主要内容外文翻译

译文,已 ，对于一般模,E

xjfxj,

x2g2,,x， n 可以通过解， n一个估计方程式的形式说明（广义最小二乘法

fx,

x,

j

.的一个综合性的定性实际上在看来GLS的方法在一定程度上与最大似然估是一致的。，在估算β和σ2上。然而，继续强调对θ的方差函数g的依赖，以后会考虑，GLS，在概念方案 调用GLS算（在θ是已知的情况下可以写成更精确如下， （一）估价β由 ，其中 是一些初步的估计，例如用OLS求njn

jfxjfxj0，令k=0.（二）形 数jg2k,,xj.n

fx,

x,

k（三）重新估计β通过求解j

j

，以便于求 k=k+1（ⅱ）CCTH 被越来越。如 能够遍历“无限的次数 就可以认为连续迭代会使k趋C=∞，则β情况下，C=∞对应 求解β中的情况ng2,,n

fx,

x,j

外）上面所给出的GLS算法。但是，现在继续考虑在步骤（i）-（iii）条中的GLS算法方面的一般方法概念，因此，这将方便证明当推广到其中θ亦被视为是实施步骤（i）及 在步骤（i）中假 是OLS，步骤（i）和（iii）在GLS算法中需要一个形为（P×1）nYfx,fx,n j其中，jOLSj，j≡1，在步骤(iii)中，j是来源于步骤(ii)的当前估计值，现在被应用到步骤(iii)中。在一般情况下，求解上式方程式中的一组固定的，已知的权重j，J=1中，n，WLS因此，实现GLS算法需要解决估计的“WLS”的形式方程的能力。因此，首j jj jj，而且很容易地看到，上式方程式可以在一个封闭的形式为β 1WLS j

jxjYjj1的解在一般情况下是不可能成立的。在某些特殊情况下，fFβ偶然的解析解，但是这是非常不寻常的。因此，方程式必须在数值求解。高斯-牛顿法的改变方法存在于非线性回归文献。另式实现这个方，β采取扩张的方法使数值*“接近”β，有 jfx,fx,*fTx,** j fx,fx,*fx,**p 2.4，fββ（XJ，β）是（P×P）导数矩性近似背后的基本假设是，使数值*“接近”β，在泰勒级数随后的条款（二次和更高）fFβ所需泰勒定理的相关性。β（0，aβ（a+1，在实际中的应用：上述所说的算法是如何在简单的相对事物实践中真正价次数。一个这样的修改是在3.6节中。如SASPROCNLIN和R/S-PLUS功能NLS（）中找到。这些都进一步，并第3.7节说明。这种一般提供用户之间的修改，其他的方法和选择的收敛准则被使，（3.3估计方法的时候通常会说基本的算法概念是比较简单的。但请记住，在实际中的应用，例如可用的，通常会变得更加复杂。，Wild（1989，14SASPROCNLIN（SASInstitute，2008）和R/S-PLUS功能NLS（）的文件（和黑斯蒂，1993年，维纳布尔斯和里普利，1999;RitzStreibig，2008外文原文ImplementationofgeneralizedleastWehaveindicatedthat,for eralE

xjfxj,

var

x2g2,,x apopularmethodforestimatingβinthemeanspecificationisgeneralizedleastsquares(GLS).Wemotivatedtheapproachfromthestandpointofsolvinganestimatingequation thenYfx,fx,n jwherethe“weights”arereplacedbyestimates.Theweightingtakesintoaccountthedifferingprecisionofeachresponsej,givingthisapproachanomnibusappeal.Infact,aswewillsee,theGLSapproachcorrespondstoumlikelihoodestimationwhentheYjhavedistributionsinacertainclass.Beforewetackletheseissues,itisworthwhiletodiscusshowthisverypopularapproaaybeimplementedinpractice.Thiswillservebothtoreinforceitsgeneralityandtointroduceustothecomputationalstrategyusedtosolveverygeneralsetsofestimatingequationsthatmaynotbesolvedinaclosedform.WewillassumefornowthatθisknowninthesensediscussedinChapter2,sothatthefocuswillbeonestimationofβandσ2only.However,wewillcontinuetohighlightdependenceofthevariancefunctiongonθ,aslaterwewillconsideraddingestimationofθtothemodel-fittingtask.GLSTheconceptualschemewewillcalltheGLSalgorithm(inthecasethatθisknown)maybewrittenmorepreciselyasfollows.（ⅰEstimate

issomeintialestimate,forexample

njn

jfxj,fxj,）From

,,xj

.Re-estimateβbyk

nnj

j

jfxj,

xj,to .Setk=k+1andreturntoContinuethroughCiterations,andadopttheCthastheIntuitively,wemightexpect(hope)thatifCwere“large,”successive

