Asymptotic Methods in Statistical Inference - Statistics Centre

146
• Outline of proof of √ -equivalence: Since is
symmetric,
∙ = − ! ³ ´ + Σ
0¸
where Σ 0 denotes the sum over indices with some
repetitions, and is ( −1 ). Since
− ! ³ ´ ³
=1+
−1´
we obtain = + (1) where the (1)
term has a finite variance (this is where some work
is required).
In particular, for =2,
with
= + −1 P ( ) −
[ ]= + [ ( 1 1 )] −
• Example: ( 1 2 ) = ( 1 − 2 ) 2 2 results in
= 2 (unbiased for [] = 2 )but
= −1 X
( − ¯) 2