Page - 267 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 277 ψ(t)= 12( √ t−1)2. Thekernel-basedMDϕDEiscalculatedusingtheGaussiankernel,andthewindow is calculatedusingSilverman’s rule. We included in the comparison theminimumdensitypower divergence (MDPD)of [14]. Theestimator isdefinedby: φˆn = arginf φ∈Φ ∫ p1+aφ (z)dz− a+1 a 1 n n ∑ i paφ(yi) = arginf φ∈Φ EPφ [ paφ ] − a+1 a EPn [ paφ ] , (22) wherea∈ (0,1]. This isaBregmandivergenceandisknowntohavegoodefficiencyandrobustness for agoodchoiceof thetradeoffparameter.Accordingtothesimulationresults in[11], thevalueofa=0.5 seems togiveagood tradeoff between robustness against outliers andagoodperformanceunder themodel.Notice that theMDPDcoincideswithMLEwhen a tends tozero. Thus,ourmethodology presentedhere in thisarticle, isapplicableonthisestimatorandtheproximalpointalgorithmcanbe usedtocalculate theMDPD.Theproximal termwillbekept thesame, i.e.,ψ(t)= 12( √ t−1)2. Remark 3 (Note on the robustness of theused estimators). In Section 3, we have proved undermild conditions that the proximal point algorithm (11) ensures the decrease of the estimated divergence. This means thatwhenwe use the dual Formulas (2) and (3), then the proximal point algorithm (11) returns at convergence the estimatorsdefinedby (4) and (5), respectively. Similarly, ifweuse thedensitypowerdivergence ofBasuet al. [14], then theproximal-pointalgorithmreturnsat convergence theMDPDdefinedby(22). The robustness properties of the dual estimators (4) and (5) are studied in [12] and [11] respectively using the influence function (IF)approach.Ontheotherhand, the robustnessproperties of theMDPDare studiedusing the IFapproach in [14]. TheMDϕDE(4)hasgenerally anunbounded IF (see [12]Section3.1),whereas the kernel-basedMDϕDE’s IFmaybebounded for example inaGaussianmodel and foranyϕ−divergencewith ϕ=ϕγwithγ∈ (0,1), see [11]Example2.Ontheotherhand, theMDPDhasgenerallyabounded IF if the tradeoff parameter a ispositive, and, inparticular, in theGaussianmodel. TheMDPDbecomesmore robustas the tradeoff parametera increases (seeSection3.3 in [14]). Therefore,weshouldexpect that theproximalpoint algorithmproduces robust estimators in the caseof thekernel-basedMDϕDEandtheMDPD,andthusobtain better results than theMLEcalculatedusing theEMalgorithm. Simulations from twomixturemodels are given below—aGaussianmixture and aWeibull mixture. TheMLEforbothmixtureswascalculatedusingtheEMalgorithm. OptimizationswerecarriedoutusingtheNelder–Meadalgorithm[22]under thestatistical tool R[23].Numerical integrations in theGaussianmixturewerecalculatedusingthedistrExIntegrate functionofpackagedistrEx. It isaslightmodificationofthestandardfunctionintegrate. Itperforms aGauss–Legendrequadraturewhen function integrate returns an error. In theWeibullmixture, we used the integral function from package pracma. Function integral includes a variety of adaptive numerical integrationmethods such asKronrod–Gauss quadrature, Romberg’smethod, Gauss–Richardsonquadrature,Clenshaw–Curtis (notadaptive)and(adaptive)Simpson’smethod. Althoughfunctionintegral is slow, itperformsbetter thanother functionsevenif the integrandhas arelativelybadbehavior. 5.1. TheTwo-ComponentGaussianMixtureRevisited We consider the Gaussian mixture (17) presented earlier with true parameters λ = 0.35, μ1=−2,μ2 = 1.5 and known variances equal to 1. Contamination was done by adding in the originalsampletothefivelowestvaluesrandomobservationsfromtheuniformdistributionU[−5,−2]. Wealsoaddedto thefive largestvalues randomobservations fromtheuniformdistributionU[2,5]. Results are summarized inTable 1. TheEMalgorithmwas initializedaccording to condition (20). Thisconditiongavegoodresultswhenweareunder themodel,whereas itdidnotalwaysresult in 267