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Entropy2016,18, 375 4.3. RealData:MovementsofAnts Fisher [3] (Example 4.4) describes adata set of thedirections 100 ants took in response to an illuminated targetplacedat 180◦ forwhich itmaybeof interest toknowwhether theants indeed (onaverage)movetowards that target (see [15] for theoriginalpublication). Thedataset isavailable asAnts_radianswithin theRpackageCircNNTSR [16]. Thecircular samplemeanfor thisdataset isabout−176.9◦; foranominal coverageprobabilityof 1−α=95%,onegetsδH≈27.3◦,δV≈20.5◦, andδA≈9.6◦ so thatall confidencesetscontain±180◦ (seeFigure6). Thedataset’s concentration isnotveryhigh,however, so thecircularpopulationmean could—accordingtoCV—alsobe−156.4◦or162.6◦. Target Figure6.Antdata ( )placedat increasingradii tovisuallyresolve ties; inaddition, thecircularmean direction( )aswellasconfidencesetsCH ( ),CV ( ), andCA ( ) areshown. 5.Discussion We have derived two confidence sets, CH and CV, for the set of circular sample means. Bothguaranteecoverageforanyfinitesamplesizewithoutmakinganyassumptionsonthedistribution of thedata (besides that theyare independentand identicallydistributed)at thecostofpotentially being quite conservative; they are non-asymptotic anduniversal in this sense. Judging from the simulations and the real data set, CV—which estimates the variance perpendicular to the mean direction—appears tobepreferableoverCH (asexpected)andsmallenoughtobeuseful inpractice. While theasymptoticconfidenceset’sopeningangle is less thanhalf (asymptoticallyabout2/3 for α = 5%) of the one forCV in our simulations and application, it has the drawback that even for a sample size of n = 100, itmay fail to give a coverage probability close to the nominal one; inaddition,onehas toassumethat thecircularpopulationmeanisunique.Ofcourse,onecouldalso devise anasymptotically justified test for the latter but thiswouldentail a correction formultiple testing(forexampleworkingwith α2 eachtime),whichwouldalsorender theasymptoticconfidence setconservative. Further improvementswouldrequiresharper“universal”massconcentration inequalities taking thefirstor thefirst twomoments intoaccount;however, this isbeyondthescopeof thisarticle. Acknowledgments: T.Hotzwishes to thank StephanHuckemann from theGeorgiaAugustaUniversity of Göttingenfor fruitfuldiscussionsconcerningthefirstconstructionofconfidenceregionsdescribedinSection2. We acknowledge support for the Article Processing Charge by the German Research Foundation and the OpenAccess Publication Fund of the TechnischeUniversität Ilmenau. F. Kelma acknowledges support by theKlausTschiraStiftung,gemeinnützigeGesellschaft,Projekt03.126.2016. AuthorContributions:Allauthorscontributedto the theoreticalandnumerical resultsaswellas to thewriting of thearticle.Allauthorshavereadandapprovedthefinalmanuscript. Conflictsof Interest: Theauthorsdeclarenoconflictof interest. 433