Concept Of Stochastic Dominance Research Articles

Statistical decision theory respecting stochastic dominance

The statistical decision theory pioneered by (Wald, Statistical decision functions, Wiley, 1950) has used state-dependent mean loss (risk) to measure the performance of statistical decision functions across potential samples. We think it evident that evaluation of performance should respect stochastic dominance, but we do not see a compelling reason to focus exclusively on mean loss. We think it instructive to also measure performance by other functionals that respect stochastic dominance, such as quantiles of the distribution of loss. This paper develops general principles and illustrative applications for statistical decision theory respecting stochastic dominance. We modify the Wald definition of admissibility to an analogous concept of stochastic dominance (SD) admissibility, which uses stochastic dominance rather than mean sampling performance to compare alternative decision rules. We study SD admissibility in two relatively simple classes of decision problems that arise in treatment choice. We reevaluate the relationship between the MLE, James–Stein, and James–Stein positive part estimators from the perspective of SD admissibility. We consider alternative criteria for choice among SD-admissible rules. We juxtapose traditional criteria based on risk, regret, or Bayes risk with analogous ones based on quantiles of state-dependent sampling distributions or the Bayes distribution of loss.

Regularization effect on model calibration

The concept of stochastic dominance plays a critical role in many economic and financial studies. However, the classical integer-degree stochastic dominance does not allow for local risk attitudes. In a 2020 paper, Huang, Tzeng and Zhao propose the concept of fractional degree stochastic dominance, which encompasses integer-degree stochastic dominance. They derive the integral conditions for fractional degree stochastic dominance. In this paper we develop the test statistics for fractional degree stochastic dominance based on a reformulation of the integral conditions. The test statistics’ asymptotic distributions are obtained and a bootstrap method for determining the critical values of the tests is also introduced. We further explore the stock preferences of the Chinese and US markets to illustrate the applicability of the test statistics for fractional degree stochastic dominance developed here.

Revisiting generalized almost stochastic dominance

We review the Generalized Almost Stochastic Dominance definition in Tsetlin et al. (Oper Res 63(2):363–377, 2015). We follow the classic concept of stochastic dominance in minimizing specific assumptions about decision makers’ risk attitudes as highlighted by Levy (Manag Sci 38(4):555–593, 1992). We refine the definition of almost stochastic dominance by Leshno and Levy (Manag Sci 48(8):1074–1085, 2002) to secure economic intuition. We then present our definitions of almost stochastic dominance with an extension to the higher degrees. Our definition satisfies both expected utility maximization property and hierarchy property while minimizing assumption and securing economic intuition.

On the properties of high-order non-monetary measures for risks

This paper investigates how welfare losses for facing high-order risk increases change when the risk environment of the decision maker is altered. To that aim, we define the nth-order utility premium as a measure of pain associated with facing the passage of one risk to a more severe one and we examine some of its properties. Changes in risk are expressed through the concept of stochastic dominance of order n. The paper investigates more particularly welfare changes of merging increases in risk, first ignoring background risks, then taking them into account. Merging increases in risk may be beneficial or not, depending on whether background risks are considered and how. The paper also provides conditions on individual preferences for superadditivity of the nth-order utility premium. The results confirm the importance and usefulness of two analytical concepts: mixed risk aversion and risk apportionment.

ALM models based on second order stochastic dominance

We propose asset and liability management models in which the risk of underfunding is modelled based on the concept of stochastic dominance. Investment decisions are taken such that the distribution of the funding ratio, that is, the ratio of asset to liabilities, is non-dominated with respect to second order stochastic dominance. In addition, the funding ratio distribution is close in an optimal sense to a user-specified target distribution. Interesting results are obtained when the target distribution is degenerate; in this case, we can obtain equivalent risk minimisation models, with risk defined as expected shortfall or as worst case loss. As an application, we consider the financial planning problem of a defined benefit pension fund in Saudi Arabia.

