Lecture7 Slide - The Department of Statistics and Applied Probability ...

22
Multivariate local linear estimation
• The limiting distribution for ˆm (x) is
{
√ d∑
nh1 · · · h d ˆm (x) − m (x) −
b α (x) = d K
2
α=1
h 2 αb α (x)
}
D
→ N {0, v(x)}
∂ 2 m
∂ 2 x α
(x) , v(x) = σ 2 (x)f −1 (x)c d K
• The Asymptotic MISE (AMISE { ˆm (x) ; h}) is
∫
σ 2 (x)dxc d d∑
∫
K
+ d2 K
∂
h 2
nh 1 · · · h d 4
αh 2 2 m
β
∂ 2 (x) ∂2 m
x α ∂ 2 (x) f(x)dx
x β
α,β=1
h opt = v (m, σ, K) n −1/(d+4) , AMISE { ˆm (x) ; h opt } = C (m, σ, K) n −4/(d+4)
• The ”curse of dimensionality”: slower convergence rate n −2/(d+4)
with high dimension d (Intuitively, why?)
ST5207 Nonparametric Regression, 10th March 2005