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A DE BRUIJN'S IDENTITY FOR DEPENDENT RANDOM VARIABLES BASED ON COPULA THEORY

Published online by Cambridge University Press:  14 October 2015

Nayereh Bagheri Khoolenjani  and
Mohammad Hossein Alamatsaz
Nayereh Bagheri Khoolenjani
Affiliation:
Department of Statistics, University of Isfahan, Isfahan 81746-73441, Iran E-mail: n.b.khoolenjani@gmail.com; alamatho@sci.ui.ac.ir.
Mohammad Hossein Alamatsaz
Affiliation:
Department of Statistics, University of Isfahan, Isfahan 81746-73441, Iran E-mail: n.b.khoolenjani@gmail.com; alamatho@sci.ui.ac.ir.
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Abstract

De Bruijn's identity shows a link between two fundamental concepts in information theory: entropy and Fisher information. In the literature, De Bruijn's identity has been stated under the assumption of independence between input signal and an additive noise. However, in the real world, the noise could be highly dependent on the main signal. The main aim of this paper is, firstly, to extend De bruijn's identity for signal-dependent noise channels and, secondly, to study how Stein and heat identities are related to De bruijn's identity. Thus, new versions of De Bruijn's identity are introduced in the case when input signal and additive noise are dependent and are jointly distributed according to Archimedean and Gaussian copulas. It is shown that in this generalized model, the derivatives of the differential entropy can be expressed in terms of a function of Fisher information. Our finding enfolds the conventional De Bruijn's identity as some remarks. Then, the equivalence among the new De Bruijn-type identity, Stein's identity and heat equation identity is established. The paper concludes with an application of the developed results in information theory.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 1 , January 2016 , pp. 125 - 140
Copyright
Copyright © Cambridge University Press 2015 

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