Seite - 269 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 277 Contaminationwasdonebyreplacing10observationsofeachsamplechosenrandomlyby10 i.i.d. observationsdrawnfromaWeibulldistributionwithshapeν=0.9andscaleσ=3.Resultsare summarizedinTable2.Notice that itwouldhavebeenbetter touseasymmetrickernels inorder to build thekernel-basedMDϕDEsince theiruse in thecontextofpositive-supporteddistributions is advised inorder toreduce thebiasatzero, see [11] foradetailedcomparisonwithsymmetrickernels. This isnot,however, thegoalof thispaper. Inaddition, theuseofsymmetrickernels in thismixture modelgavesatisfactoryresults. Simulationsresults inTable2confirmoncemore thevalidityofourproximalpointalgorithmand theclear robustnessofboth thekernel-basedMDϕDEandtheMDPD. Table2.Themeanandthestandarddeviationof theestimatesandtheerrorscommitted ina100-run experimentofatwo-componentWeibullmixture. Thetruesetofparameter isλ=0.35,ν1=1.2,ν2=2. EstimationMethod λ sd(λ) μ1 sd(μ1) μ2 sd(μ2) TVD sd(TVD) WithoutOutliers ClassicalMDϕDE 0.356 0.066 1.245 0.228 2.055 0.237 0.052 0.025 NewMDϕDE–Silverman 0.387 0.067 1.229 0.241 2.145 0.289 0.058 0.029 MDPD a=0.5 0.354 0.068 1.238 0.230 2.071 0.345 0.056 0.029 EM(MLE) 0.355 0.066 1.245 0.228 2.054 0.237 0.052 0.025 With10%Outliers ClassicalMDϕDE 0.250 0.085 1.089 0.300 1.470 0.335 0.092 0.037 NewMDϕDE–Silverman 0.349 0.076 1.122 0.252 1.824 0.324 0.067 0.034 MDPD a=0.5 0.322 0.077 1.158 0.236 1.858 0.344 0.060 0.029 EM(MLE) 0.259 0.095 0.941 0.368 1.565 0.325 0.095 0.035 6.Conclusions We introduced in this paper a proximal-point algorithm that permits calculation of divergence-basedestimators.Westudiedthe theoretical convergenceof thealgorithmandverified it ina two-componentGaussianmixture.Weperformedseveral simulationswhichconfirmedthat the algorithmworks and is away to calculate divergence-based estimators. We also applied our proximalalgorithmonaBregmandivergenceestimator (theMDPD),andthealgorithmsucceededto produce theMDPD.Further investigationsabout theroleof theproximal termandacomparisonwith directoptimizationmethods inorder toshowthepracticaluseof thealgorithmmaybeconsidered in a futurework. Acknowledgments:Theauthorsaregrateful toLaboratoiredeStatistiqueThéoriqueetAppliquée,Université PierreetMarieCurie, forfinancial support. AuthorContributions:MichelBroniatowskiproposeduseofaproximal-pointalgorithminorder tocalculate the MDϕDE.MichelBroniatowskiproposedbuildingaworkbasedonthepaperof [2].DiaaAlMohamadproposed thegeneralization inSection2.3 andprovidedall of the convergence results in Section3. DiaaAlMohamad also conceived the simulations. Finally, Michel Broniatowski contributed to improving the textwritten by DiaaAlMohamad.Bothauthorshavereadandapprovedthefinalmanuscript. Conflictsof Interest:Theauthorsdeclarenoconflictof interest. References 1. McLachlan,G.J.;Krishnan,T.TheEMAlgorithmandExtensions;Wiley:Hoboken,NJ,USA,2007. 2. Tseng,P. AnAnalysisof theEMAlgorithmandEntropy-LikeProximalPointMethods. Math.Oper. Res. 2004,29, 27–44. 3. Chrétien,S.;Hero,A.O.GeneralizedProximalPointAlgorithmsandBundle Implementations.Available online: http://www.eecs.umich.edu/techreports/systems/cspl/cspl-316.pdf (acceesedon25July2016). 4. Goldstein,A.;Russak, I. Howgoodare theproximalpointalgorithms? Numer. Funct. Anal.Optim. 1987, 9, 709–724. 269