Transformed statistical distance measures and the fisher information matrix

Most multivariate statistical techniques are based upon the concept of distance. The purpose of this paper is to introduce statistical distance measures, which are normalized Euclidean distance measures, where the covariances of observed correlated measurements x1,…,xn and entries of the Fisher information matrix (FIM) are used as weighting coefficients. The measurements are subject to random fluctuations of different magnitudes and have therefore different variabilities. A rotation of the coordinate system through a chosen angle while keeping the scatter of points given by the data fixed, is therefore considered. It is shown that when the FIM is positive definite, the appropriate statistical distance measure is a metric. In case of a singular FIM, the metric property depends on the rotation angle. The introduced statistical distance measures, are matrix related, and are based on m parameters unlike a statistical distance measure in quantum information, which is also related to the Fisher information and where the information about one parameter in a particular measurement procedure is considered. A transformed FIM of a stationary process as well as the Sylvester resultant matrix are used to ensure the relevance of the appropriate statistical distance measure. The approach used in this paper is such that matrix properties are crucial for ensuring the relevance of the introduced statistical distance measures.