Our main focus is on the performance of linear allocation rules: performance
is measured by the magnitude of mis-allocated probabilities. At the "interim"
stage, we can assess the predictive probabilities about various probabilities
of error at the "end" of the experiment. For example, the "actual" error
rate for Fisher's sample linear discriminant can be estimated at the interim
stage and at the end of the experiment. At the interim stage, it may be of
interest to assess the chances that this error rate will be less than or
greater than .01, .05, .2 etc. after more observations are taken. If it is
assessed that the estimated "actual" error rate will be greater than .2 at
the "end" of the experiment with predictive probability .99, this may be
grounds to terminate the experiment at the interim stage or perhaps to
consider additional variables that might aid in lowering the error rate.
On the other extreme, if the "actual" error rate is estimated to be less
than .01 at the interim stage, and if the predictive probability that it
will remain that low is high, it may be deemed unnecessary to observe more
data or perhaps continue the experiment with enthusiasm. Similar
considerations will be made with respect to the "true" error rate defined
for the population linear discriminant.
The approach taken is Bayesian. Since the problem involves the Mahalnobis
measure of divergence D^2, which crops up in testing the similarity of two
multivariate normal populations, we initially discuss this problem in
sections 2 and 3. Section 4 considers the effect of a potential training
sample increase on the "true" errors of classification. The effect of the
training sample increase on the "actual" errors of classification is addressed
in section 5. The results are exhibited by an example presented in section
6.
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