主要內(nèi)容外文翻譯

譯文,已 ，對于一般模,E

xjfxj,

x2g2,,x， n 可以通過解， n一個估計方程式的形式說明（廣義最小二乘法

fx,

x,

j

.的一個綜合性的定性實際上在看來GLS的方法在一定程度上與最大似然估是一致的。，在估算β和σ2上。然而，繼續(xù)強調(diào)對θ的方差函數(shù)g的依賴，以后會考慮，GLS，在概念方案 調(diào)用GLS算（在θ是已知的情況下可以寫成更精確如下， （一）估價β由 ，其中 是一些初步的估計，例如用OLS求njn

jfxjfxj0，令k=0.（二）形 數(shù)jg2k,,xj.n

fx,

x,

k（三）重新估計β通過求解j

j

，以便于求 k=k+1（ⅱ）CCTH 被越來越。如 能夠遍歷“無限的次數(shù) 就可以認為連續(xù)迭代會使k趨C=∞，則β情況下，C=∞對應(yīng) 求解β中的情況ng2,,n

fx,

x,j

外）上面所給出的GLS算法。但是，現(xiàn)在繼續(xù)考慮在步驟（i）-（iii）條中的GLS算法方面的一般方法概念，因此，這將方便證明當推廣到其中θ亦被視為是實施步驟（i）及 在步驟（i）中假 是OLS，步驟（i）和（iii）在GLS算法中需要一個形為（P×1）nYfx,fx,n j其中，jOLSj，j≡1，在步驟(iii)中，j是來源于步驟(ii)的當前估計值，現(xiàn)在被應(yīng)用到步驟(iii)中。在一般情況下，求解上式方程式中的一組固定的，已知的權(quán)重j，J=1中，n，WLS因此，實現(xiàn)GLS算法需要解決估計的“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個封閉的形式為β 1WLS j

jxjYjj1的解在一般情況下是不可能成立的。在某些特殊情況下，fFβ偶然的解析解，但是這是非常不尋常的。因此，方程式必須在數(shù)值求解。高斯-牛頓法的改變方法存在于非線性回歸文獻。另式實現(xiàn)這個方，β采取擴張的方法使數(shù)值*“接近”β，有 jfx,fx,*fTx,** j fx,fx,*fx,**p 2.4，fββ（XJ，β）是（P×P）導(dǎo)數(shù)矩性近似背后的基本假設(shè)是，使數(shù)值*“接近”β，在泰勒級數(shù)隨后的條款（二次和更高）fFβ所需泰勒定理的相關(guān)性。β（0，aβ（a+1，在實際中的應(yīng)用：上述所說的算法是如何在簡單的相對事物實踐中真正價次數(shù)。一個這樣的修改是在3.6節(jié)中。如SASPROCNLIN和R/S-PLUS功能NLS（）中找到。這些都進一步，并第3.7節(jié)說明。這種一般提供用戶之間的修改，其他的方法和選擇的收斂準則被使，（3.3估計方法的時候通常會說基本的算法概念是比較簡單的。但請記住，在實際中的應(yīng)用，例如可用的，通常會變得更加復(fù)雜。，Wild（1989，14SASPROCNLIN（SASInstitute，2008）和R/S-PLUS功能NLS（）的文件（和黑斯蒂，1993年，維納布爾斯和里普利，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“l(fā)arge,”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