Page - 273 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 whose inverse is the so called theKaniadakis’κ-logarithm logk : (0,∞)→R,which isgivenby logκ(u)= { uκ−u−κ 2κ , ifκ =0, ln(u), ifκ=0. It is clear thatexpκ(·) satisfies (a1)and (a2). Letu0: T→ (0,∞)beanymeasurable function forwhich∫ Texpκ(u0)dμ<∞.Wewill showthatu0 satisfies expression (1). Foranyu∈Randα≥1,wecanwrite expκ(αu)=α 1/|κ|(|κ|u+ √ 1/α2+ |κ|2u2)1/|κ| ≤α1/|κ|(|κ|u+ √ 1+ |κ|2u2)1/|κ| =α1/|κ|expκ(u), whereweused thatexpκ(·)= exp−κ(·). Then,weconclude that ∫ Texpκ(αu0)dμ<∞ for allα≥0. Fixany measurable functionc: T→R such that∫T ϕ(c)dμ=1. For eachλ>0,wehave∫ T expκ(c+λu0)dμ≤ 1 2 ∫ T expκ(2c)dμ+ 1 2 ∫ T expκ(2λu0)dμ ≤21/|κ|−1 ∫ T expκ(c)dμ+2 1/|κ|−1 ∫ T expκ(λu0)dμ <∞, which shows that expκ(·) satisfies (a3). Therefore, the Kaniadakis’ κ-exponential expκ(·) is an example of ϕ-function. Therestrictionthat ∫ T ϕ(c)dμ=1canbeweakened,asasserted in thenext result. Lemma1. Let c˜: T→Rbeanymeasurable functionsuchthat∫T ϕ(c˜)dμ<∞. Then,∫T ϕ(c˜+λu0)dμ<∞ for allλ>0. Proof. Notice that if ∫ T ϕ(c˜)dμ≥1, then ∫ T ϕ(c˜−αu0)dμ=1forsomeα>1. Fromthedefinitionof u0, it follows that ∫ T ϕ(c˜+λu0)dμ= ∫ T ϕ(c+(α+λ)u0)dμ<∞,where c= c˜−αu0. Nowassume that ∫ T ϕ(c˜)dμ < 1. Consider any measurable set A ⊆ T with measure 0 < μ(A) < μ(T). Letu: T→ [0,∞)beameasurable function supportedon A satisfying ϕ(c˜+u)1A = [ϕ(c˜)+α]1A, whereα=(1−∫T ϕ(c˜)dμ)/μ(A). Defining c=(c˜+u)1A+ c˜1T\A,wesee that∫T ϕ(c)dμ=1. Bythe definitionofu0,wecanwrite∫ T ϕ(c˜+λu0)dμ≤ ∫ T ϕ(c+λu0)dμ<∞, foranyλ>0, which is thedesiredresult. AsaconsequenceofLemma1,condition(a3)canbereplacedbythe followingone: (a3’) Thereexistsameasurable functionu0: T→ (0,∞) suchthat∫ T ϕ(c+λu0)dμ<∞, forallλ>0, (2) foreachmeasurable function c: T→R forwhich∫T ϕ(c)dμ<∞. Without the equivalence between conditions (a3) and (a3’), we could not generalize Rényidivergence in themannerwepropose. In fact,ϕ-functionscouldbedefineddirectly in terms of (a3’), without mentioning (a3). We chose to begin with (a3) because this condition appeared initially in [12]. 273