Asymptotic Methods in Statistical Inference - Statistics Centre

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(i) the t-test ( = ¯() = ), for which
( ) → Φ(∆ − )with =1,
(ii) the sign test, with =2(0). The ARE of
the sign test to the t-test is ( =1)
à ! 2
= =(2(0)) 2
This can be arbitrarily large or small; for ∼
(0 2 )itis =2 ≈ 637 . For the Laplace
(() =5exp(−||), 2 =2)itis2.
(iii) An alternative procedure is the one-sample
Wilcoxon test: Rank the | | and sum the ranks
(rather than merely the signs) of the positive .
Let be this sum and define = ³
2
´; large
values support . For alternatives = ∆ √
it can be shown (and will be in Lecture 16) that
√ ( − ( ))
q
13
→ (0 1)
where () = ( 1 + 2 0) and 1 2
are distributed independently, and symmetrically
around , withd.f. ( ≤ ) = ( − ). Thus
(0) = 12 —defining the rejection region — and