Stochastic dominance tests | Vose Software

# Stochastic dominance tests

Stochastic dominance tests are a statistical means of determining the superiority of one distribution over another. There are several types (or degrees) of stochastic dominance. We have never found any particular use for any but the first and second order tests described here. It would be a very rare problem where the distributions of two options can be selected for no better reason than an very marginal ordering provided by a statistical test. In the real world, there are usually far more persuasive reasons to select one option over another: option A would expose us to a greater chance of losing money than B; or a greater maximum loss; or would cost more to implement; we feel more comfortable with option A because we've done something similar before; option B will make us more strategically placed for the future; option B is based on an analysis with fewer assumptions; etc.

### First order stochastic dominance

Consider options A, B that have the cumulative distribution functions FA(x) and FB(x),where it is desirable to maximise the value of x.

If:

FA(x) FB(x) for all x

then option A dominates option B. That amounts to saying that the cdf of option A is to the right of that of option B in an ascending plot:

Looked at another way, option A is superior to option B because for any cumulative probability value, it gives a higher profit.  First order stochastic dominance is intuitive and makes virtually no assumptions about the decision-makers utility function, only that it is continuous and monotonically increasing with increasing x.

### Second order stochastic dominance

If:

then option A has second order stochastic dominance over option B.

This figure shows option A having second (but not first) order stochastic dominance over option B. The function D(z) is also plotted to show it is always positive.

However, in this scenario, option A does not have second order stochastic dominance over option B because D(z) dips below zero.

Second order stochastic dominance makes the assumption that the decision-maker has a risk averse utility function over the entire range of the variable. This assumption is not very restrictive and will almost always apply. However, as already mentioned, it should virtually never be necessary to resort to second order (or higher) dominance tests since the decision-maker would find other, more important, differences between the available options.

ModelRisk provides a function VoseDominance that will produce a matrix of first and second order stochastic dominance test results for a set of generated outputs.