Page - 352 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 computationalcomplexity.Unfortunately, eigenfunctionsof theLaplacianoperatorareknownonDn butnotoncompactsub-domains. This iswhythestudyof thismethodhasnotbeendeepened. 3.3. Kernels LetK :R+→R+beamapwhichverifies the followingproperties: (i) ∫ RdK(||x||)dx=1; (ii) ∫ Rd xK(||x||)dx=0; (iii) K(x>1)=0; (iv) sup(K(x))=K(0). Let p ∈ Dn. Generally, given apoint p on aRiemannianmanifold, expp defines an injective applicationonlyonaneighborhoodof0. On theSiegel space, expp is injectiveon thewhole space. Whenthe tangentspaceTpDn isendowedwith the local scalarproduct, ||u||= d(p,expp(u)), where ||.|| is theEuclideandistanceassociatedwiththelocalscalarproductandd(., .) is theRiemannian distance. The correspondingLebesguemeasureonTpDn is noted Lebp. Let exp∗p(Lebp)denote the push-forwardmeasureofLeppby expp. The functionθpdefinedby: θp : q → θp(q)= dvoldexp∗p(Lebp) (q) (6) is thedensityof theRiemannianmeasureonDnwithrespect to theLebesguemeasureLebp after the identificationofDn andTpDn inducedby expp (seeFigure1). Figure1.M isaRiemannianmanifold,andTxM is its tangentspaceatx. Theexponentialapplication inducesavolumechangeθxbetweenTxMandM. GivenKandapositiveradius r, theestimatorof f proposedby[1] isdefinedby: fˆk= 1 k∑i 1 rn 1 θxi(x) K ( d(x,xi) r ) . (7) Thecorrective factor θxi(x) −1 isnecessary since thekernelKoriginally integrates toonewith respect to theLebesguemeasureandnotwithrespect to theRiemannianmeasure. It canbenoticed that thisestimator is theusualkernelestimator in thecaseofEuclideanspace.Whenthecurvatureof thespace isnegative,which is thecaseof theSiegel space, thedistributionplacedovereachsample xihasxi as intrinsicmean. Thefollowingtheoremprovidesconvergencerateof theestimator. It isa directadaptationofTheorem3.1of [1]. Theorem1. Let (M,g) be aRiemannianmanifold of dimensionn andμ itsRiemannianvolumemeasure. LetXbearandomvariable taking itsvalues inacompact subsetCof (M,g). Let0< r≤ rinj,where rinj is the infimumof the injectivity radiusonC.Assumethe lawofXhasa twicedifferentiabledensity f with respect to 352