Asymptotic Methods in Statistical Inference - Statistics Centre

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• Note that an unbiased estimate of
= [ ( 1 )] is ³ 1 ´
with
distinct indices 1 . Consider those ordered
-tuples ( 1 )forwhich1≤ 1 ≤
. Thereare ³ ´
of these. (Reason: Anyoneof
them can be permuted in ! ways to give a unique,
unordered -tuple ( 1 ) with no repetitions;
there are in total
( − 1) ···( − +1)=!( − )!
of these.)
Averaging the corresponding values of gives another
unbiased estimator:
= ³ 1 X
³ 1 ´
´
( 1 )
with the sum being over these ordered -tuples.
Any such statistic is called the U-statistic corresponding
to .
— e.g. =1: = [ ()] = −1 P
=1 ( )