Optimal PID controller tuning using stochastic programming techniques

We argue that stochastic programming provides a powerful framework to tune and analyze the performance limits of controllers. In particular, stochastic programming formulations can be used to identify controller settings that remain robust across diverse scenarios (disturbances, set‐points, and modeling errors) observed in real‐time operations. We also discuss how to use historical data and sampling techniques to construct operational scenarios and inference analysis techniques to provide statistical guarantees on limiting controller performance. Under the proposed framework, it is also possible to use risk metrics to handle extreme (rare) events and stochastic dominance concepts to conduct systematic benchmarking studies. We provide numerical studies to illustrate the concepts and to demonstrate that modern modeling and local/global optimization tools can tackle large‐scale applications. The proposed work also opens the door to data‐based controller tuning strategies that can be implemented in real‐time operations. © 2017 American Institute of Chemical Engineers AIChE J, 64: 2997–3010, 2018

On Consistency of the Omega Ratio with Stochastic Dominance Rules

Omega ratios have been introduced in Keating and Shadwick (2002) as a performance measure to compare the performance of different investment opportunities. It does not have some of the drawbacks of the famous Sharpe ratio. In particular, it is consistent with first order stochastic dominance. Omega ratios also have an interesting relation to expectiles, which found increasing interest recently as risk measures. There is some confusion in the literature about consistency with respect to second order stochastic dominance. In this paper we clarify this and extend it to a consistency result with respect to stochastic dominance of order 1 + γ recently introduced in Muller, Scarsini, Tsetlin, and Winkler (2017) and generalizing the classical concepts of stochastic dominance of first and second order. Several examples illustrate the usefulness of this result. Finally, some consistency results for even more general stochastic dominance rules are shown, including the concept of ϵ-almost stochastic dominance introduced by Leshno and Levy (2002).

On the Properties of Non-Monetary Measures for Risks

This paper investigates how welfare losses for facing risks change as the risk environment of the decision-maker is altered. To that aim, we define the risk apportionment of order n (RAn) utility premium as a measure of pain associated with facing the passage from one risk to a riskier one. Changes in risks are expressed through the concept of stochastic dominance of order n. Three configurations of risk exposures are considered. The paper first shows how the RAn utility premium is modified when initial wealth becomes riskier. Second, the paper provides conditions on individual preferences for superadditivity of the RAn utility premium. Third, the paper investigates welfare changes of merging increases in risks. These results offer new interpretations of the sign of higher derivatives of the utility function.

Assessment of optimal distribution network reconfiguration results using stochastic dominance concepts

Abstract In the recent research on power and energy system optimization, many deterministic and heuristic solvers have been proposed. Each proposal claims that the new solver is better than the previous ones on the basis of performance indicators, which are often limited to the best solution found or to simple statistics (mean, median, standard deviation). This paper introduces a new and more significant performance indicator based on the concept of first-order stochastic dominance. This indicator can generally compare the solutions of a given optimization problem for which the global optimum is not known. The optimal discrete distribution network reconfiguration for a real-scale system was taken as an example problem, to show the characteristics of the proposed indicator. The results obtained show the effectiveness of the proposed indicator to limit the acceptability of “better” solvers to the ones that actually exhibit enhanced performance with respect to incrementally improved benchmarks.

On qualitative multi-attribute group decision making and its consensus measure: A probability based perspective

This paper focuses on qualitative multi-attribute group decision making (MAGDM) with linguistic information in terms of single linguistic terms and/or flexible linguistic expressions. To do so, we propose a new linguistic decision rule based on the concepts of random preference and stochastic dominance, by a probability based interpretation of weight information. The importance weights and the concept of fuzzy majority are incorporated into both the multi-attribute and collective decision rule by the so-called weighted ordered weighted averaging operator with the input parameters expressed as probability distributions over a linguistic term set. Moreover, a probability based method is proposed to measure the consensus degree between individual and collective overall random preferences based on the concept of stochastic dominance, which also takes both the importance weights and the fuzzy majority into account. As such, our proposed approaches are based on the ordinal semantics of linguistic terms and voting statistics. By this, on one hand, the strict constraint of the uniform linguistic term set in linguistic decision making can be released; on the other hand, the difference and variation of individual opinions can be captured. The proposed approaches can deal with qualitative MAGDM with single linguistic terms and flexible linguistic expressions. Two application examples taken from the literature are used to illuminate the proposed techniques by comparisons with existing studies. The results show that our proposed approaches are comparable with existing studies.

