Asymptotic Methods in Statistical Inference - Statistics Centre

109
We have lim →∞ { ∗ () − 1 ()} 0( ∗ is
asymptotically Pitman closer) if
min ³ 1 −1´ − −−1 1 − −
This holds if
(i) 1, or if
(ii) ≤ 1and −1 − −−1 1 − − ,i.e.the
function
() = −1 ³ − −´ + − − 1
has () 0(01). Equivalently, with
= 1, we need
(i) ,or
(ii) 1 ≤ and () = (−1)(−(−1))
0.
The second condition holds for ∈ ( − 1], and
so the requirement is merely − 1. Equivalently,
= log( − 1) ≈ 541.