New results and tests for stochastic dominance between linear combinations

TL;DR

This paper extends theoretical results on stochastic dominance for convex combinations of i.i.d. variables, especially with infinite mean, and introduces nonparametric tests to empirically verify stochastic orderings between linear combinations.

Contribution

It broadens the class of distributions for which stochastic dominance results hold and develops two nonparametric testing procedures for stochastic orderings.

Findings

Proposed tests control size asymptotically under the null hypothesis.

Both tests are consistent against non-dominance alternatives.

Monte Carlo experiments demonstrate good finite-sample performance.

Abstract

Convex combinations of i.i.d. random variables without a finite mean can behave in a strikingly different way from the finite-mean case: as the weight vector becomes more balanced, the resulting combination may become stochastically larger, rather than less dispersed. Existing results establish stochastic dominance between pairs of linear combinations-or between a convex combination and the underlying variable-under shape restrictions on the distribution and structural assumptions on the weights. We expand the class for which the general result can be derived. Nonetheless, two practical limitations remain: (i) the sufficient conditions vary across results, and (ii) being non-necessary, they exclude many relevant configurations. Moreover, under a statistical perspective, where the true distribution of the data is assumed to be unknown, these conditions cannot be checked. Motivated by this gap, we develop nonparametric procedures to test whether two linear combinations are stochastically ordered. We propose two complementary approaches: a least-favorable calibration and a bootstrap-based method.We show that both tests control size asymptotically under the null of stochastic dominance and are consistent against alternatives of non-dominance. Monte Carlo experiments illustrate the finite-sample performance of the proposed procedures across a range of models and weight configurations.