Abstract
In this survey paper, a summary of results which are to be found in a series of papers, is presented. The subject of interest is focused on matrix algebraic properties of the Fisher information matrix (FIM) of stationary processes. The FIM is an ingredient of the Cram´er-Rao inequality, and belongs to the basics of asymptotic estimation theory in mathematical statistics. The FIM is interconnected with the Sylvester, Bezout and tensor Sylvester matrices. Through these interconnections it is shown that the FIM of scalar and multiple stationary processes fulfill the resultant matrix property. A statistical distance measure involving entries of the FIM is presented. In quantum information, a different statistical distance measure is set forth. It is related to the Fisher information but where the information about one parameter in a particular measurement procedure is considered. The FIM of scalar stationary processes is also interconnected to the solutions of appropriate Stein equations, conditions for the FIM to verify certain Stein equations are formulated. The presence of Vandermonde matrices is also emphasized.
Highlights
In this survey paper, a summary of results derived and described in a series of papers, is presented.It concerns some matrix algebraic properties of the Fisher information matrixEntropy 2014, 16 of stationary processes
Through these interconnections it is shown that the Fisher information matrix (FIM) of scalar and multiple stationary processes fulfill the resultant matrix property
These factored forms of the FIM enable us to show that the FIM of scalar and multiple stationary processes fulfill the resultant matrix property
Summary
A summary of results derived and described in a series of papers, is presented. The derived interconnections are obtained by developing the necessary factorizations of the FIM in terms of the Sylvester, Bezout and tensor Sylvester matrices These factored forms of the FIM enable us to show that the FIM of scalar and multiple stationary processes fulfill the resultant matrix property. The singularity conditions of the appropriate Fisher information matrices and Sylvester, Bezout and tensor Sylvester matrices coincide, these results are described in [4,5,6]. The general and more detailed results are set forth in [12] and [13] In this survey paper it is shown that the FIM of linear stationary processes form a class of structured matrices. A statistical distance measure is expressed in terms of entries of a FIM
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