balance between return and risk. Despite their usefulness for portfolio optimization, they cannot efficiently capture the specific differences and complexity of real-world financial markets. Several modern approaches attempt to overcome these limitations by using nonlinear models, machine learning, and advanced risk measures. In this study, we propose a novel strategy for optimizing portfolios that incorporates second-order stochastic dominance constraints and solves them with neural networks. we show that portfolios subject to second-order stochastic dominance constraints outperform their traditional counterparts, especially in tail-risk situations.

Testing out-of-sample portfolio performance using second-order stochastic dominance constrained optimization approach

Second-order Stochastic Dominance (SSD) criterion can be used to support portfolio decision making under risk and uncertainty. In this paper, we develop novel robust SSD criteria to capture the strength of dominance and portfolio optimization models utilizing these criteria to identify portfolios whose in-sample SSD dominance over a given benchmark is likely to hold also out-of-sample. The developed models can incorporate incomplete probability information by allowing a set of feasible state probabilities. We also show that these portfolio optimization models can be formulated as linear programming problems. We report results from applying these SSD-based portfolio optimization models with different sets of state probabilities in an empirical application, with a focus on evaluating the out-of-sample portfolio performance of the optimized portfolios.

Adjustable light robust optimization with second order stochastic dominance constraints

Robust optimization suffers from excessive conservatism and infeasibility due to stringent requirements on all constraints within the uncertainty set, rendering robust solutions impractical. The light robust approach has been proposed to mitigate these challenges. This method aims to identify a solution that minimizes the weighted sum of all constraint violations while adhering to a predetermined limit on the deterioration of expected return. Drawing inspiration from light robustness, our paper introduces the adjustable light robust optimization models with second-order stochastic dominance constraints. We empirically examine their feasibility, out-of-sample performance, and the dynamic trade-off between return and robustness.

ODC and ROC Curves, Comparison Curves and Stochastic Dominance

SummaryWe discuss two novel approaches to inter‐distributional comparisons in the classical two‐sample problem. Our starting point is properly standardised and combined, very popular in several areas of statistics and data analysis, ordinal dominance and receiver characteristic curves, denoted by ODC and ROC, respectively. The proposed new curves are termed the comparison curves. Their estimates, being weighted rank processes on (0,1), form the basis of inference. These weighted processes are intuitive, well‐suited for visual inspection of data at hand and are also useful for constructing some formal inferential procedures. They can be applied to several variants of two‐sample problem. Their use can help improve some existing procedures both in terms of power and the ability to identify the sources of departures from the postulated model. To simplify interpretation of finite sample results, we restrict attention to values of the processes on a finite grid of points. This results in the so‐called bar plots (B‐plots), which readably summarise the information contained in the data. What is more, we show that B‐plots along with adjusted simultaneous acceptance regions provide principled information about where the model departs from the data. This leads to a framework that facilitates identification of regions with locally significant differences.We show an implementation of the considered techniques to a standard stochastic dominance testing problem. Some min‐type statistics are introduced and investigated. A simulation study compares two tests pertinent to the comparison curves to well‐established tests in the literature and demonstrates the strong and competitive performance of the former in many typical situations. Some real data applications illustrate simplicity and practical usefulness of the proposed approaches. A range of other applications of considered weighted processes is briefly discussed too.

Investments in transmission lines and storage units considering second-order stochastic dominance constraints

Increasing intermittent renewable generation to meet the climate goals entails a deep transformation of current power systems. The transmission system must adapt to ensure a rapid and flexible response to the changes in the energy flows. Flexibility can be provided by reinforcing the interconnection among all the market agents and/or by installing facilities with fast response to the changes. In this work, we study the investment problem of a central planner that seeks to expand the transmission network and install storage units considering decarbonizing measures. We propose a two-stage stochastic problem with uncertainty on the demand growth and including representative days to characterize the hourly demand and renewable power variability. To obtain better expansion strategies according to a limit on the carbon emissions, second-order stochastic dominance constraints are imposed. Numerical analyses based on the power system of the Canary island (Spain) are provided. The results show that to install storage units is key to efficiently integrate renewable generation. The investments in the transmission system are mostly in lower-voltage lines. The formulation with stochastic dominance constraints results in higher second-stage investments, allowing a better adaptation to the demand growth evolution.

Law-invariant return and star-shaped risk measures

This paper presents novel characterization results for classes of law-invariant star-shaped functionals. We begin by establishing characterizations for positively homogeneous and star-shaped functionals that exhibit second- or convex-order stochastic dominance consistency. Building on these characterizations, we proceed to derive Kusuoka-type representations for these functionals, shedding light on their mathematical structure and intimate connections to Value-at-Risk and Expected Shortfall. Furthermore, we offer representations of general law-invariant star-shaped functionals as robustifications of Value-at-Risk. Notably, our results are versatile, accommodating settings that may, or may not, involve monotonicity and/or cash-additivity. All of these characterizations are developed within a general locally convex topological space of random variables, ensuring the broad applicability of our results in various financial, insurance and probabilistic contexts.

Conditions for extrapolating differences in consumption to differences in welfare

AbstractWe characterize conditions under which a better consumption distribution implies higher welfare. Specifically, here “better consumption” means first‐order stochastic dominance, and “higher welfare” means higher expected utility for every subpopulation of individuals with the same utility function. Although this implication seems natural, we first provide a counterexample wherein better consumption risk allocation outweighs lower consumption. We then show that higher expected utility results from higher consumption in different settings, including fixed dependence (fixed copula) between consumption and individual risk preferences, or alternatively using the rank invariance assumption from the treatment effects literature. These are discussed in several real‐world examples.

Estimating Probability Weighting Functions through Option Pricing Bounds

Abstract This paper proposes a novel approach to estimating the probability weighting function (PWF) of investors in the option market. We match observed option prices to the option pricing bounds under stochastic dominance rules. Using 1-month S&P 500 index option data, we find that investors could subjectively employ an inverse S-shaped probability weighting function, which increases the weights on extreme returns and asymmetrically assigns greater weights to extremely low returns than to extremely high returns. Our findings suggest that the inverse S-shaped nature of the PWFs is robust across various estimation specifications, such as adopting an alternative methodology to construct the return distribution, and employing option data with different times to maturity. (JEL G12)

Increasing schedule reliability in the multiple depot vehicle scheduling problem with stochastic travel time

The multiple depot vehicle scheduling problem (MDVSP) is one of the most studied problems in public transport service planning. It consists of assigning buses to each timetabled trip while respecting vehicle availability at each depot. Although service quality, and especially reliability, is the core of most transport agencies, the MDVSP is more often than not solved solely in a cost-efficient way. This work introduces a data-driven model to the reliable MDVSP with stochastic travel time (R-MDVSP-STT). The reliability of a schedule is assessed and accounted for by propagating delays using the probability mass function of the travel time of each timetabled trip. We propose a heuristic branch-and-price algorithm to solve this problem and a labeling algorithm with a stochastic dominance criterion for the associated subproblems. The solutions obtained are compared based on three metrics — under normal and extraordinary circumstances. Computational results on real-life instances show that our method can efficiently find good trade-offs between operational costs and reliability, improving the reliability of the solutions with little cost increase.