Page - 432 - in Differential Geometrical Theory of Statistics

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