Asymptotic Methods in Statistical Inference - Statistics Centre

106
This conclusion (smaller mse ⇒ Pitman closer) need
not hold without these assumptions. The conclusion
smaller mse ⇔ smaller variance need not hold either.
Suppose 1 are i.i.d. (0). Consider estimates
= () of . With = ( +1)
we have unbiasedness. We compare (+1) to 1
with ³ respect to both mse and Pitman closeness. From
() ≤ ´
=() we calculate that h i =
+ () then obtain the bias, variance and mse:
µ
( ) =
+1 − 1
à µ !
( ) = 2 2
+2 − 2
+1
∙
( ) = 2
+2 − 2 +1¸
2
+1
Note mse is minimized by =( +2)( +1). Both
this and =( +1) satisfy ( − 1) → 1; we shall
handle both cases by comparing 1 with ∗,where
( ∗ − 1) → 1. For arbitrary , werewritethemse
as
h
( ( − 1))
2 +1
( )=
− 2 ( − 1) + 2i 2
( +1)( +2)