Asymptotic Methods in Statistical Inference - Statistics Centre

87
We exhibit 1 () 2 () → 1− and ()
→0
such that
→ 1
→∞
³ √
1): 1 () ≤ || ≤ 2 || − 1| ´
≤ 2 ()
2): (| − 1| ) ≥ ()
Then (9.2) is, uniformly in , ≤
() = max{| 1 () − (1 − )| | 2 () − (1 − )|}
+2 (1 − ())
Now () → max {| 1 () − (1 − )| | 2 () − (1 − )|}
as →∞, and this may be made arbitrarily small.
For 1) note that
³
|| ≤ 2
√
|| − 1| ´
is maximized by choosing aslargeaspossible(=
1+), yielding 2 () =2Φ ³ 2
√ 1+´
− 1, and
similarly 1 () =2Φ ³ √
2 1 − ´
− 1. Here we use
the independence of and . Both these bounds
→ 1 − as → 0.