### Abstract

We give a short proof of the following result. Let (X, Y) be any distribution on N×{0, 1}, and let (X_{1}, Y_{1}), ..., (X_{n}, Y_{n}) be an i.i.d. sample drawn from this distribution. In discrimination, the Bayes error L* = inf_{g} P{g(X)≠Y} is of crucial importance. Here we show that without further conditions on the distribution of (X, Y), no rate-of-convergence results can be obtained. Let φ_{n}(X_{1}, Y_{1}, ..., X_{n}, Y_{n}) be an estimate of the Bayes error, and let {φ_{n}(.)} be a sequence of such estimates. For any sequence {a_{n}} of positive numbers converging to zero, a distribution of (X, Y) may be found such that E{|L*-φ_{n}(X_{1}, Y_{1}, ..., X_{n}, Y_{n})|}≥a_{n} infinitely often.

Original language | English |
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Pages (from-to) | 643-645 |

Number of pages | 3 |

Journal | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Volume | 21 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jan 1 1999 |

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### ASJC Scopus subject areas

- Software
- Computer Vision and Pattern Recognition
- Computational Theory and Mathematics
- Artificial Intelligence
- Applied Mathematics

### Cite this

*IEEE Transactions on Pattern Analysis and Machine Intelligence*,

*21*(7), 643-645. https://doi.org/10.1109/34.777375