Seite - 312 - in Differential Geometrical Theory of Statistics

entropy Article ASequenceofEscortDistributionsand GeneralizationsofExpectationson q-ExponentialFamily HiroshiMatsuzoe DepartmentofComputerScienceandEngineering,NagoyaInstituteofTechnology,Nagoya466-8555, Japan; matsuzoel@nitech.ac.jp;Tel.: +81-52-735-5143 AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 26October2016;Accepted: 19December2016;Published: 25December2016 Abstract: In the theoryofcomplexsystems, longtailedprobabilitydistributionsareoftendiscussed. Forsuchaprobabilitydistribution,adeformedexpectationwithrespect toanescortdistribution is moreuseful thanthestandardexpectation. In thispaper,bygeneralizingsuchescortdistributions, a sequence of escort distributions is introduced. As a consequence, it is shown that deformed expectationswithrespect tosequentialescortdistributionseffectivelyworkforanomalousstatistics. 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 expectationwith respect to the thirdescortdistribution. Keywords: escortdistribution;escortexpectation; statisticalmanifold;deformedexponential family; Tsallis statistics; informationgeometry MSC:53A15;53B50;62F99;94A14 1. Introduction Long tailedprobabilitydistributionsand their relatedprobabilitydistributionsare important objects in anomalous statistical physics (cf. [1–3]). For such long tailed probability distributions, the standardexpectationdoesnot exist ingeneral. Therefore, thenotionof escortdistributionhas been introduced[4]. Sinceanescortdistributiongivesasuitableweight for tailprobability, theescort expectationwhich is the expectationwith respect to anescortdistribution ismoreuseful than the standardone. Inanomalousstatistics, adeformedexponential functionandadeformedlogarithmfunctionplay essential roles. In fact, adeformedexponential family isan importantstatisticalmodel inanomalous statistics. Suchastatisticalmodel isdescribedbysuchadeformedexponential function. Inparticular, thesetofallq-normaldistributions(orStudent’s t-distributions,equivalently) isaq-exponential family, which isdescribedbyaq-deformedexponential function[5] (seealso [6,7]). Ontheotherhand,ageneralizedscore function isdefinedfromadeformedlogarithmfunction. In thepreviousworks, theauthorshowedthatadeformedscore function isunbiasedwithrespect to theescortexpectation[8,9]. This implies thatadeformedscore function is regardedasanestimating functiononadeformedexponential family. Inaddition, in informationgeometry, it is knownthat adeformedexponential familyhasastatisticalmanifoldstructure. Thenadeformedscore function is regarded as a tangent vector on this statisticalmanifold [6,10]. Therefore, properties of escort expectationsarecloselyrelatedtogeometric structuresonadeformedexponential family. Entropy2017,19, 7 312 www.mdpi.com/journal/entropy