Seite - 342 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 Restrictingattentionnowto r,s∈{1,2}, aswemay,andrequiringm≥ s so thatF[m−s](μ)givenby(4) isdefined, (8)gives: ε [m] 1,1(μ)=∑ ∞ x=m+1xp(x)=μ∑ ∞ x=mp(x)=μ{1−F[m−1](μ)}, ε [m] 2,1(μ)=∑ ∞ x=m+1(x+1)p(x)= ε [m] 1,1(μ)+{1−F[m](μ)}, ε [m] 1,2(μ)=∑ ∞ x=m+1x 2p(x)=∑∞x=m+1{x(x−1)+x}p(x) =μ2{1−F[m−2](μ)}+ε[m]1,1(μ) and: ε [m] 2,2(μ)=∑ ∞ x=m+1(x+1) 2p(x)=∑∞x=m+1{x2+(x+1)+x}p(x) = ε [m] 1,2(μ)+ε [m] 2,1(μ)+ε [m] 1,1(μ). Accordingly, for given μ, each ε[m]r,s (μ)decreases strictly to zerowithmproviding—to anydesired accuracy—truncateandboundapproximations foreachofν,τ, andρ. In thisconnection,wenote that theupper tailprobabilities involvedherecanbeboundedbystandardChernoffarguments. References 1. Critchley,F.;Marriott,P.Computational InformationGeometry inStatistics: Theoryandpractice.Entropy 2014,16, 2454–2471. 2. Marriott,P.;Sabolova,R.;VanBever,G.;Critchley,F.Geometryofgoodness-of-fit testinginhighdimensional lowsample sizemodelling. InGeometric Science of Information: Second InternationalConference,GSI 2015, Palaiseau,France,October28–30,2015,Proceedings;Nielsen,F.,Barbaresco,F.,Eds.; Springer: Berlin,Germany, 2015;pp. 569–576. 3. Amari, S.-I.; Nagaoka, H.Methods of Information Geometry; Translations ofMathematicalMonographs; AmericanMathematicalSociety: Providence,RI,USA,2000. 4. Cressie,N.;Read,T.R.C.Multinomialgoodness-of-fit tests. J.R.Stat. Soc. B1984,46, 440–464. 5. Read,T.R.C.;Cressie,N.A.C.Goodness-of-FitStatistics forDiscreteMultivariateData; Springer:NewYork,NY, USA,1988. 6. Kass,R.E.;Vos,P.W.GeometricalFoundationsofAsymptotic Inference; JohnWiley&Sons, Inc.:Hoboken,NJ, USA,1997. 7. Agresti,A.CategoricalDataAnalysis, 3rded.;Wiley:Hoboken,NJ,USA,2013. 8. Amari,S.-I.Differential-geometricalmethods instatistics. InLectureNotes inStatistics; Springer:NewYork, NY,USA,1985;Volume28. 9. Barndorff-Nielsen,O.E.; Cox,D.R.Asymptotic Techniques forUse inStatistics; Chapman&Hall: London, UK,1989. 10. Anaya-Izquierdo,K.;Critchley,F.;Marriott, P.Whenarefirst-orderasymptoticsadequate? Adiagnostic. STAT2014,3, 17–22. 11. Lauritzen,S.L.GraphicalModels;ClarendonPress:Oxford,UK,1996. 12. Geyer,C.J.Likelihoodinference inexponential familiesanddirectionsof recession.Electron. J.Stat. 2009, 3, 259–289. 13. Fienberg, S.E.; Rinaldo, A.Maximum likelihood estimation in log-linearmodels. Ann. Stat. 2012, 40, 996–1023. 14. Eguchi,S.;Copas, J.Localmodeluncertaintyandincomplete-databias. J.R.Stat. Soc. B2005,67, 1–37. 15. Copas, J.;Eguchi,S.Likelihoodforstatisticallyequivalentmodels. J.R.Stat. Soc. B2010,72, 193–217. 342