Asymptotic Methods in Statistical Inference - Statistics Centre

131
• A closely related estimate, and one with somewhat
simpler asymptotic properties, is the ‘Ustatistic’.
It differs from the V-statistic in disallowing
repetitions such as ( ).
• There is a symmetric functional with the same
expectation : first note that for
= [ ( 1 )], ( 1 ) can be
replaced by ³ 1 ´
for any permutation
( 1 ) of (1) without altering the expected
value (since 1 are i.i.d.), or by
∗ ( 1 )= 1 !
X
³ 1 ´
where the sum is over all ! permutations of (1).
Thus also = [ ∗ ( 1 )] where ∗ is
symmetric, i.e. invariant under permutations of
its arguments. e.g. if =2then
∗ ( 1 2 ) = 1 2 [ ( 1 2 )+ ( 2 1 )]
= ∗ ( 2 1 )
We will assume from now on that is symmetric.