Four families of measures of entropy
type: Item
instance of: scholarly article
title: Four families of measures of entropy
zbMATH Open document ID: 0589.62007
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The following four new parametric measures of entropy for a probability distribution \(P=(p_ 1,p_ 2,...,p_ n)\) are proposed \[ H_ a(P)=- \sum^{n}_{i=1}p_ i \ln p_ i+a^{-1}\sum^{n}_{i=1}((1+ap_ i)\ln (1+ap_ i)-ap_ i], \] \[ H_ b(P)=-\sum^{n}_{i=1}p_ i \ln p_ i+b^{-1}\sum^{n}_{i=1}(1+bp_ i)l\quad n (1+bp_ i)-b^{- 1}(1+b)\ln (1+b), \] \[ H_ c(P)=-\sum^{n}_{i=1}p_ i \ln p_ i+c^{-2}\sum^{n}_{i=1}[(1+cp_ i\quad) \ln (1+cp_ i)-cp_ i], \] \[ H_ d(P)=-\sum^{n}_{i=1}p_ i \ln p_ i+d^{- 2}\sum^{n}_{i=1}(1+dp_ i) \ln (1+dp_ i)-d^{-2}(1+d)\ln (1+d), \] where a, b, c, d are real numbers greater than -1. Each of them is a continuous concave function of \(p_ 1,p_ 2,...,p_ n\) which is maximum when all the probabilities are equal and which is minimum when one of the probabilities is unity and all others are zero. Each includes Shannon's measure as a limiting case. However unlike Shannon's measure, these are non-additive, but have the advantage of greater flexibility in applications due to the presence of the parameters. The variations of these entropies with the parameters are also studied. The main advantage is that when they are maximized subject to linear constraints by using Lagrange's method in accordance with the maximum entropy principle, automatically they give non-negative probabilities. In particular when \(H_ a(P)\) is maximized subject to the average energy constraint, this gives rise to Maxwell-Boltzmann, Fermi-Dirac, Bose- Einstein distributions as well as to a new category of intermediate statistical distributions.
Mathematics Subject Classification ID: 62B10
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 3947397
zbMATH Keywords: parametric measures of entropy
zbMATH Keywords: continuous concave function
zbMATH Keywords: Shannon's measure
zbMATH Keywords: non-additive
zbMATH Keywords: linear constraints
zbMATH Keywords: Lagrange's method
zbMATH Keywords: maximum entropy principle
zbMATH Keywords: Maxwell-Boltzmann
zbMATH Keywords: Fermi-Dirac
zbMATH Keywords: Bose-Einstein
author: J. N. Kapur
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3947397