Stochastic Dominance | Vose Software

# Stochastic Dominance

### Introduction

The Stochastic Dominance window
from ModelRisk

Stochastic dominance tests are a statistical means of determining the superiority of one distribution over another. There are several types (or orders) of stochastic dominance. In virtually every case, the first- and second-order are all you need.

##### 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.

##### Second-order Stochastic Dominance

If

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

Stochastic dominance is great in principle but tends to be rather onerous to apply in practice particularly if one is comparing several possible options. ModelRisk has the facility to compare as many options as you wish. First of all one simulates say 5000 iterations of the outcome of each possible option and imports these into contiguous columns in a spreadsheet. Then these are fed into the ModelRisk interface.

Selecting an output location allows one to insert the stochastic dominance matrix in the spreadsheet as an array, which will show all the dominance combinations and update if the simulation output arrays are altered.

Note that first order dominance is stronger than second order dominance, so 1d implies 2d as well.

Stochastic dominance are explained more in-depth here.

### Window elements

In the {Data} field comes the array with the data to be analysed.

In the {TitleArray} field you can (optionally) specify an array in your spreadsheet that contains labels for the data. If no array is selected, these will be named A,B,C...

You can specify the orientation of the data: whether datapoints are arranged in columns (this is usually the case) or rows.

Below the graphs, a "dominance matrix" is shown. For every pair of datasets in the {data} field, it shows which one dominates over the other. 1d means first order dominance, 2d means second order dominance, as explained in the introduction above.

For explanations about other fields, buttons, graphs and summary statistics tables in this window, see

### Useful tips and tricks

The output of ModelRisk windows always corresponds to VoseFunctions (the functions ModelRisk adds to Excel) being entered into one or more spreadsheet cells.

You can always re-open the window for a ModelRisk function that is in a spreadsheet cell by using View Function. Select the spreadsheet cell and then select View Function from the ModelRisk menu/toolbar/ribbon.