Comparison Between Wilcoxon and LeastSquares Regression MethodsDereck Shen Benjamin Andrews22s 166 ProjectDecember 6, 2009IntroductionISimulated data was used to compare two regressionproceduresIWilcoxon method and Least Squares methodIWhen error terms follow different distributions, how doesthis affect regression coefficient estimates?The Least Squares Regression MethodIParametric regression methodIAssumptions include independent error terms,homoscedasticity, normal error distributionIEstimated by R function lmIA linear model with one regressor:yi= β0+ β1xi+ iIThe least square estimators:ˆβ1=Pni=1(xi− ¯x)(yi− ¯y)Pni=1(xi− ¯x)2ˆβ0= ¯y −ˆβ1¯xThe Wilcoxon Regression MethodIRank-based regression methodILess sensitive to outliersIEstimated using R function wwfit by Professor JoeMcKean of Western Michigan. Functions are available onhis website.ILet ei(β1) = yi− β1xiIThe Wilcoxon estimators are:ˆβ1minimizesnXi=1R(ei(β1)) −n + 12ei(β1)where R(ei(β1)) denotes the rank of ei(β1) among {ej(β1)}ˆβ0is the median of {ei(ˆβ1)}MethodsITrue Model: Yi= 1 + 2xi+ iISample size of n = 30 for each data set under allconditionsIThe xivalues were generated from a uniform distribution(0,5)Ixivalues were fixed for all simulation conditionsIAcceptable error ±.005, so S = 40, 000.IThe error terms, , are generated in four different ways.Four ScenariosIThe error terms are i.i.d. with distribution N(0,1).IThe error terms are i.i.d. wtih distribution exp(λ=.5).IThe error terms are i.i.d. with a contaminated normaldistribution (80% chance the error is from N(0,1)distribution and 20% chance the error is from N(0,10)distribution). Generated by R function rcn written byProfessor McKean.IThe error terms follow a multivariate normal distribution.Marginal distributions are all N(0,1). Error terms arecorrelated.ResultsTable: Scenario 1 i∼ N(0, 1)ˆβLS0ˆβLS1ˆβW0ˆβW1True Value 1.0000 2.0000 1.0000 2.0000Mean 1.0008 1.9998 1.0008 1.9998Bias 0.0008 -0.0002 0.0008 -0.0002SD 0.3082 0.1306 0.3414 0.1348MSE 0.0950 0.0171 0.1166 0.0182Results cont.Table: Scenario 2 i∼ exp(.5)ˆβLS0ˆβLS1ˆβW0ˆβW1True Value 1.0000 2.0000 1.0000 2.0000Mean 2.9991 2.0022 2.4229 2.0082Bias 1.9991 0.0022 1.4229 0.0082SD 0.6192 0.2612 0.5077 0.1851MSE 4.3799 0.0682 2.2825 0.0343Results cont.Table: Scenario 3 i∼ Contaminated NormalˆβLS0ˆβLS1ˆβW0ˆβW1True Value 1.0000 2.0000 1.0000 2.0000Mean 0.9974 2.0016 0.9977 2.0012Bias -0.0026 0.0016 -0.0023 0.0012SD 0.5155 0.2180 0.4268 0.1764MSE 0.2658 0.0475 0.1821 0.0311Results cont.Table: Scenario 4 i∼ MV NˆβLS0ˆβLS1ˆβW0ˆβW1True Value 1.0000 2.0000 1.0000 2.0000Mean 0.9996 1.9994 0.9997 1.9995Bias -0.0004 -0.0006 -0.0003 -0.0005SD 0.3186 0.1624 0.3557 0.1679MSE 0.1015 0.0264 0.1265 0.0282DiscussionIWhen errors were normally distributed, both the LeastSquares and Wilcoxon estimates were very similar.IWhen i∼ exp(.5), the slope term estimates,ˆβ1, werefairly accurate for both the Least Squares and Wilcoxonmethods. The intercept terms,ˆβ0, were quite biasedthough the Wilcoxon estimate was less biased than theLeast Squares estimate. MSE for the intercept terms weresmaller for the Wilcoxon method though both methods hadrelatively high values.IMSE values were slightly lower for the Wilcoxon methodcompared to the Least Squares method when the datacontained outliers.IWhen the error terms were correlated, both methodsseemed to perform equally well. Results were similar tothose in Scenario 1 where errors were independent andnormally
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