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Entropy 2014, 16(4), 2023-2055; doi:10.3390/e16042023

Matrix Algebraic Properties of the Fisher Information Matrix of Stationary Processes

Rothschild Blv. 123 Apt.7, 65271 Tel Aviv, Israel
Received: 12 February 2014 / Revised: 11 March 2014 / Accepted: 24 March 2014 / Published: 8 April 2014
(This article belongs to the Special Issue Information Geometry)

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.
Keywords: Bezout matrix; Sylvester matrix; tensor Sylvester matrix; Stein equation; Vandermonde matrix; stationary process; matrix resultant; Fisher information matrix Bezout matrix; Sylvester matrix; tensor Sylvester matrix; Stein equation; Vandermonde matrix; stationary process; matrix resultant; Fisher information matrix
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Klein, A. Matrix Algebraic Properties of the Fisher Information Matrix of Stationary Processes. Entropy 2014, 16, 2023-2055.

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