  We introduce a 2-dimensional stochastic dominance (2DSD) index to
characterize both strict and almost stochastic dominance. Based on this index,
we derive an estimator for the minimum violation ratio (MVR), also known as the
critical parameter, of the almost stochastic ordering condition between two
variables. We determine the asymptotic properties of the empirical 2DSD index
and MVR for the most frequently used stochastic orders. We also provide
conditions under which the bootstrap estimators of these quantities are
strongly consistent. As an application, we develop consistent bootstrap testing
procedures for almost stochastic dominance. The performance of the tests is
checked via simulations and the analysis of real data.
