▁on ▁the ▁small ▁interval ▁, ▁where ▁it ▁is ▁equal ▁to ▁zero . ▁It ▁is ▁not ▁difficult ▁to ▁see ▁that ▁this ▁estim ator ▁is ▁consistent ▁for ▁ θ , ▁and ▁its ▁asympt otic ▁distribution ▁is ▁▁▁ ▁for ▁any ▁α ▁ ∈ ▁R . ▁Thus ▁this ▁estim ator ▁has ▁the ▁same ▁asympt otic ▁distribution ▁as ▁ ▁for ▁all ▁, ▁whereas ▁for ▁ ▁the ▁rate ▁of ▁convergence ▁becomes ▁arbitr arily ▁fast . ▁This ▁estim ator ▁is ▁sup ere fficient , ▁as ▁it ▁sur pass es ▁the ▁asympt otic ▁behavior ▁of ▁the ▁efficient ▁estim ator ▁ ▁at ▁least ▁at ▁one ▁point ▁. ▁In ▁general , ▁sup ere ffic iency ▁may ▁only ▁be ▁att ained ▁on ▁a ▁subset ▁of ▁measure ▁zero ▁of ▁the ▁parameter ▁space ▁ Θ . ▁ ▁Example ▁▁ ▁Suppose ▁x 1 , ▁… , ▁x n ▁is ▁an ▁Independent ▁and ▁Ident ically ▁Dist ributed ▁( I ID ) ▁random ▁sample ▁from ▁normal ▁distribution ▁ ▁with ▁unknown ▁mean ▁but ▁known ▁variance . ▁Then ▁the ▁common ▁estim ator ▁for ▁the ▁population ▁mean ▁ θ ▁is ▁the ▁arithmetic ▁mean ▁of ▁all ▁observations : ▁. ▁The ▁corresponding ▁H od ges ’ ▁estim ator ▁will ▁be ▁, ▁where ▁ 1 { … } ▁denotes ▁the ▁indicator ▁function . ▁ ▁The ▁mean ▁square ▁error ▁( scale d ▁by ▁n ) ▁associated ▁with ▁the ▁regular ▁estim ator ▁x ▁is ▁constant ▁and ▁equal ▁to ▁ 1 ▁for ▁all ▁ θ ’ s . ▁At ▁the ▁same ▁time ▁the ▁mean ▁square ▁error ▁of ▁the ▁H od ges ’ ▁estim ator ▁ ▁behav es ▁er