A stochastic dominance based approach to consumer-oriented Kansei evaluation with multiple priorities

Nowadays, with the increasing of aesthetic products, it becomes more and more important and quite difficult for consumers to choose their preferred products, especially the ones whose artistic and aesthetic aspects play a crucial role in consumer purchase decisions. Taking Kansei as one quality aspect of products, consumer-oriented Kansei evaluation focuses on evaluation of existing commercial products based on consumers’ Kansei preferences. This paper proposes a stochastic dominance based approach to consumer-oriented Kansei evaluation with multiple priorities. Particularly, given a consumer’s preferences, the concept of stochastic dominance is used to build an evaluation function for each Kansei attribute. Then, the importance weights captured by a priority hierarchy of Kansei attributes, together with the fuzzy majority, are incorporated into the aggregation of individual stochastic dominance degrees into an overall one. An application to the hand-painted Kutani cups in Ishikawa, Japan, is conducted to illustrate the effectiveness and efficiency of the proposed approach. It is seen that the proposed approach outperforms the existing research in terms of easy of use and better decision-support to the consumers.

On Ambiguity Apportionment

This paper investigates the notion of changes in ambiguity over loss probabilities in the smooth ambiguity model developed by Klibanoff, Marinacci and Mukerji (2005). Changes in ambiguity over loss probabilities are expressed through the specific concept of stochastic dominance of order n defined by Ekern (1980). We characterize conditions on the function capturing attitudes towards ambiguity under which an individual always considers one situation to be more ambiguous than another in a model of two states of nature. We propose an intuitive interpretation of the properties of this function in terms of preferences for harms disaggregation over probabilities, also labelled ambiguity apportionment.

On ambiguity apportionment

This paper investigates the notion of changes in ambiguity over loss probabilities in the smooth ambiguity model developed by Klibanoff, Marinacci and Mukerji (Econometrica 73:1849–1892, 2005). Changes in ambiguity over loss probabilities are expressed through the specific concept of stochastic dominance of order n defined by Ekern (Econ Lett 6:329–333, 1980). We characterize conditions on the function capturing attitudes towards ambiguity under which an individual always considers one situation to be more ambiguous than another in a model of two states of nature. We propose an intuitive interpretation of the properties of this function in terms of preferences for harms disaggregation over probabilities, also labelled ambiguity apportionment.

A Nonparametric Test for Stochastic Dominance Using Total Time on Test Transform

SYNOPTIC ABSTRACTThe concept of stochastic dominance has found important applications in the theory and applications of economics and statistics. It refers to a set of relations that may hold between a pair of distributions of random variables. In this article, we present a new definition of stochastic dominance using total time on test transform. The proposed definition based on total time on test is expressed in terms of quantile functions. The total time on test transform of order S is a generalization of the well-known Sth order stochastic dominance. A simple nonparametric test is developed to test the proposed stochastic dominance. It is shown that the distribution of the test statistic is asymptotically normal. Simulation studies are carried out to assess the efficiency of the test. We apply the proposed test to a real-life dataset of personal expenditures.

The Hellinger distance in Multicriteria Decision Making: An illustration to the TOPSIS and TODIM methods

Due to the difficulty in some situations of expressing the ratings of alternatives as exact real numbers, many well-known methods to support Multicriteria Decision Making (MCDM) have been extended to compute with many types of information. This paper focuses on the information represented as probability distribution. Many of the methods that deal with probability distribution use the concept of stochastic dominance, which imposes very strong restrictions to differentiate two probability distributions, or uses the probability distributions to obtain a quantity that will be used to rank the alternatives. This paper brings the Hellinger distance concept to the MCDM context to assist the models to deal with probability distributions in a direct way without any transformation. Transformations in the data or summary quantities may miss represent the original information. For direct comparisons among probability distributions we use the stochastic dominance degree (SDD). We illustrate how simple it can be to adapt the existing methods to deal with probability distributions through the Hellinger distance and SDD by adapting the TOPSIS and TODIM (an acronym in Portuguese of Interactive and Multicriteria Decision Making) methods.

Stochastic Dominance under the Nonlinear Expected Utilities

In 1947, von Neumann and Morgenstern introduced the well-known expected utility and the related axiomatic system (see von Neumann and Morgenstern (1953)). It is widely used in economics, for example, financial economics. But the well-known Allais paradox (see Allais (1979)) shows that the linear expected utility has some limitations sometimes. Because of this, Peng proposed a concept of nonlinear expected utility (see Peng (2005)). In this paper we propose a concept of stochastic dominance under the nonlinear expected utilities. We give sufficient conditions on which a random choiceXstochastically dominates a random choiceYunder the nonlinear expected utilities. We also provide sufficient conditions on which a random choiceXstrictly stochastically dominates a random choiceYunder the sublinear expected utilities.

