Zollanvari Amin, Genton Marc G
Department of Statistics and Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, USA.
Sankhya Ser A. 2013 Aug 1;75(2). doi: 10.1007/s13171-013-0029-9.
We provide a fundamental theorem that can be used in conjunction with Kolmogorov asymptotic conditions to derive the first moments of well-known estimators of the actual error rate in linear discriminant analysis of a multivariate Gaussian model under the assumption of a common known covariance matrix. The estimators studied in this paper are plug-in and smoothed resubstitution error estimators, both of which have not been studied before under Kolmogorov asymptotic conditions. As a result of this work, we present an optimal smoothing parameter that makes the smoothed resubstitution an unbiased estimator of the true error. For the sake of completeness, we further show how to utilize the presented fundamental theorem to achieve several previously reported results, namely the first moment of the resubstitution estimator and the actual error rate. We provide numerical examples to show the accuracy of the succeeding finite sample approximations in situations where the number of dimensions is comparable or even larger than the sample size.
我们提供了一个基本定理，在已知共同协方差矩阵的假设下，该定理可与柯尔莫哥洛夫渐近条件结合使用，以推导多元高斯模型线性判别分析中实际错误率的著名估计量的一阶矩。本文研究的估计量是代入式和平滑再代入误差估计量，在柯尔莫哥洛夫渐近条件下，此前尚未对这两者进行过研究。这项工作的结果是，我们给出了一个最优平滑参数，该参数使平滑再代入成为真实误差的无偏估计量。为了完整性，我们进一步展示了如何利用所给出的基本定理来得到几个先前报道的结果，即再代入估计量的一阶矩和实际错误率。我们提供了数值示例，以展示在维度数量与样本大小相当甚至大于样本大小的情况下，后续有限样本近似的准确性。
