Seite - 266 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 277 amount andsuitablyadjustingλ, thevalueof hi(x|φ)wouldbeunchanged. Weexplore thismore thoroughlybywritingthecorrespondingequations. Letussuppose,absurdly, that fordistinctφandφ′, wehaveDψ(φ|φ′)=0. BydefinitionofDψ, it isgivenbyasumofnonnegative terms,which implies thatall termsneedtobeequal tozero. Thefollowing linesareequivalent∀i∈{1,··· ,n}: hi(0|λ,μ1,μ2) = hi(0|λ′,μ′1,μ′2), λe−12(yi−μ1)2 λe−12(yi−μ1)2+(1−λ)e−12(yi−μ2)2 = λ′e−12(yi−μ′1)2 λ′e− 1 2(yi−μ′1)2+(1−λ′)e−12(yi−μ′2)2 , log ( 1−λ λ ) − 1 2 (yi−μ2)2+ 12(yi−μ1) 2 = log ( 1−λ′ λ′ ) − 1 2 (yi−μ′2)2+ 1 2 (yi−μ′1)2. Lookingat this setofnequationsasanequalityof twopolynomialsonyofdegree1atnpoints, wededuce thataswehave twodistinctobservations, say,y1 andy2, the twopolynomialsneedtohave thesamecoefficients. Thus, thesetofnequations isequivalent to the followingtwoequations:{ μ1−μ2 = μ′1−μ′2 log ( 1−λ λ ) + 12μ 2 1− 12μ22 = log ( 1−λ′ λ′ ) + 12μ ′ 1 2− 12μ′22. (21) These twoequationswith threevariableshavean infinitenumberofsolutions. Take, forexample, μ1=0, μ2=1, λ= 23, μ ′ 1= 1 2, μ ′ 2= 3 2, λ ′= 12. Remark2. Theprevious conclusion canbe extended to any two-componentmixture of exponential families having the form: pφ(y)=λe∑ m1 i=1θ1,iy i−F(θ1)+(1−λ)e∑m2i=1θ2,iyi−F(θ2). Onemaywrite the correspondingn equations. Thepolynomial of yi hasadegreeof atmostmax(m1,m2). Thus, if onedisposesofmax(m1,m2)+1distinctobservations, the twopolynomialswill have the samesetof coefficients. Finally, if (θ1,θ2)∈Rd−1withd>max(m1,m2), thenassumptionA3doesnothold. Unfortunately, we have no an information about the difference between consecutive terms ‖φk+1−φk‖except for thecaseofψ(t)= ϕ(t)=−log(t)+ t−1whichcorresponds to theclassical EMrecurrence: λk+1= 1 n n ∑ i=1 hi(0|φk), μk+11 = ∑ni=1yihi(0|φk) ∑ni=1hi(0|φk) μk+11 = ∑ni=1yihi(1|φk) ∑ni=1hi(1|φk) . Tseng[2]hasshownthatwecanprovedirectly thatφk+1−φk converges to0. 5. SimulationStudy We summarize the results of 100 experiments on 100 samples by giving the average of the estimates and theerror committed, and the correspondingstandarddeviation. The criterionerror is the totalvariationdistance (TVD),which is calculatedusing theL1distance. Indeed, theScheffé Lemma(see [20] (Page129)) states that: sup A∈Bn(R) ∣∣∣Pφ(A)−PφT(A)∣∣∣= 12 ∫ R ∣∣∣pφ(y)−pφT(y)∣∣∣dy. TheTVDgivesameasureof themaximumerrorwemaycommitwhenweuse theestimated model in lieuof the truedistribution.Weconsider theHellingerdivergence forestimatorsbasedon ϕ−divergences,whichcorresponds toϕ(t)= 12( √ t−1)2.Ourpreferenceof theHellingerdivergence is thatwehopetoobtainrobustestimatorswithout lossofefficiency(see [21]).Dψ is calculatedwith 266