åni1Xi0Xk2Kix136mllxYk0Xk0x136m1llxwhereYkåni1YiKixåni1KixXkåni1XiKixåni1KixandKixKhx0XiSolveWearegoingtosolvethosetwoproblemstogethe ID: 828766
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1 EXERCISE14WEIYULI1.SupposefXi,Yigarebiva
EXERCISE14WEIYULI1.SupposefXi,YigarebivariaterandomsampleswithYi=m(Xi)+ui,wherem()isanunknownsmoothfunction,anduisatisesE[uijXi]=0,Var(uijXi)=s2(Xi),a.s.(1)Solvethelocallinearestimationofm(x),anditsasymptoticbiasandvariance(mainterms).(2)Solvethelocallinearestimationˆm(1)ll(x)oftherstderivativeofm(x),andprovethatˆm(1)ll(x)=åni=1(Yi�¯Yk)(Xi�¯Xk)Ki,x åni=1(Xi�¯Xk)2Ki,x,ˆmll(x)=¯Yk�(¯Xk�x)ˆm(1)ll(x),where¯Yk=åni=1YiKi,x/åni=1Ki,x,¯Xk=åni=1XiKi,x/åni=1Ki,xandKi,x=Kh(x�Xi).Solve.Wearegoingtosolvethosetwoproblemstogether,sincethenotationˆmll(x)in(2)isexactlywhatwefocusonin(1).WithkernelKandbandwidthh,considertheminimizationproblemminm,bnåi=1(Yi�m�(Xi�x)b)2Ki,x,whereKi,x=Kh(x�Xi)=1 hK(x�Xi h).LetM(x)=m(x)b(x),X=0B@1X1�x......1Xn�x1CA,W=diag(K1,x,,Kn,x)andY=0B@Y1...Yn1CA,thentheminimizationproblemcanberegardedasaleastsquareproblemwithsolutionˆM(x)=(X0WX)�1X0WY:=ˆmll(x)ˆm(1)l
2 l(x),whichconsistsofthelocallineare
l(x),whichconsistsofthelocallinearestimationsofm(x)anditsrstderivative.WecanexplicitlywriteX0WX=åiKi,xåi(Xi�x)Ki,xåi(Xi�x)Ki,xåi(Xi�x)2Ki,x=åiKi,x1¯Xk�x¯Xk�xåi(Xi�x)2Ki,x/åiKi,x. Date:2019/11/25.liweiyu@mail.ustc.edu.cn.1 2WEIYULIFurthernoticingthatåi(Xi�x)2Ki,x=åi(Xi�¯Xk)+(¯Xk�x)2Ki,x=åi(Xi�¯Xk)2Ki,x+2(¯Xk�x)åi(Xi�¯Xk)Ki,x+(¯Xk�x)2nåi=1Ki,x=åi(Xi�¯Xk)2Ki,x+(¯Xk�x)2nåi=1Ki,xsimpliesthematrixasX0WX=åiKi,x 1¯Xk�x¯Xk�xåi(Xi�¯Xk)2Ki,x åiKi,x+(¯Xk�x)2!.Similarly,wehaveX0WY=åiYiKi,xåi(Xi�x)YiKi,x=åiKi,x¯Ykåi(Xi�x)YiKi,x/åiKi,x=åiKi,x ¯Ykåi(Yi�¯Yk)(Xi�¯Xk)Ki,x åiKi,x+¯Yk(¯Xk�x)!.Atrickweuseistoeliminatepartofthesecondrowsthatis(¯Xk�x)-proportionaltotherstrows.Inspecic,multiplyingthenonsingularA=(åiKi,x)�110�(¯Xk�x)1,wehaveAX0WX= 1¯Xk�x0åi(Xi�¯Xk)2Ki,x åiKi,x!,AX0WY= ¯Ykåi(Yi�¯Yk)(Xi&
3 #0;¯Xk)Ki,x åiKi,x!.Consequently,
#0;¯Xk)Ki,x åiKi,x!.Consequently,ˆmll(x)ˆm(1)ll(x)=�AX0WX�1AX0WY= 1¯Xk�x0åi(Xi�¯Xk)2Ki,x åiKi,x!�1 ¯Ykåi(Yi�¯Yk)(Xi�¯Xk)Ki,x åiKi,x!