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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
Differential Geometrical Theory of Statistics
- Title
- Differential Geometrical Theory of Statistics
- Authors
- Frédéric Barbaresco
- Frank Nielsen
- Editor
- MDPI
- Location
- Basel
- Date
- 2017
- Language
- English
- License
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Size
- 17.0 x 24.4 cm
- Pages
- 476
- Keywords
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Categories
- Naturwissenschaften Physik