Seite - 235 - in Differential Geometrical Theory of Statistics

entropy Article ExplicitFormulaofKoszul–VinbergCharacteristic FunctionsforaWideClassofRegularConvexCones HideyukiIshi GraduateSchoolofMathematics,NagoyaUniversity,Nagoya464-8602, Japan;hideyuki@math.nagoya-u.ac.jp; Tel.: +81-52-789-4877 AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 16September2016;Accepted: 20October2016;Published: 26October2016 Abstract:TheKoszul–Vinbergcharacteristic functionplaysafundamentalroleinthetheoryofconvex cones.Wegiveanexplicitdescriptionof the functionandrelated integral formulas foranewclass ofconvexcones, includinghomogeneousconesandconesassociatedwithchordal (decomposable) graphs appearing in statistics. Furthermore, we discuss an application tomaximum likelihood estimationforacertainexponential familyoveraconeof thisclass. Keywords:convexcone;homogeneouscone;graphicalmodel;Koszul–Vinbergcharacteristic function 1. Introduction LetΩbeanopenconvexcone inavectorspaceZ. TheconeΩ is said toberegular ifΩcontains no straight line,which is equivalent to the conditionΩ∩ (−Ω) = {0}. In this paper,we always assumethataconvexcone isopenandregular. ThedualconeΩ∗withrespect toan innerproduct (·|·) onZ isdefinedby: Ω∗ := { ξ∈Z ; (x|ξ)>0 (∀x∈Ω\{0})} . Then,Ω∗ is again a regular open convex cone, and we have (Ω∗)∗ = Ω. The Koszul–Vinberg characteristic functionϕΩ :Ω→R>0 definedby: ϕΩ(x) := ∫ Ω∗ e−(x|ξ)dξ (x∈Ω) playsa fundamental role in the theoryof regularconvexcones [1–4]. Inparticular, ϕΩ is an important function in the theoryofconvexprogramming[5], and ithas alsobeenstudied recently in connectionwith thermodynamics [6,7]. Thereare several (notmany) classesofcones forwhichanexplicit formulaof theKoszul–Vinbergcharacteristic function isknown. Amongthem, theclassofhomogeneouscones [8–10]andtheclassofconesassociatedwithchordal graphs[11]areparticularly fruitful researchobjects. In thispaper,wepresentawideclassofcones, includingbothof them,andgiveanexplicit expressionof theKoszul–Vinbergcharacteristic function (Section3).Moreover,weget integral formulas involvingthecharacteristic functionsandtheso-called generalizedpower functions,whichare expressedas someproductofpowersofprincipalminors of real symmetricmatrices (Section4).After investigatingthemultiplicativeLegendre transformof generalizedpowerfunctions inSection5,westudyamaximumlikelihoodestimatorforaWishart-type naturalexponential familyconstructedfromthe integral formula (Section6). A regular open convex coneΩ ⊂ Z is said to be homogeneous if the linear automorphism groupGL(Ω) := {α∈GL(Z) ; αΩ=Ω} acts onΩ transitively. The conePn of positive definite n×n real symmetricmatrices isa typical exampleofhomogeneouscones. It isknown[12–16] that everyhomogeneouscone is linearly isomorphic toaconePn∩Zwithanappropriate subspaceZ of thevector spaceSym(n,R)ofalln×n real symmetricmatrices,whereZ admitsaspecificblock Entropy2016,18, 383 235 www.mdpi.com/journal/entropy