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Entropy2016,18, 375
Table1.Results forsimulation1(twopointsofequalmassat±10◦)basedon10,000repetitionswith
n=400observationseach: averageobservedδH,δV, andδA (withcorrespondingstandarddeviation),
aswellas frequency(withcorrespondingstandarderror)withwhichμ=1wascoveredbyCH,CV,
andCA, respectively; thenominalcoverageprobabilitywas1−α=95%.
ConfidenceSet Meanδ (±s.d.) CoverageFrequency(±s.e.)
CH 8.2◦ (±0.0005◦) 100.0%(±0.0%)
CV 2.4◦ (±0.0025◦) 100.0%(±0.0%)
CA 1.0◦ (±0.0019◦) 94.8%(±0.2%)
Table 1 shows the results based on 10,000 repetitions for a nominal coverage probability of
1−α=95%: the average δH is about 3.5 times larger than δV,which is about twice as large as δA.
Asexpected, theasymptoticsareratherprecise in this situation:CAdidcover the truemeaninabout
95%of thecases,which implies that theotherconfidencesetsarequiteconservative; indeedCH and
CV covered the truemean inall repetitions. Onemayalsonote that theanglesvariedonlya little
betweenrepetitions.
4.2. Simulation2: ThreePointsPlacedAsymmetrically
Secondly,weconsiderasituationwhichhasbeendesignedtoshowthatevenaconsiderably large
sample size (n= 100)doesnotguaranteeapproximate coverage for theasymptotic confidence set
CA: thedistributionofZ is concentratedonthreepoints,ξj= exp(θjπi/180), j=1,2,3withweights
ωj=P(Z= ξj)chosensuchthatEZ= |EZ|=0.9 (implyingasmallvarianceandμ=1),ω1=1%
andArgξ1> 0,whileArgξ2,Argξ3< 0. Innumbers, θ1≈ 25.8, θ2≈−0.3, and θ3≈−179.7 (in ◦)
whileω2≈94%,andω3≈5%(seeFigure5).
0 θ1=25.8◦
θ2=−0.3◦θ3=−179.7◦
EZ
Figure5.ThreepointsplacedasymmetricallywithdifferentmassesandtheirEuclideanmean.
Theresultsbasedon10,000repetitionsareshowninTable2whereanominalcoverageprobability
of1−α=90%wasprescribed.Clearly,CAwith itscoverageprobabilityof less than64%performs
quitepoorlywhile theothers are conservative; δV ≈ 5◦ still appears small enough tobeuseful in
practice, though.
Table2.Results forsimulation2 (threepointsplacedasymmetrically)basedon10,000repetitionswith
n=100observationseach: averageobservedδH,δV, andδA (withcorrespondingstandarddeviation),
aswellas frequency(withcorrespondingstandarderror)withwhichμ=1wascoveredbyCH,CV,
andCA, respectively; thenominalcoverageprobabilitywas1−α=90%.
ConfidenceSet Meanδ (±s.d.) CoverageFrequency(±s.e.)
CH 16.5◦ (±0.85◦) 100.0%(±0.0%)
CV 5.0◦ (±0.38◦) 100.0%(±0.0%)
CA 0.4◦ (±0.28◦) 62.8%(±0.5%)
432
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