Lecture3 Slide - The Department of Statistics and Applied Probability ...
Lecture3 Slide - The Department of Statistics and Applied Probability ...
Lecture3 Slide - The Department of Statistics and Applied Probability ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
24<br />
Step one, continued<br />
•<br />
≤<br />
[ f (x)<br />
nh<br />
∫<br />
n −(2+δ) 2 1+δ 1<br />
h 1+δ<br />
K 2 (v) dv {1 + u (1)}] −(2+δ)/2<br />
× n×<br />
∫<br />
max f (t)<br />
t∈[a,b]<br />
K 2+δ (v) dv {1 + u (1)}<br />
≤ Cn (2+δ)/2+1−(2+δ) h (2+δ)/2−(1+δ) = Cn −δ/2 h −δ/2 = C (nh) −δ/2<br />
• Lyapunov CLT shows<br />
V −1<br />
n<br />
n∑<br />
i=1<br />
{ f (x)<br />
nh<br />
∫<br />
ξ in<br />
D<br />
→ N (0, 1) ,<br />
{ f (x)<br />
nh<br />
∫<br />
} −1/2 ∑ n<br />
K 2 (v) dv<br />
i=1<br />
ξ in<br />
D<br />
→ N (0, 1)<br />
K 2 (v) dv} −1/2 [<br />
ˆf (x) − f (x) − Bias<br />
{<br />
ˆf (x)<br />
}] D→ N (0, 1)<br />
ST5207 Nonparametric Regression, 27th January 2005