observedFisherinformationanduseitasanotherapproximationtoFisherinformationOfcoursewestilldontknowsoweneedasecondpluginusingJxxinsteadofJxThepluginprincipleappliesheretooTheerrormade ID: 115564
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Sowhatisgenerallydoneistostartagoodlocaloptimizationalgorithmata\good"startingpointandtakethesolutionproducedbythealgorithmtobetheMLE(ifthealgorithmconvergestoasolution).Technically,whatisrequiredofthestartingpointtobe\good"isthatitobeysthesquarerootlaw:itsestimationerrorgoestozerolikeaconstantdividedbythesquarerootofthesamplesize.Generally,onejustusesthebestestimatoronecancalculateasthestartingpoint.1.3ExpectedFisherInformationBecausethelogfunctionismonotone,maximizingthelikelihoodisthesameasmaximizingtheloglikelihoodlx()=logLx():(3)Formanyreasonsitismoreconvenienttouseloglikelihoodratherthanlikeli-hood.Thederivativesoftheloglikelihoodfunction(3)areveryimportantinlikeli-hoodtheory.Themomentsofloglikelihoodderivativessatisfysomeimportantidentities.Notethatifthelikelihoodisgivenby(2),thentheloglikelihoodisgivenbylx()=logh(x)+logf(x)(4)andthederivatives(thesearederivativeswithrespectto)donotinvolvethelogh(x)termbecauseitdoesnotcontain.Sothesederivativesarewelldenedandthesameregardlessofwhath(x)weuse.Alsonotethatthemaximizerof(4)nomatterhowdened(localorglobalmaximizer)doesnotdependonh(x).ThustheMLEisthesame(ifdened)regardlessofwhath(x)weuse.First,therstderivativehasexpectationzeroErlx() =0;(5)whererfdenotesthevectorofpartialderivativesofascalarfunctionfofavectorvariable,oftencalledthegradientoff,andrf(x)denotesthevalueofthegradientatthepointx.Second,thevarianceoftherstderivativeisminustheexpectationofthesecondvarrlx() =�Er2lx() ;(6)wherer2fdenotesthematrixofsecondpartialderivativesofascalarfunctionfofavectorvariable,oftencalledthehessianoff,andr2f(x)denotesthevalueofthehessianatthepointx.Eithersideof(6)iscalledtheexpectedFisherinformation(orjust\Fisherinformation"withno\expected"whenitisclearwhatismeant)andisdenotedI().2 observedFisherinformationanduseitasanotherapproximationtoFisherin-formation.Ofcourse,westilldon'tknow,soweneedasecond\plug-in"usingJx(^x)insteadofJx().Theplug-inprincipleappliesheretoo.TheerrormadebyapproximatingI()byJx(^x)isnegligiblecomparedtotheerrorapproxi-matingtheactualsamplingdistributionoftheMLEbyNormal�;I()�1.1.6SummaryofTheoryTheasymptoticapproximationtothesamplingdistributionoftheMLE^xismultivariatenormalwithmeanandvarianceapproximatedbyeitherI(^x)�1orJx(^x)�1.2MaximumLikelihoodEstimationinR2.1TheCauchyLocation-ScaleFamilyThe(standard)CauchyDistributionisthecontinuousunivariatedistributionhavingdensityf(x)=1 1 1+x2;�1x1:(7)ThestandardCauchydistributionhasnoparameters,butitinducesatwo-parameterlocation-scalefamilyhavingdensitiesf;(x)=1 fx� (8)Iffisanydistributionhavingmeanzeroandvariance1,thenf;hasmeanandvariance2.ButtheCauchydistributionhasneithermeannorvariance.Thuswecallthelocationparameterandthescaleparameter.SincethestandardCauchydistributionisclearlysymmetricaboutzero,theCauchy(;)distributionissymmetricabout.Henceisthepopulationmediananda\good"estimateisthesamplemedian.Arobustscaleestimatoranalogoustothesamplemedianistheinterquartilerange(IQR).TheIQRofthestandardCauchydistributionis�qcauchy(3/4)-qcauchy(1/4)[1]2ThusthepopulationIQRoftheCauchy(;)distributionis2,andhencea\good"estimateofisthesampleIQRdividedby2.2.2MaximumLikelihood2.2.1OneParameterTheRfunctionnlmminimizesarbitraryfunctionswritteninR.Sotomax-imizethelikelihood,wehandnlmthenegativeoftheloglikelihood(foranyfunctionf,minimizing�fmaximizesf).4 �nsim-100ကmu-0ကmu.hat-double(nsim)ကmu.twiddle-double(nsim)ကfor(iin1:nsim){+xsim-rcauchy(n,location=mu)+mu.start-median(xsim)+out-nlm(mlogl,mu.start,x=xsim)+mu.hat[i]-out$estimate+mu.twiddle[i]-mu.start+}ကmean((mu.hat-mu)^2)[1]0.06203118ကmean((mu.twiddle-mu)^2)[1]0.08242236Thetwonumbersreportedfromthesimulationarethemeansquareerrors(MSE)ofthetwoestimators.Theirratioကmean((mu.hat-mu)^2)/mean((mu.twiddle-mu)^2)[1]0.7526013istheasymptoticrelativeeciency(ARE)oftheestimators.NowweseetheMLEismoreaccurate,astheorysaysitmustbe.2.2.3TwoParametersMinustheloglikelihoodforthetwo-parameterCauchycanbewrittenကmlogl3-function(theta,x){+sum(-dcauchy(x,location=theta[1],scale=theta[2],log=TRUE))+}andtheMLEcalculatedbyကtheta.start-c(median(x),IQR(x)/2)ကtheta.start[1]-0.19550620.7125899ကout-nlm(mlogl3,theta.start,x=x)ကtheta.hat-out$estimateကtheta.hat[1]-0.18092990.76055616 2.3.3ExpectedFisherInformationRhasafunctionderivthatdoesderivativesofRexpressions.Butitisn'tverysophisticated.Itwon'tcalculatelikelihoodderivativeshere.Solet'sdothederivativesbypencilandpaper.First,theloglikelihooditselflx(;)=nlog()�nXi=1log�2+(xi�)2Therstderivativesare@lx(;) @=nXi=12(xi�) 2+(xi�)2@lx(;) @=n �nXi=12 2+(xi�)2Rdoesn'tdoanalyticintegralsatall.Butitdoesdonumericalintegrals,whichisallweneedtodoFisherinformation.�theta.hat[1]-0.18092990.7605561�mu-theta.hat[1]ကsigma-theta.hat[2]ကgrad1-function(x)2*(x-mu)/(sigma^2+(x-mu)^2)ကgrad2-function(x)(1/sigma-2*sigma/(sigma^2+(x-mu)^2))ကfish.exact-matrix(NA,2,2)ကfish.exact[1,1]-integrate(function(x)grad1(x)^2*dcauchy(x,+mu,sigma),-Inf,Inf)$valueကfish.exact[2,2]-integrate(function(x)grad2(x)^2*dcauchy(x,+mu,sigma),-Inf,Inf)$valueကfish.exact[1,2]-integrate(function(x)grad1(x)*grad2(x)*+dcauchy(x,mu,sigma),-Inf,Inf)$valueကfish.exact[2,1]-fish.exact[1,2]ကround(n*fish.exact,10)[,1][,2][1,]25.931570.00000[2,]0.0000025.93157ကfish[,1][,2][1,]32.507337270.01480913[2,]0.0148091319.348190968