Moment inequalities for higher-order (inverse) stochastic dominance

TL;DR

This paper advances higher-order stochastic dominance by proving necessary moment inequalities for the inverse higher-order order ($n$-ISD) with $n>2$, linking the $n$-ISD relation to the minimum order statistics via $\mu^X_{1:k}$ inequalities, and by showing how background risk can strengthen $n$-SD beyond moment-based comparisons. It develops both integral- and asymptotic-approximation tools, and proves a smoothing/convolution result: if $\gamma = \int (F_Y^{[n]}-F_X^{[n]}) > 0$, there exists an independent $Z$ such that $X+Z >_n Y+Z$; this yields corollaries for $n=1,2$ related to mean and variance improvements. The work deepens understanding of the interpretation and robustness of $n$-SD and $n$-ISD and provides new analytical tools for risk assessment and welfare analysis in economics and finance.

Abstract

Stochastic dominance has been studied extensively, particularly in the finance and economics literature. In this paper, we obtain two results. First, necessary conditions for higher-order inverse stochastic dominance are developed. These conditions, which involve moment inequalities of the minimum order statistics, are analogous to the ones obtained by Fishburn (1980b) for usual higher-order stochastic dominance. Second, we investigate how background risk variables influence usual higher-order stochastic dominance. The main result generalizes the ones in Pomatto et al. (2020) from the first-order and second-order stochastic dominance to the higher-order.

Figures (1)

• Figure 1: Representations of $F_X^{[-2]}(p)$ and $\widetilde{F}_X^{[-2]}(p)$

Theorems & Definitions (23)

• Theorem 2.1
• Theorem 2.2
• Corollary 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• Proposition 2.6
• proof
• Example 3.1
• Example 3.2