Page - 315 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 3. EscortDistributionsandGeneralizationsofExpectations Inanomalousstatistics, ageneralizedexpectation, calledanescortexpectation, isoftendiscussed since the standard expectation does not exist in general (cf. [2,5,6]). In this section, we recall generalizationsofexpectationsandintroduceasequential structureofescortdistributions. LetSqbeaq-exponential family. Foragiven p(x;θ)∈Sqwedefinetheq-escortdistributionPq(x;θ) of p(x;θ)andthenormalizedq-escortdistributionPescq (x;θ)by Pq(x;θ) := Pq,(1)(x;θ) := {p(x;θ)}q, Pescq (x;θ) := 1 Zq(p) {p(x;θ)}q, where Zq(p)= ∫ Ω {p(x;θ)}qdx, respectively. For a q-exponential familySq = {pq(x;θ)}, the set ofnormalizedescortdistributions Sq′={Pescq (x;θ)} isaq′-exponential familywithq′=(2q−1)/q. Example 2. Let tν(x;μ,σ) be a univariate Student’s t-distribution with degree of freedom ν. Then its normalized escort distribution is also a univariate Student’s t-distribution with degree of freedom ν+2. In fact, fromEquation (4), adirect calculationshows that q′= 2q−1 q = ν+5 ν+3 . This implies that thedegree of freedomν′= ν+2. Therefore,weobtaina sequenceof escortdistributions froma givenStudent’s t-distribution tν: tν→ tν+2→ tν+4→··· . This sequence is called a τ-sequence, and the procedure to obtain from a given t-distribution to another t-distribution throughanescortdistribution is called theτ-transformation [16]. Foragiven pq(x;θ)∈Sq,wecandefinetheescortofanescortdistribution P˜q(x;θ) :=Pq,(2)(x;θ) := q{Pq(x;θ)}q ′ = q{pq(x;θ)}2q−1. We call P˜q(x;θ) the second escort distributionof pq(x;θ). The coefficient qbefore {pq(x;θ)}2q−1 comes from considerations ofU-information geometry [17]. Wewill discuss in the latter part of Section5. Similarly, we can define the n-th escort distribution Pq,(n)(x;θ) from the sequence of escortdistributions: Pq,(n)(x;θ) :={q(2q−1) · · ·((n−1)q−(n−2))}{pq(x;θ)}nq−(n−1). (5) Let f(x)beafunctiononΩ. Theq-expectationEq,p[f(x)]andthenormalizedq-expectationEescq,p[f(x)] withrespect to p(x;θ)∈Sq aredefinedby Eq,p[f(x)] := ∫ Ω f(x)Pq(x;θ)dx, Eescq,p[f(x)] := ∫ Ω f(x)Pescq (x;θ)dx, respectively.Wedenoteby E˜q,p[f(x)] theexpectationwithrespect to thesecondescortdistribution P˜q(x;θ), that is, E˜q,p[f(x)] := ∫ Ω f(x)P˜q(x;θ)dx = q ∫ Ω f(x){pq(x;θ)}2q−1dx. 315