Page - 363 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 outsideW1, andvanishes inU. Defineg′onM′by g′= f1g+ f2g0 onNand g′= f2g0 outsideofN. Since f1+ f2> 0, g′ ispositivedefiniteeverywhereonM′. Since f1 vanishesoutside ofW2,g′ is smoothonM′. Finally, since f1=1and f2=0onU,g′= gonU. We cannowproveTheorem1. LetX be a randomvariable as in Theorem1. Following the notations of the theorem and the lemma, letU = { x∈M,d(x,C)< rinj } . U is open, relatively compactandcontainsC. Let (M′,g′)beas in the lemma.Let fˆ and fˆ ′bethekerneldensityestimators definedonMandM′, respectively. Theorem3.1of [1]provides thedesiredresults for fˆ ′. For r≤ rinj, thesupportandthevaluesonthesupportof fˆ ′and fˆ coincide. Thus, thedesiredresultalsoholdsfor fˆ . References 1. Pelletier,B.KerneldensityestimationonRiemannianmanifolds.Stat. Probab. Lett. 2005,73, 297–304. 2. Hendriks,H.NonparametricestimationofaprobabilitydensityonaRiemannianmanifoldusingFourier expansions.Ann. Stat. 1990,18, 832–849. 3. Asta,D.M.KernelDensityEstimationonSymmetricSpaces. InGeometricScienceof Information; Springer: Berlin/Heidelberg,Germany,2015;Volume9389,pp. 779–787. 4. Barbaresco, F.Robust statistical radarprocessing inFréchetmetric space: OS-HDR-CFARandOS-STAP processing insiegelhomogeneousboundeddomains. InProceedingsof the201112thInternationalRadar Symposium(IRS),Leipzig,Germany,7–9Septerber2011. 5. Barbaresco,F. InformationGeometryofCovarianceMatrix: Cartan-SiegelHomogeneousBoundedDomains, Mostow/BergerFibrationandFréchetMedian. InMatrix InformationGeometry;Bhatia,R.,Nielsen,F.,Eds.; Springer: Berlin/Heidelberg,Germany,2012;pp. 199–256. 6. Barbaresco,F. InformationgeometrymanifoldofToeplitzHermitianpositivedefinitecovariancematrices: Mostow/Berger fibration and Berezin quantization of Cartan-Siegel domains. Int. J. Emerg. Trends SignalProcess. 2013,1, 1–87. 7. Berezin,F.A.Quantization incomplexsymmetric spaces. Izv.Math. 1975,9, 341–379. 8. Lenz,R.SiegelDescriptors for ImageProcessing. IEEESignalProcess. Lett. 2016,25, 625–628. 9. Barbaresco, F. Robust Median-Based STAP in Inhomogeneous Secondary Data: Frechet Information Geometry ofCovarianceMatrices. In Proceedings of the 2ndFrench-Singaporian SONDRAWorkshop onEMModeling,NewConceptsandSignalProcessingForRadarDetectionandRemoteSensing,Cargese, France,25–28May2010. 10. Degurse, J.F.; Savy, L.; Molinie, J.P.; Marcos, S. A Riemannian Approach for Training Data Selection in Space-TimeAdaptive ProcessingApplications. In Proceedings of the 2013 14th International Radar Symposium(IRS), Dresden,Germany,19–21 June2013;Volume1,pp. 319–324. 11. Degurse, J.F.; Savy,L.;Marcos,S. InformationGeometry for radardetection inheterogeneousenvironments. InProceedingsof the33rdInternationalWorkshoponBayesianInferenceandMaximumEntropyMethods inScienceandEngineering,Amboise,France,21–26September2014. 12. Barbaresco,F.Koszul InformationGeometryandSouriauGeometricTemperature/CapacityofLieGroup Thermodynamics.Entropy2014,16, 4521–4565. 13. Barbaresco,F.NewGenerationofStatisticalRadarProcessingbasedonGeometricScienceof Information: InformationGeometry,MetricSpacesandLieGroupsModelsofRadarSignalManifolds. InProceedingsof the4thFrench-SingaporianRadarWorkshopSONDRA,Lacanau,France,23May2016. 14. Jeuris, B.; Vandebril, R. The Kahler mean of Block-Toeplitz matrices with Toeplitz structured block. SIAMJ.MatrixAnal.Appl. 2015,37, 1151–1175. 15. Huckemann,S.;Kim,P.;Koo, J.;Munk,A.Mobiusdeconvolutiononthehyperbolicplanwithapplication to impedancedensityestimation.Ann. Stat. 2010,38, 2465–2498. 16. Asta,D.;Shalizi,C.Geometricnetworkcomparison. 2014,arXiv:1411.1350. 363