Page - 268 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 277 goodestimates (theproportionconvergedtowards0or1)whenoutlierswereadded,andthus theEM algorithmwasreinitializedmanually. Table1.Themeanandthestandarddeviationof theestimatesandtheerrorscommitted ina100run experimentofa two-componentGaussianmixture. The truesetofparameters isλ=0.35,μ1=−2, μ2=1.5. EstimationMethod λ sd(λ) μ1 sd(μ1) μ2 sd(μ2) TVD sd(TVD) WithoutOutliers ClassicalMDϕDE 0.349 0.049 –1.989 0.207 1.511 0.151 0.061 0.029 NewMDϕDE–Silverman 0.349 0.049 –1.987 0.208 1.520 0.155 0.062 0.029 MDPD a=0.5 0.360 0.053 –1.997 0.226 1.489 0.135 0.065 0.025 EM(MLE) 0.360 0.054 –1.989 0.204 1.493 0.136 0.064 0.025 With10%Outliers ClassicalMDϕDE 0.357 0.022 –2.629 0.094 1.734 0.111 0.146 0.034 NewMDϕDE–Silverman 0.352 0.057 –1.756 0.224 1.358 0.132 0.087 0.033 MDPD a=0.5 0.364 0.056 –1.819 0.218 1.404 0.132 0.078 0.030 EM(MLE) 0.342 0.064 –2.617 0.288 1.713 0.172 0.150 0.034 Figure 1 shows the values of the estimated divergence for both Formulas (2) and (3) on a logarithmicscaleateach iterationof thealgorithm. Figure1.Decreaseof the (estimated)Hellingerdivergencebetweenthe truedensityandtheestimated model at each iteration in theGaussianmixture. Thefigure to the left is the curveof thevaluesof thekernel-baseddualFormula (3). Thefigure to theright is thecurveofvaluesof theclassicaldual Formula (2).Valuesare takenata logarithmicscale log(1+x). Concerning our simulation results, the total variation of all four estimationmethods is very closewhenweareunder themodel.Whenweaddedoutliers, theclassicalMDϕDEwasassensitive as themaximumlikelihoodestimator. Theerrorwasdoubled. Both thekernel-basedMDϕDEand theMDPDare clearly robust since the total variationof theseestimatorsunder contaminationhas slightly increased. 5.2. TheTwo-ComponentWeibullMixtureModel Weconsider a two-componentWeibullmixturewithunknownshapes ν1 = 1.2,ν2 = 2anda proportionλ=0.35. Thescalesareknownanequal toσ1=0.5,σ2=2. Thedesity functionisgivenby: pφ(x)=2λα1(2x)α1−1e−(2x) α1+(1−λ)α2 2 (x 2 )α2−1 e−( x 2) α2 . (23) 268