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entropy Article TheInformationGeometryofSparse Goodness-of-FitTesting PaulMarriott 1,*,RadkaSabolová2,GermainVanBever3 andFrankCritchley2 1 DepartmentofStatisticsandActuarialScience,UniversityofWaterloo,200UniversityAvenueWest, Waterloo,ONN2L3G1,Canada 2 SchoolofMathematicsandStatistics,TheOpenUniversity,WaltonHall,MiltonKeynes, BuckinghamshireMK76AA,UK;radka.sabolova@open.ac.uk(R.S.); f.critchley@open.ac.uk(F.C.) 3 DepartmentofMathematics&ECARES,Université libredeBruxelles,AvenueF.D.Roosevelt42, 1050Brussels,Belgium;gvbever@ulb.ac.be * Correspondence: pmarriot@uwaterloo.ca;Tel.: +1-519-888-4567 AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 31August2016;Accepted: 19November2016;Published: 24November2016 Abstract: Thispapertakesaninformation-geometricapproachtothechallengingissueofgoodness-of-fit testinginthehighdimensional, lowsamplesizecontextwhere—potentially—boundaryeffectsdominate. Themaincontributionsof thispaperare threefold: first,wepresentandprove twonewtheoremson thebehaviourofcommonlyusedtest statistics in thiscontext; second,weinvestigate—inthenovel environmentof theextendedmultinomialmodel—the linksbetweeninformationgeometry-based divergences andstandardgoodness-of-fit statistics, allowingus to formalise relationshipswhich havebeenmissing in the literature;finally,weusesimulationstudies tovalidateandillustrateour theoretical resultsandtoexplorecurrentlyopenresearchquestionsabout thewaythatdiscretisation effects can dominate sampling distributions near the boundary. Novelly accommodating these discretisationeffects contrasts sharplywith theessentially continuousapproachof skewnessand othercorrectionsflowingfromstandardhigher-orderasymptoticanalysis. Keywords: extendedmultinomialmodels;goodness-of-fit testing; informationgeometry 1. Introduction Westartbyemphasising the threefoldachievementsof thispaper, spelledout indetail in termsof thepaper’s sectionstructurebelow. First,wepresentandprovetwonewtheoremsonthebehaviour ofsomestandardgoodness-of-fit statistics in thehighdimensional, lowsamplesizecontext, focusing onbehaviour “near the boundary”of the extendedmultinomial family. Wealso comment on the methodsofproofwhichallowexplicit calculationsofhigherordermoments in thiscontext. Second, workingagainexplicitly in theextendedmultinomial context,wefillahole in the literatureby linking information-geometric-baseddivergences and standard goodness-of-fit statistics. Finally, weuse simulationstudies toexplorediscretisationeffects that candominate samplingdistributions“near theboundary”. Indeed,we illustrate andexplorehow—in thehighdimensional, lowsample size context—alldistributionsareaffectedbyboundaryeffects. Wealsouse these simulation results to explorecurrentlyopenresearchquestions.Ascanbeseen, theoverarchingthemeis the importance of working in the geometry of the extended exponential family [1], rather than the traditional manifold-basedstructureof informationgeometry. Inmore detail, the paper extends and builds on the results of [2], andweuse notation and definitionsconsistentlyacross these twopapers. Bothpapers investigate the issueofgoodness-of-fit testing inthehighdimensionalsparseextendedmultinomialcontext,usingthetoolsofComputational InformationGeometry (CIG) [1]. Entropy2016,18, 421 325 www.mdpi.com/journal/entropy