
The Misdetect Rate
Now we will repeat the experiment we just did but in a slightly different form.
Each point of the first set has
a closest interpoint distance to the second set. There are ten points in the first set.
Likewise each point in the second set has a closest interpoint distance to the first set.
There are also ten points in the second set. Hence there are a total of twenty closest
interpoint distances associated with the point sets. We next explore as test statistics
the arithmetic mean, the harmonic mean,
the geometric mean, and the maximum interpoint distance of these twenty closest interpoint distances.
The table below shows the misdetect rate
when we perform a hypothesis test at the 1% significance level.
Test Statistic  Misdetect Rate  Power 
Min  .4097  .5903 
Arithmetic Mean  .1923  .8097 
Geometric Mean  .0245  .9755 
Harmonic Mean  .1365  .8635 
Max  .9061  .0939 
Table showing the misdetect rate for different test statistics for a 1% significance level test
We see an interesting change. If we work with all the 100 interpoint distances, then the harmonic
mean was the best with a misdetection rate of .11. But if we let each point find its best point in the other set, and we work
with these twenty best interpoint distances, the geometric mean provides the smallest misidentification
rate of .0245.
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