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A new stochastic dominance criterion for dependent random variables with applications

F. Belzunce, C. Martínez-Riquelme

TL;DR

This paper introduces a weak joint stochastic dominance criterion $X \le_{\text{st:wj}} Y$ that accounts for the dependence structure between paired random variables, addressing shortcomings of traditional stochastic dominance and related tests in non-normal or dependent settings. It establishes the theoretical relationship between st:wj and existing orders, and develops a nonparametric, KS-type test with a Gaussian-process limit for practical inference, including handling discrete and ordinal data. An empirical finance application demonstrates how the new criterion can guide portfolio choices when standard dominance fails, highlighting its potential for improving asset allocation decisions. The work also discusses properties, limitations, and directions for future research, notably multivariate extensions and transformation behavior.

Abstract

In this paper we develop a new tool for the comparison of paired data based on a new criterion of stochastic dominance that takes into account the dependence structure of the random variables under comparison. This new procedure provides a more detailed comparison of dependent random variables and overcomes some difficulties of standard techniques like Student's t and Wilcoxon-Mann-Whitney tests for non normal data. This tool provides an alternative to the usual stochastic dominance criterion which only considers the marginal distributions in the comparison. We show how this new tool can be fruitfully used for the comparison of paired asset returns.

A new stochastic dominance criterion for dependent random variables with applications

TL;DR

This paper introduces a weak joint stochastic dominance criterion that accounts for the dependence structure between paired random variables, addressing shortcomings of traditional stochastic dominance and related tests in non-normal or dependent settings. It establishes the theoretical relationship between st:wj and existing orders, and develops a nonparametric, KS-type test with a Gaussian-process limit for practical inference, including handling discrete and ordinal data. An empirical finance application demonstrates how the new criterion can guide portfolio choices when standard dominance fails, highlighting its potential for improving asset allocation decisions. The work also discusses properties, limitations, and directions for future research, notably multivariate extensions and transformation behavior.

Abstract

In this paper we develop a new tool for the comparison of paired data based on a new criterion of stochastic dominance that takes into account the dependence structure of the random variables under comparison. This new procedure provides a more detailed comparison of dependent random variables and overcomes some difficulties of standard techniques like Student's t and Wilcoxon-Mann-Whitney tests for non normal data. This tool provides an alternative to the usual stochastic dominance criterion which only considers the marginal distributions in the comparison. We show how this new tool can be fruitfully used for the comparison of paired asset returns.
Paper Structure (6 sections, 8 theorems, 40 equations, 6 figures, 1 table)

This paper contains 6 sections, 8 theorems, 40 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Stochastic dominance among two dependent random variables
  3. A non parametric asymptotic test for the weak joint stochastic dominance
  4. Monte Carlo results
  5. An application in finance
  6. Conclusions and future research

Key Result

Theorem 2.7

Given a bivariate random vector $(X,Y)$, if $X\le_{\textup{st:j}}Y$, then $X\le_{\textup{st:wj}}Y$.

Figures (6)

  • Figure 1: A discrete distribution on $\mathbb R^2$ where each point has the same probability equal to 1/6.
  • Figure 2: Joint density functions for $(X,Y)\sim N(\mu,V)$, where $\mu=(1,2)$, the variances are 2 and 1, the correlation is 0.8 (on the left) and -0.8 (on the right), where $t=1.2$.
  • Figure 3: Survival functions for $X-Y$ and $Y-X$ according to cases 4, 5, 6, 7 and 8.
  • Figure 4: Scatter plot for Twitter and Facebook weekly returns (on left side), Q-Q plot for Twitter and Facebook returns (on the right side, above) and Q-Q plot for the difference between Twitter (X) and Facebook (Y) returns (on the right side, below).
  • Figure 5: Scatter plot for Amazon and Facebook weekly returns (on left side), Q-Q plot for Amazon and Facebook returns (on the right side, above) and Q-Q plot for the difference between Amazon (X) and Facebook (Y) returns (on the right side, below).
  • ...and 1 more figures

Theorems & Definitions (23)

  • Example 1.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • Theorem 2.7
  • proof
  • Theorem 2.9
  • proof
  • ...and 13 more