A new variance estimator for parameters of semiparametric ge(4)

APPENDIX

andvar(β )intermsoftheindividualsmoothingmatriceslikeEquationsEquationsforβ

(2.8)and(2.10),butthatapplywithanynumberofsplinescanbeobtainedrecursivelyas

(suchasEquation(2.8))andvar(β )(i.e.,Equationfollows.Startwithexpressionsforβ

(2.10))thatapplywithJ 1splines,intermsofsmoothingmatricesS1,S2,...,SJ 1,

andmatrixA(then×nmatrix 2l/ η ηt,asinEquation(2.8)).Expressionsforβ

)thatapplywithoneadditionalspline(say,SJ)isobtainedbysubstituting:(I var(β

SiSJ) Si(I SJ)inplaceofSifori=1,2,...,J 1andA(I SJ)forAthroughout.

256W.D.FLANDERS,M.KLEIN,ANDP.TOLBERT

Thisrecursiveapproachisjusti edbywritingthesystemofJ+1linearEquationsin

andf ...,f ,comparabletoEquations(2.4)–(2.7);eliminatingf ;andunknownsβ1JJrearrangingtoobtainareducedsystemofJequationsinJunknowns.Thenewsystemofequationshasthesameformastheoriginal,providedweidentify(I SiSJ) Si(I SJ)withSifori=1,2,...,J 1andA(I SJ)withAthroughout.

ACKNOWLEDGMENTS

ThisworkwassupportedbygrantsfromtheU.S.EnvironmentalProtectionAgency(R82921301-0)andfromtheNationalInstituteofEnvironmentalHealthSciences(R01ES11294).

[ReceivedApril2004.RevisedNovember2004.]

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