Lecture3 Slide - The Department of Statistics and Applied Probability ...

23
Step one, continued
•
≤
[ f (x)
nh
∫
n −(2+δ) 2 1+δ 1
h 1+δ
K 2 (v) dv {1 + u (1)}] −(2+δ)/2
× n×
∫
max f (t)
t∈[a,b]
K 2+δ (v) dv {1 + u (1)}
≤ Cn (2+δ)/2+1−(2+δ) h (2+δ)/2−(1+δ) = Cn −δ/2 h −δ/2 = C (nh) −δ/2
• Lyapunov CLT shows
V −1
n
n∑
i=1
ξ in
D
→ N (0, 1) ,
{ f (x)
nh
∫
} −1/2 ∑ n
K 2 (v) dv
i=1
ξ in
D
→ N (0, 1)
ST5207 Nonparametric Regression, 27th January 2005