Strategic rehabilitation planning of piped water networks using multi-criteria decision analysis

To overcome the difficulties of strategic asset management of water distribution networks, a pipe failure and a rehabilitation model are combined to predict the long-term performance of rehabilitation strategies. Bayesian parameter estimation is performed to calibrate the failure and replacement model based on a prior distribution inferred from three large water utilities in Switzerland. Multi-criteria decision analysis (MCDA) and scenario planning build the framework for evaluating 18 strategic rehabilitation alternatives under future uncertainty. Outcomes for three fundamental objectives (low costs, high reliability, and high intergenerational equity) are assessed. Exploitation of stochastic dominance concepts helps to identify twelve non-dominated alternatives and local sensitivity analysis of stakeholder preferences is used to rank them under four scenarios. Strategies with annual replacement of 1.5–2% of the network perform reasonably well under all scenarios. In contrast, the commonly used reactive replacement is not recommendable unless cost is the only relevant objective. Exemplified for a small Swiss water utility, this approach can readily be adapted to support strategic asset management for any utility size and based on objectives and preferences that matter to the respective decision makers.

How does pricing power affect a firm’s sourcing decisions from unreliable suppliers?

We study sourcing decisions of price-setting and price-taking firms with two unreliable suppliers, where a price-setting firm sets the retail price after the supply uncertainty is resolved and a price-taking firm takes the retail price as given. We investigate the impacts of market conditions, suppliers’ wholesale prices and their reliabilities on the optimal sourcing decisions of price-setting and price-taking firms, and examine how a firm’s pricing power affects these impacts. We define a supplier’s reliability in terms of the “size” or the “variability” of his random capacity using the concepts of stochastic dominance. We find that the supplier reliability affects the optimal sourcing decisions differently for price-setting and price-taking firms. Specifically, with a price-setting firm, a supplier can win a larger order by increasing his reliability, it is not always so with a price-taking firm.

Stochastically weighted stochastic dominance concepts with an application in capital budgeting

The problem of comparing random vectors arises in many applications. We propose three new concepts of stochastically weighted dominance for comparing random vectors X and Y. The main idea is to use a random vector V to scalarize X and Y as VTX and VTY, and subsequently use available concepts from stochastic dominance and stochastic optimization for comparison. For the case where the distributions of X, Y and V have finite support, we give (mixed-integer) linear inequalities that can be used for random vector comparison as well as for modeling of optimization problems where one of the random vectors depends on decisions to be optimized. Some advantages of the proposed new concepts are illustrated with the help of a capital budgeting example.

Using Expert Judgments to Explore Robust Alternatives for Forest Management under Climate Change

We develop and apply a judgment-based approach to selecting robust alternatives, which are defined here as reasonably likely to achieve objectives, over a range of uncertainties. The intent is to develop an approach that is more practical in terms of data and analysis requirements than current approaches, informed by the literature and experience with probability elicitation and judgmental forecasting. The context involves decisions about managing forest lands that have been severely affected by mountain pine beetles in British Columbia, a pest infestation that is climate-exacerbated. A forest management decision was developed as the basis for the context, objectives, and alternatives for land management actions, to frame and condition the judgments. A wide range of climate forecasts, taken to represent the 10-90% levels on cumulative distributions for future climate, were developed to condition judgments. An elicitation instrument was developed, tested, and revised to serve as the basis for eliciting probabilistic three-point distributions regarding the performance of selected alternatives, over a set of relevant objectives, in the short and long term. The elicitations were conducted in a workshop comprising 14 regional forest management specialists. We employed the concept of stochastic dominance to help identify robust alternatives. We used extensive sensitivity analysis to explore the patterns in the judgments, and also considered the preferred alternatives for each individual expert. The results show that two alternatives that are more flexible than the current policies are judged more likely to perform better than the current alternatives on average in terms of stochastic dominance. The results suggest judgmental approaches to robust decision making deserve greater attention and testing.