Asymptotic Methods in Statistical Inference - Statistics Centre

Thus in (3.2), () () isthec.f.of ()(), i.e.
() =
⎡
Z ∞
−∞ ⎢
⎣
1+ 3
6 √ 3()
+ 3 4 4 ()+ 2 3 6()
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⎤
33
⎥
⎦ ()+ ³ −1´
• By uniqueness of c.f.s, an expansion for the density
of is
() = ()+ 3
6 √ 3()()
+ 3 4 4 ()+ 2 3 6()
()+ ³ −1´
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and an expansion for the d.f. is, since
()() = h (−1) (−1) () i 0 £
= −−1 ()() ¤ 0 ,
() = Φ() − 3
6 √ 2()()
− 3 4 3 ()+ 2 3 5()
()+ ³ −1´
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The first term gives the classical CLT for normalized
averages of i.i.d.s: → Φ. The second
refines this; even this term vanishes if the are
symmetrically distributed ( 3 =0).