Asymptotic Methods in Statistical Inference - Statistics Centre

129
15. Expectation functionals; U- and V-statistics
• Nonparametric estimation. Let 1 ∼
,where is an unknown member of a class F of
distributions. We wish to estimate a ‘parameter’
= ( ); we call a functional since it maps
F into R. The ‘plug-in’ estimator is ˆ = ( ˆ ),
where ˆ is the empirical distribution function
(e.d.f.):
ˆ () = #of ’ ≤
∼
( ())
→ ()
— e.g. = −1 (12) (population median); ˆ =
ˆ
−1 (12) = min n | ˆ () ≥ 12 o is (one definition
of) the sample median.
• We will investigate the asymptotic properties of
such estimators.