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9.1()(rank)9.1()X1;X2;¢¢¢;XnX(1)·X(2)·¢¢¢·X(n)X(k)9.2()F(x)rFr(x)=P(X(r)x)=P(rXix)=nXi=rCinFi(x)[1¡F(x)]n¡i=Fr(x)=n!(r¡1)!(n¡r)!F(x)Z0tr¡1(1¡t)n¡rdtfr(x)=n!(r¡1)!(n¡r)!Fr¡1(x)[1¡F(x)]n¡rf(x)9.2[21].9.3X1;X2;¢¢¢;XnF(x)F(x)FT(X1;X2;¢¢¢;XnF2FTF(distribution-free)H09.4X1;X2;¢¢¢;XnF(x)F(x)FT(X1;X2;¢¢¢;XnF2FTF(distribution-free)9.5Xµ0ª=Ã(X¡µ0)-154-Ã(t)=½1t00t0ªXµ09.1X1;X2;¢¢¢;XnXiFi(x)iFi(µ0)Fi(µ0)=p0ª=Ã(Xi¡µ0(1¡p0)B=nXi=1ªi=nXi=1Ã(Xi)H0:F(0)=12H1:F(0)6=129.6Z1;Z2;¢¢¢;ZnF(z)Z(1)·Z(2)·¢¢¢·Z(N)Ri=r;Zi=Z(r)i=1;2;¢¢¢;NRiZiRiiZi(Z(1);Z(2);¢¢¢;Z(N)9.7()x1;¢¢¢;xnx1;¢¢¢;xnrixirixixi;¢¢¢;xnX1;¢¢¢;XnRiXi(X1;¢¢¢;Xn)R=(R1;¢¢¢;Rn)(X1;¢¢¢;Xn)9.2R=(R1;R2;¢¢¢;RN)R=f(r1;¢¢¢;rN)j(r1;¢¢¢;rN)(1;¢¢¢;N)gRRRS(R)PfR=rg=1N!PfRi=rg=1NPfRi=r;Rj=sg=1N(N¡1)tm;n(d)1;2;¢¢¢;m+nndtm;n(d)tm;n(d)=tm;(n¡1)(d¡m¡n)+t(m¡1);n(d)ti;0(0)=1;ti;0(d)=0t0;j(j(j+1)2)=1;t0;j(d)=0-155-9.3Histogram9.8jX1j;¢¢¢;jXnj(R+;¢¢¢;R+n)R+iXiªiR+iXiª1;¢¢¢;ªn;R+S(ª1;¢¢¢;ªn;R+)0ª1;¢¢¢;ªnS(ª1;¢¢¢;ªn)p0µ0(µ0Fi(µ0)=p0)9.3HistogramStatisticalPropertiesthehistogramasanestimatoroftheunknownpdff(x)theoriginx0=0somepointx2Bj=[(j¡1)h;jh)thedensityestimateassignedbythehistogramtoxis^fh(x)=1nhnXi=1I(Xi2Bj)BiasIsthehistogramanunbiasedestimator?Ef^fh(x)g=1nhnXi=1EfI(Xi2Bj)gXiisi:i:d=)theindicatorfunctionIisi:i:dEf^fh(x)g=1nhnE(I(Xi2Bj))I(Xi2Bj)=(0pis1¡Rjh(j¡1)hf(u)du1pisRjh(j¡1)hf(u)duhenceitisBernoullidistributedEfI(Xi2Bi)g=PfI(Xi2Big=Zh(j¡1)hf(u)duthenE(^fh(x))=1hZh(j¡1)hf(u)duthelasttermisnotingeneralequaltothef(x)hence,thehistogramisingeneralnotanunbiasedestimatorofthef(x):Bias(^fh(x))=E(^fh(x)¡f(x))=1hZh(j¡1)hf(u)du¡f(x)=1hZh(j¡1)h[f(u)¡f(x)]dua¯rst-orderTaylorapproximationoff(x)¡f(u)aroundthecentermj=(j¡12)hyields:Bias(^fh(x))¼f0(mj)(mj¡x)ifmjisequaltothex?-156-VarianceVarf^fh(x)g=1n2h2nZBjf(u)duÃ1¡ZBjf(u)du!¼1nhf(x)integral:Zbaf(x)dx=lim¸!0nXi=1f(»i)¢xiMeansquarederrorVarianceandbiasvaryoppositedirectionswithh,wehavetosettlefor¯ndingthevalueofhthatyieldstheoptimalcompromisebetweenvarianceandbiasreduction.themeansquarederror(MSE)ofthehistogram:MSEf^fh(x)g=Ef[^fh(x)¡f(x)]2gwhichcanbewrittenasthesumofthevarianceandthesquaredbias:MSEf^fh(x)g=Var(^fh(x))+[Bias(^fh(x))]2Meanintegratedsquarederrortheapplicationofthe!MSEisdi±cultinpracticesincethederivedformulafortheMSEdependsontheunknowndensityfunctionfbothinthevarianceandthesquaredbiasterm.themostwidelyusedglobalmeasureofestimationaccuracyisthemeanintergratedsquarederror:MISE(^fh(x))=EhR+1¡1f^fh(x)¡f(x)g2dxi=R+1¡1E[f^fh(x)¡f(x)g2]dx=R+1¡1MSEdxMISE(^fh(x))¼R1nhf(x)dx+RPjI(x2Bj)f(j¡12)h¡xg2[f0f(j¡12)hg]2dx=1nh+PjRBjf(j¡12)h¡xg2[f0f(j¡12)hg]2dx¼1nh+Pj[f0f(j¡12)hg]2²RBjfx¡(j¡12)hg2dx¼1nh+h212Rff0(x)g2dxh!01nh+h212kf0k22theasymptoticofMSE:AMSE(^fh(x))=1nh+h212°°f0°°22optimalBinwidthdi®erentiatingAMSEwithrespecttohgives:@AMSE(^fh(x))@h=¡1nh2+16hkf0k22=0henceh0=Ã6nkf0k22!13»n¡13assumingastandardnormdistribution:f(x)='(x)=1p2¼exp(¡x22)-157-9.4Nonparametricdensityestimationandf0(x)=(¡x)f(x)kf0k22=k(¡x)f(x)k22=fRx2f2(x)dxg12£2=1p2¼Rx21p2¼exp(¡x22£2)dx=1p2¼£1p2£Rx21p2¼£1p12£exp(¡x22£12)dx=14p¼Dependenceofthehistogramontheoriginthispropertystronglycon°ictswiththegoalofnonparametricstatisticstoletthedataspeakforthemselvesASH:averagedshiftedhistogram^fh(x)=1nnXi=1f1MhM¡1Xl=0XjI(Xi2Bjl)I(x2Bjl)g9.4NonparametricdensityestimationmotivationandderivationeveniftheASHseemedtosolvethechoice-of-an-originproblemthehistogramretainssomeundesirableproperties:(1)thehistogramassignseachxin[mj¡h2;mj+h2)thesameestimatorofthepdf,namely^fh(mj).Thisseemstobeoverlyrestrictive.(2)Thehistogramisnotacontinuousfunctionbuthasjumpsattheboundariesofthebins.fobservationsthatfallintoasmallintervalcontainingxgfobservationsthatfallintoasmallintervalaroundxgweconsiderintervalsoftheform[x¡h;x+h):^fh(x)=12hn#fXi2[x¡h;x+h)gtheformulacanberewrittenifweuseaweightingfunction,calledtheuniformkernelfunction:K(u)=12I(juj·1)andletu=x¡Xihwecanwrite:^fh(x)=1nhnPi=1K(x¡Xih)=1nhnPi=112I(¯¯x¡Xih¯¯·1)maybeweshouldgivemoreweighttocontributionsfromobservationsveryclosetoxthantothosecomingfromobservationsthataremoredistant.forinstant,Epanechnikokernel:K(²)=34(1¡u2)I(juj·1)-158-basedonarandomsamplefX1;X2;¢¢¢;Xngfromf^fh(x)=1nnXi=1Kh(x¡Xi)whereKh(²)=1hK(²h)hcontrolsthesmoothnessoftheestimateandthechoiceofthehisacrucialproblem.therescaledkernelfunctionissimply:1nhK(x¡Xih)=1nK(x¡Xi)Z1nhK(x¡Xih)dx=1nhZhK(u)du=1nStatisticPropertiesBiasBiasf^fh(x)g=Ef^fh(x)g¡f(x)=1nnPi=1EfKh(x¡Xi)g¡f(x)=EfKh(x¡X)g¡f(x)=R1hK(x¡uh)f(u)du¡f(x)asecond-Taylorexpansionoff(u)aroundx:Biasf^fh(x)g=h22f00(x)¹2(K)+o(h2)¹2(K)=Zs2K(s)dsthebiasisproportionaltoh2.moreover,thebiasisdependonthecurvatureofthedensityatx.VarianceVarf^fh(x)g=Varf1nnPi=1Kh(x¡Xi)g=1nVarfKhg=1nfEK2h¡(EKh)2gusingEKh=f(x)+o(h)andEK2h=ZK2hf(u)duandvariablesubstitutionandTaylorexpansionargumentsinthederivationofthebias:Varf^fh(x)g=1nhkKk22f(x)+o(1nh)kKk22isshorthandforRK2(s)ds,thesquaredL2normofK.MSEMSEf^fh(x)g=h44f00(x)2¹2(K)2+1nhkKk22f(x)+o(h4)+o(1nh)-159-9.5hopt(x)dependsonxandisthusalocalbandwidth.AMSEAMSE=ZMSEdxAMSE=RMSEdx=1nhkKk22+h44f¹2(K)g2kf00k22di®erentiati