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The reference interval in higher-order stochastic dominance

Ruodu Wang, Qinyu Wu

TL;DR

This paper shows that higher-order stochastic dominance ($n$SD) rankings can depend on the chosen reference interval, with strict strengthening as the interval contracts when $n>3$, and that mean-preserving variants are interval-invariant only when $n-m\le 3$. The authors establish rigorous equivalences between interval-based and real-line formulations for $n$SD and its mean-preserving extensions, and prove that backward implications fail for $n\ge 4$ via explicit counterexamples and constructive sequences. They unify and extend these results to the $(n,m)$-mean preserving framework, showing backward implications hold for $n-m\le 3$ and that the risk-increase family remains interval-consistent. The findings highlight robustness concerns and practical implications for applying higher-order stochastic dominance in economics and finance, emphasizing careful interval selection and interpretation of higher-order risk preferences.

Abstract

Given two random variables taking values in a bounded interval, we study whether one dominates the other in higher-order stochastic dominance depends on the reference interval in the model setting. We obtain two results. First, the stochastic dominance relations get strictly stronger when the reference interval shrinks if and only if the order of stochastic dominance is larger than three. Second, for mean-preserving stochastic dominance relations, the reference interval is irrelevant if and only if the difference between the degree of the stochastic dominance and the number of moments is no larger than three. These results highlight complications arising from using higher-order stochastic dominance in economic applications.

The reference interval in higher-order stochastic dominance

TL;DR

This paper shows that higher-order stochastic dominance (SD) rankings can depend on the chosen reference interval, with strict strengthening as the interval contracts when , and that mean-preserving variants are interval-invariant only when . The authors establish rigorous equivalences between interval-based and real-line formulations for SD and its mean-preserving extensions, and prove that backward implications fail for via explicit counterexamples and constructive sequences. They unify and extend these results to the -mean preserving framework, showing backward implications hold for and that the risk-increase family remains interval-consistent. The findings highlight robustness concerns and practical implications for applying higher-order stochastic dominance in economics and finance, emphasizing careful interval selection and interpretation of higher-order risk preferences.

Abstract

Given two random variables taking values in a bounded interval, we study whether one dominates the other in higher-order stochastic dominance depends on the reference interval in the model setting. We obtain two results. First, the stochastic dominance relations get strictly stronger when the reference interval shrinks if and only if the order of stochastic dominance is larger than three. Second, for mean-preserving stochastic dominance relations, the reference interval is irrelevant if and only if the difference between the degree of the stochastic dominance and the number of moments is no larger than three. These results highlight complications arising from using higher-order stochastic dominance in economic applications.

Paper Structure

This paper contains 6 sections, 7 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs
  4. Proof of Proposition \ref{['th-main']}
  5. Proof of Theorem \ref{['prop-nSD']}
  6. Proof of Theorem \ref{['prop-mainMN']}

Key Result

Proposition 1

Let $a,b\in\mathbb{R}$ with $a<b$ and $n\in\mathbb{N}$. For $X,Y\in\mathcal{X}_{[a,b]}$, we have and (i) $\Rightarrow$ (iv) always holds true. But (iv) $\Rightarrow$ (i) holds for all $X,Y\in \mathcal{X}_{[a,b]}$ if and only if $n\le 3$.

Figures (1)

  • Figure 1: $\mathbb{E}[(\eta - Y)_+^3] - \mathbb{E}[(\eta - X)_+^3]$

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Example 1
  • Theorem 1
  • Theorem 2
  • Lemma 1: Proposition 6 of OR01
  • proof : Proof of Step (a)
  • proof : Proof of Step (b)
  • proof : Proof of Step (c)
  • ...and 5 more