Page - 314 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 In thispaper,wefocusontheq-exponential case.However,manyresults for theq-exponential family canbegeneralized for the χ-exponential family (cf. [6,8]). We remark that a q-exponential familyandaχ-exponential familyhavefurthergeneralizations. See [15], forexample. Example1 (Student’s t-distribution (cf. [2,6,7])). Fix anumber q (1< q< 1+2/d, d∈N), and set ν = −d−2/(1−q). We define a d-dimensional Student’s t-distribution with degree of freedom ν or a q-Gaussiandistributionby pq(x;μ,Σ) := Γ ( 1 q−1 ) (πν) d 2Γ ( ν 2 )√ det(Σ) [ 1+ 1 ν t(x−μ)Σ−1(x−μ) ] 1 1−q , whereX= t(X1, . . . ,Xd) is a randomvector onRd,μ= t(μ1, . . . ,μd) is a locationvector onRd andΣ is a scalematrixonSym+(d). Forsimplicity,weassumethatΣ is invertible.Otherwise,weshouldchooseasuitable basis{vα}onSym+(d) suchthatΣ=∑αwαvα. Then, thesetof allStudent’s t-distributions isaq-exponential family. In fact, settingparametersby zq= (πν) d 2Γ ( ν 2 )√ det(Σ) Γ ( 1 q−1 ) , R˜= zq−1q (1−q)d+2Σ −1, and θ=2R˜μ, (2) wehave pq(x;μ,Σ) = 1 zq [ 1+ 1 ν t(x−μ)Σ−1(x−μ) ] 1 1−q = [( 1 zq )1−q − 1−q (1−q)d+2 ( 1 zq )1−q t(x−μ)Σ−1(x−μ) ] 1 1−q = expq [ −t(x−μ)R˜(x−μ)+ lnq 1zq ] = expq [ d ∑ i=1 θixi− d ∑ i=1 R˜iix2i −2∑ i<j R˜ijxixj− 14 tθR˜−1θ+ lnq 1 zq ] . Since θ ∈ Rd and R˜ ∈ Sym+(d), the set of all Student’s t-distributions is a d(d+3)/2-dimensional q-exponential family. Thenormalizationψ(θ) isgivenby ψ(θ)= 1 4 tθR˜−1θ− lnq 1zq . A univariate Student’s t-distribution is awell-knownprobability distribution in elementary statistics.Wedenote itby tν(x;μ,σ) := 1 Zq expq [ − (x−μ) 2 (3−q)σ2 ] , (3) where μ ∈ R is a locationparameter, σ ∈ R++ is a scale parameter, and Zq is the normalization definedby Zq= √ 3−q q−1Beta ( 3−q 2(q−1), 1 2 ) σ. In thiscase, thedegreeof freedomisν=(3−q)/(q−1). Conversely, theparameterq isgiveby q= ν+3 ν+1 . (4) 314