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<p><b>Abstract</b>—We give a short proof of the following result. Let <tmath>$(X,Y)$</tmath> be any distribution on <tmath>${\cal N} \times \{0,1\}$</tmath>, and let <tmath>$(X_1,Y_1),\ldots,(X_n,Y_n)$</tmath> be an i.i.d. sample drawn from this distribution. In discrimination, the Bayes error <tmath>$L^* = \inf_g {\bf P}\{g(X) \not= Y \}$</tmath> is of crucial importance. Here we show that without further conditions on the distribution of <tmath>$(X,Y)$</tmath>, no rate-of-convergence results can be obtained. Let <tmath>$\phi_n (X_1,Y_1,\ldots,X_n,Y_n)$</tmath> be an estimate of the Bayes error, and let <tmath>$\{ \phi_n(.) \}$</tmath> be a sequence of such estimates. For any sequence <tmath>$\{a_n\}$</tmath> of positive numbers converging to zero, a distribution of <tmath>$(X,Y)$</tmath> may be found such that <tmath>${\bf E} \left\{ | L^* - \phi_n (X_1,Y_1,\ldots,X_n,Y_n) | \right\} \ge a_n$</tmath> infinitely often.</p>
Discrimination, statistical pattern recognition, nonparametric estimation, Bayes error, lower bounds, rates of convergence.
András Antos, László Györfi, Luc Devroye, "Lower Bounds for Bayes Error Estimation", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 21, no. , pp. 643-645, July 1999, doi:10.1109/34.777375
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