Say that given an estimator T ^ , of a statistic T, we have that n ( T ^ − T ) → D N ( 0 , V a r ), where n is the sample size. Consider now − T ^ , does the asymptotic result hold also in this case? If yes/no, under which conditions?

Question

Say that given an estimator             T      ^      , of a statistic   T, we have that       n    (            T      ^        −  T  )      →                              D            N    (  0  ,  V  a  r  ), where   n is the sample size. Consider now   −            T      ^      , does the asymptotic result hold also in this case? If yes/no, under which conditions?

toliwask · Accepted Answer

If       X    n        →    D    X then   −      X    n        →    D    −  X. I think you can prove this directly using one of the many equivalent definitions of convergence in distribution, or use a hammer like Slutsky&#039;s theorem.Thus       n    (  (  −            T      ^        )  −  (  −  T  )  )      →    D        N    (  0  ,  V  ).

Say that given an estimator T^, of a statistic T, we have that sqrt n (T^−T)->DN(0,Var), where n is the sample size. Consider now −T^, does the asymptotic result hold also in this case? If yes/no, under which conditions?

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