The quality of training sample estimates of the Bhattacharyya coefficient

The quality, in terms of the bias and variance, of estimates of the Bhattacharyya coefficient based on n training samples from two classes described by multivariate Gaussian distributions is considered. The case where the classes are described by a common covariance matrix, as well as the case where each class is described by a different covariance matrix, is analyzed. Expressions for the bias and the variance of estimates of the Bhattacharyya coefficient are derived, and numerical examples are used to show the relationship between these parameters, the number of training samples, and the dimensionality of the observation space