=0@1�(¯Xk�x)åiKi,x åi(Xi�¯Xk)2Ki,x0åiKi,x åi(Xi�¯Xk)2Ki,x1A ¯Ykåi(Yi�¯Yk)(Xi�¯Xk)Ki,x åiKi,x! EXERCISE143=0@¯Yk�(¯Xk�x)åi(Yi�¯Yk)(Xi�¯Xk)Ki,x åi(Xi�¯Xk)2Ki,xåi(Yi�¯Yk)(Xi�¯Xk)Ki,x åi(Xi�¯Xk)2Ki,x1A,whichcompletestheproofin(2).FromthetheoreminPage27ofLec14.pdf,asymptoticbiasandvarianceofˆmll(x)arebias(ˆmll(x))=k21 2h2m00(x),Var(ˆmll(x))=k02s2(x) nhf(x),wheref()isthedensityofX1.2.ConsidertheexponentialgeneralizedlinearmodelYjX=xExp(l(x)),l(x)=eb0+b1x.Usinglocallikelihoodestimation,writeanestimatefunctiondependingonthesam-pleX,Y,estimatepointx,bandwidthhandkernelK.Generateasimulateddataset,useyourfunctiontoestimate,andchoosetheoptimalbandwidthbycross-validation.Solve.Notethatinlocalregression,themodelhastheexpressionYi=m(Xi)+
4 ei,wherem(x)=E(YjX=x)=1 l(x)=e�b0�
ei,wherem(x)=E(YjX=x)=1 l(x)=e�b0�b1x.Ontheotherhand,thelinkfunctionisg(m)=�log(m)inGLMtting.Thelog-likelihoodofbisl(b)=nåi=1log�f(y)=nåi=1log�l(Xi)�l(Xi)Yi.Thelocallog-likelihoodofbaroundxisthenlx,h(b)=nåi=1[log(l(Xi�x))�l(Xi�x)Yi]Kh(x�Xi)=nåi=1hb0+b1(Xi�x)�eb0+b1(Xi�x)YiiKh(x�Xi).Maximizinglx,h(b)yieldsˆb=(ˆb0,ˆb1)0,thuswehaveˆml(x;h,p)=g�1(ˆb0)=e�ˆb0.Foroptimalbandwidth,usecross-validation,thatis,maximizeLCV(h)=nåi=1l(Yi,ˆl�i(Xi)).Fromtheabovetheoreticalanalysis,weprovidethecodesfollowingthematerialLec14.r:##Initializationn200truebetac(2,2)#truebeta_0andbeta_1lambdafunction(x)exp(truebeta[1]+truebeta[2]*x)likfunction(y,beta)beta-exp(beta)*y#likelihoodfunctionwith$\lambda=e^\hat{\beta}$set.seed(0) 4WEIYULI##GenerateadatasetXrunif(n=n,-3,3)Yrexp(n=n,rate=lambda(X))##Setbandwidthandevaluationgridh0.1xseq(-3,3,l=501)##Optimizetheweightedlog-likelihoodexplicitlysuppressWarnings(fitNlmsapply(x,functio
5 n(x){Kdnorm(x=x,mean=X,sd=h)nlm(f=functi
n(x){Kdnorm(x=x,mean=X,sd=h)nlm(f=function(beta){sum(K*(Y*exp(beta[1]+beta[2]*(X-x))-(beta[1]+beta[2]*(X-x))))},p=c(0,0))$estimate[1]}))##ExactLCVhseq(0.1,1,by=.1)suppressWarnings(LCVsapply(h,function(h){sum(sapply(1:n,function(i){Kdnorm(x=X[i],mean=X[-i],sd=h)lik(Y[i],nlm(f=function(beta){sum(K*(Y[-i]*exp(beta[1]+beta[2]*(X[-i]-X[i]))-(beta[1]+beta[2]*(X[-i]-X[i]))))},p=c(0,0))$estimate[1])}))}))plot(h,LCV,type="o")abline(v=h[which.max(LCV)],col=2)##Comparetheoptimalbandwidthwiththenon-optimalonehh[which.max(LCV)]suppressWarnings(fitNlm.optsapply(x,function(x){Kdnorm(x=x,mean=X,sd=h)nlm(f=function(beta){sum(K*(Y*exp(beta[1]+beta[2]*(X-x))-(beta[1]+beta[2]*(X-x))))},p=c(0,0))$estimate[1]})) EXERCISE145 plot(x,1/lambda(x),type="l",lwd=2,log="y")lines(x,exp(-fitNlm),col=2,lwd=2,lty=2)#inverseoflinkfunctionglines(x,exp(-fitNlm.opt),col=3,lwd=2,lty=2)legend("topright",legend=c("true","non-optimalbw","optimalbw"),col=1:3,lwd=c(2,2,2),lty=c(1,2,2)) Wecanobservethatoptimalbandwidthgivesbettertting.