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Any Discrimination Rule Can Have an Arbitrarily Bad Probability of Error for Finite Sample Size
February 1982 (vol. 4 no. 2)
pp. 154-157
Luc Devroye, School of Computer Science, McGill University, Montreal, P.Q., Canada.
Consider the basic discrimination problem based on a sample of size n drawn from the distribution of (X, Y) on the Borel sets of Rdx {O, 1}. If 0 < R*< is a given number, and '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 (Rn) of the discrimination rule is larger than R* + 'On for infinitely many n. We give a formal proof of this result, which is a generalization of a result by Cover [1].
Citation:
Luc Devroye, "Any Discrimination Rule Can Have an Arbitrarily Bad Probability of Error for Finite Sample Size," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 4, no. 2, pp. 154-157, Feb. 1982, doi:10.1109/TPAMI.1982.4767222
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