两总体的minimax判别

THE MINIMAX DISCRIMINATION FOR TWO POPULATIONS

设两总体π1,π2对某σ-有限测度μ分别有密度函数f_1(x)和f_2(x),证明了关于π1,π2的minimax判别的存在性,唯一性(在某种意义下),给出了minimax判别的一般形式。作为定理的应用,若π1,π2具有相同协方差矩阵的正态分布,则其minimax判别恰好是建立在马氏距离基础上的距离判别。

Let two populations π1, π2 have density functions f1(x) and f2(x) respectively, with respect to a sigma-finite measure μ. The existance and uniquenees of minimax discrimination in some sense are shown. The general form of minimax discrimination is also given. As an application of the theorem, It is shown that if π1, π2 have normal distributions with equal covariance matrix, then the minimax discrimination is just distance discrimination based on Mahalanobis distance.