Any Discrimination Rule Can Have an Arbitrarily Bad Probability of Error for Finite Sample Size

Consider the basic discrimination problem based on a sample of size n drawn from the distribution of (X, Y) on the Borel sets of R
^d
x {0, 1}. If 0 ⩽ R*. < ½ is a given number, and ø
_n
→ 0 is an arbitrary positive sequence, then for any discrimination rule one can find a distribution for (X, Y), not depending upon n, with Bayes probability of error R* such that the probability of error (R
_n
) of the discrimination rule is larger than R* + ø
_n
for infinitely many n. We give a formal proof of this result, which is a generalization of a result by Cover [1]. Furthermore, sup all distributions of (X, Y) with R* = 0 R
_n
⩾ ½. Thus, any attempt to find a nontrivial distribution-free upper bound for R
_n
will fail, and any results on the rate of convergence of R
_n
to R* must use assumptions about the distribution of (X, Y).