kwouldbemoreandmoresimilar.Ifwecoulditerate“forever,”wewouldhopethatsuccessiveiterateswouldcoincide,sothatthealgorithmcouldbesaidtohave“converged.”Wewilldenotethisasthecase“C=∞.”IfC=∞,thentheβvalueappearinginthe“weights”andthatintherestofthemustcoincide.Thus,thecaseC=∞correspondstothecasewhereweareng2,,n

fx,

x,in

j

Aswewillsee,solving(3.2)mayinfactbeimplementedusinganapproachdifferentfrom(andmoredirectthan)theGLSalgorithmgivenabove.However,wewillcontinuefornowtothinkofteralapproachconceptuallyintermsoftheGLSalgorithminsteps(i)–(iii),asthiswillproveconvenientwhenwegeneralizetothecasewhereθisalsotakentobeunknownandestimated.Implementingsteps(i)and

0

instep(i),bothsteps(i)and(iii)intheGLSalgorithmsolutionofa(p×1)setofestimatingequationsofthenYfx,fx,n jwherethe

areasetoffixed,knownconstants.Inthecaseof

j≡1forallj,course;instep(iii),the

arethecurrentestimatedvaluesfromstep(ii),whicharefixedin(iii).Ingeneral,solving(3.3)inthecaseofasetoffixed,knownweights1,...,n,correspondstothemethodof

,jThus,implementationoftheGLSalgorithmrequirestheabilitytosolveestimatingequationsofthe“WLS”form.Wethusfocus onhowthismaybecarriedout.j Notethatiff( ,β)werealinearfunctionofβ,i.e.,fx,xT，j

x,

and iseasytoseethat(3.3)maybesolvedinaclosedform Inparticular,undertheseconditions,itiseasytoverifythatthesolution 1WLS jj jWhenlinearityoffdoesnothold,andfisageneralnonlinearfunctionofβ,thenitisclearthlosedformsolutionisnolongerpossibleingeneral.Insomespecialcases,theformsoffandfβmayfortuitouslyadmitanyticalsolution,butthisisveryunusual.Accordingly,(3.3)mustbesolvednumerically.Thebasicmethodfornumericalsolutionoftheequationmaybederivedindifferentways.Hereisoneway,avariantofanideacalledtheGauss-Newtonmethodinthenonlinearregressionliterature.Wewilldiscussanotherwaytomotivatethismethodshortly.ByaTaylorseriesexpansion,wemayapproximatef(xfunctionsofβ.Takingtheexpansionsaboutsome

,β)andfβ(xj,β)by*“closeto”β,we jfx,fx,*fTx,** j fx,fx,*fx,**p SeeSection2.4foranoverviewofthisnotation;here,fββ(xj,β)isthe(p×p)matrixofsecondpartialderivatives.Theunderlyingassumptionbehindthelinearapproximationisthat,for*“closeto”β,thesubsequentterms(quadraticandhigher)intheTaylorseriesaresufficientlysmallastobe“negligible.”NotealsothattheseexpressionscarryimplicitassumptionsabouttheexistenceofpartialderivativesoffandfβrequiredfortherelevanceofTaylor’stheorem.SUMMARY:Thisargumentsuggeststhat,toimplementsolutionoftheestimatingequationwitownweights,onewouldbeginwithastartingvalueβ(0)anda=0,andwouldobtainsuccessiveupdatesβ(a+1),declaringasolutiontotheequationtobereachedwhentwosuccessiveiteratesβ(a)andβ(a+1)are“sufficientlyclose”insomesense.Wewilldiscussselectionofstartingvalues,convergencecriteriafordeclaringthesolutionhasbeenreached,andotherissuesmomentarily.Thesuccessofthisprocedureobviouslydependsontherelevanceoftheapproximationsmade.REALISTICIMPLEMENTATION:Thealgorithmasdescribedaboveissimplisticrelativetohowthingsareactuallyimplementedinpractice.Severalmodificationstothebasicalgorithm,choicesofconvergencecriteria,andsoonhavebeensuggestedtoimproveperformance;thatis,toofferbetterassurancethatthetruesolutionisfoundandtodecreasethecomputationtime(numberofiterationsandfunctionevaluations).OnesuodificationisdiscussedinSection3.6.ModificationstothebasicalgorithmgivenhereandalternativealgorithmsthatgofindingthesolutioninotherwaysareavailableinsoftwaresuchasSASprocnlinandthefunctionnls().Thesearediscussedfurtherandi