ABSTRACT
Thus,itfollowsfromthedescriptionabovethattheexpectedblocklength E*L 1undertheSBmethodisp-1•Weshallassumethatpgoestozeroat acertainratewithn.Thisensuresthattheexpectedblocksizeunderthe SBtendstoinfinitywiththesamplesize,justasdotheblocksizesofother blockbootstrapmethods.However,unlikeblockbootstrapmethods basedonnonrandomblocklengths,sincetheblocklengthvariables L 1,•••,L 11undertheSBmethodarerandom,weneedtoresamplea randomnumberofblockstogenerateasetofnbootstrapsamples undertheSBmethod.LetK=inf{k2::l:L1+···+LK2::n}.Then, theSBmethodselectstheKblocks~(11 ,L 1),•••,:!J(h,LK).Notethat therearealtogetherN 1=L 1+···+LKelementsintheresampledblocks :!J(I1,L 1),•••,21J(IK,LK).Arrangingtheseelementsinaseries,wegetthe bootstrapobservationsXt,...,Xt,.TodefinetheSBversionofthe randomvariableT11=t11 (X1,•••,X11 ;8),wemayusethefirstnorallN 1 oftheX1's.Thus,theSBversionsofT"aregivenby
f~=t 11 (Xi,...,X~;B11 ), wheree,isanestimatorof()basedonXI'. ..'XII'SincethedifferenceNl-11 isnegligiblecomparedton,T~andf~tendtohaveverysimilarlarge sampleproperties.Fortechnicalreasons,wewoulduseT~astheSBversion ofT11intherestofthepaper.TheSBestimatoroftheunknownsampling distributionG11ofT,isthendefinedastheconditionaldistributionG11 ,say ofT~,givenX1,•••,Xn-Thus,forafunctionalA(-)ofthesamplingdistributionG,,theSBestimatorofA(G11 )isgivenbyA(G11 ).Forexample,theSB estimatorsofE(T11 )andvar(T11 )arerespectivelygivenbyE*T~=
Bn = H(X11 ), where H be a smooth real valued function on ~d. Though it looks like a very restrictive model, considering suitable transformations of the original observations, many important population parameters and their commonly used estimators may be expressed in this form. We illustrate the scope of the smooth function model with the following examples.
