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Entropy 2017, 19(1), 7; doi:10.3390/e19010007

A Sequence of Escort Distributions and Generalizations of Expectations on q-Exponential Family

Department of Computer Science and Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan
Academic Editors: Frédéric Barbaresco and Frank Nielsen
Received: 26 October 2016 / Revised: 16 December 2016 / Accepted: 19 December 2016 / Published: 25 December 2016
(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)
View Full-Text   |   Download PDF [257 KB, uploaded 28 December 2016]

Abstract

In the theory of complex systems, long tailed probability distributions are often discussed. For such a probability distribution, a deformed expectation with respect to an escort distribution is more useful than the standard expectation. In this paper, by generalizing such escort distributions, a sequence of escort distributions is introduced. As a consequence, it is shown that deformed expectations with respect to sequential escort distributions effectively work for anomalous statistics. In particular, it is shown that a Fisher metric on a q-exponential family can be obtained from the escort expectation with respect to the second escort distribution, and a cubic form (or an Amari–Chentsov tensor field, equivalently) is obtained from the escort expectation with respect to the third escort distribution. View Full-Text
Keywords: escort distribution; escort expectation; statistical manifold; deformed exponential family; Tsallis statistics; information geometry escort distribution; escort expectation; statistical manifold; deformed exponential family; Tsallis statistics; information geometry
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Matsuzoe, H. A Sequence of Escort Distributions and Generalizations of Expectations on q-Exponential Family. Entropy 2017, 19, 7.

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