Asymptotic Methods in Statistical Inference - Statistics Centre

149
By using the fact that | 2 − 2 | ≤ 2| − | if
|| || ≤ 1weobtain
=
≤
≤
≤
| ( ) − ( )|
¯
¯
h
( () − ()) 2 − ( () − ()) 2i¯¯¯
h¯¯¯( () − ()) 2 − ( () − ()) 2¯¯¯
2 [| () − ()|]
2 ( );
it follows that (·) iscontinuousat .
i
• Example 2. ( )= []. Let and be
any d.f.s with finite means; put
=(1− ) +
for 0 1. Let → 0, then
( )= ( ) ≤ → 0
but ( ) − ( ) = (( ) − ( )) need
not → 0. For instance if ( )=0and( )=
then ( ) − ( ) →∞. Thus this ‘mean
functional’ is not continuous. A consequence is
that ³ ˆ ´
= ¯ need not be consistent for []
in the presence of outliers.