We propose a new quadratic discriminant function. First, we show that the eigenvalues of a covariance matrix obtained from samples are biased. Then, we describe how to rectify them and how to use the rectified eigenvalues. In order to derive the relation between sample eigenvalues and true eigenvalues, we analyze the biases of the expected sample eigenvalues by using the perturbation method and obtain approximate simultaneous linear equations to rectify sample eigenvalues in multidimensional normal cases. Finally, we show by a Monte Carlo method that our discriminant function works effectively in eight-dimensional normal cases especially in the case of a small sample size.
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