Nonparametric tests for stochastic dominance between linear combinations of risks

Abstract

Weighted aggregations of i.i.d. risks 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. 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 establish their asymptotic validity under the null of stochastic dominance and their consistency against the alternative of non-dominance. Monte Carlo experiments illustrate the finite-sample performance of the proposed procedures across a range of models and weight configurations. An application to insurance claim-severity data illustrates how the tests can be used to assess whether pooling leads to a stochastically larger loss.