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entropy Article Non-AsymptoticConfidenceSetsforCircularMeans† ThomasHotz*,‡,FlorianKelma‡ andJohannesWieditz ‡ Institut fürMathematik,TechnischeUniversität Ilmenau,98684Ilmenau,Germany; florian.kelma@tu-ilmenau.de (F.K.); johannes.wieditz@tu-ilmenau.de (J.W.) * Correspondence: thomas.hotz@tu-ilmenau.de;Tel.: +49-3677-69-3627 † Thispaper isanextendedversionofourpaperpublishedinProceedingsof the2ndInternational ConferenceonGeometricScienceof Information,Palaiseau,France,28–30October2015;Nielsen,F., Barbaresco,F.,Eds.;LectureNotes inComputerScience,Volume9389;Springer InternationalPublishing: Cham,Switzerland,2015;pp. 635–642. ‡ Theseauthorscontributedequally to thiswork. AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 15 July2016;Accepted: 13October2016;Published: 20October2016 Abstract: Themeanofdataon theunit circle isdefinedas theminimizerof the average squared Euclideandistancetothedata. BasedonHoeffding’smassconcentrationinequalities,non-asymptotic confidencesetsforcircularmeansareconstructedwhichareuniversal inthesensethattheyrequireno distributionalassumptions. Theseare thencomparedwithasymptotic confidencesets insimulations andforarealdataset. Keywords: directionaldata; circularmean;universal confidence sets; non-asymptotic confidence sets;massconcentration inequalities;Hoeffding’s inequality MSC:62H11;62G15 1. Introduction Inapplications,dataassumingvaluesonthecircle, i.e., circulardata, arise frequently,examples beingmeasurementsofwinddirections, or timeof theday thatpatientsareadmitted toahospital unit. We refer to the literature, e.g., [1–5], for anoverviewof statisticalmethods for circular data, inparticular theonesdescribed in this section. Here,wewill concernourselveswith thearguablysimplest statistic, themean.However,given thatacircledoesnotcarryavectorspacestructure, i.e., there isneitheranaturaladditionofpointson thecirclenorcanonedivide thembyanaturalnumber,whatshouldthemeaningof“mean”be? Inorder tosimplify theexposition,wespecificallyconsider theunit circle in thecomplexplane, S1= {z∈C : |z|= 1}, andweassumethedatacanbemodelledas independent randomvariables Z1, . . . ,Zn which are identically distributed as the randomvariable Z taking values in S1. In the literature,however, thecircle isoftentakento lie in therealplaneR2, i.e.,whilewedenote thepoint onthecirclecorrespondingtoanangleθ∈ (−π,π]byexp(iθ)= cos(θ)+ isin(θ)∈Conemaytake it tobe (cosθ,sinθ)∈R2. Ofcourse,C isarealvectorspace,sotheEuclideansamplemeanZ¯n= 1n∑ n k=1Zk∈C iswell-defined. However,unlessallZk take identicalvalues, itwill (by thestrict convexityof theclosedunitdisc) lie inside thecircle, i.e., itsmodulus |Z¯n|willbe less than1. Though Z¯n cannotbe takenasameanonthe circle, if Z¯n =0,onemightsaythat it specifiesadirection; this leads to the ideaofcalling Z¯n/|Z¯n| the circularsamplemeanof thedata. Entropy2016,18, 375 424 www.mdpi.com/journal/entropy