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

6
Multivariate nonparametric estimation
• Let { Y i , X T i
} n
i=1 = {Y i, X i1 , ..., X id } n i=1
Y = m (X) + σ (X) ε, X = (X 1 , ..., X d )
where the noise satisfies E (ε|X) = 0, var (ε|X) = 1.
• How to estimate multivariate function m?
be i.i.d. sample from model
• We will discuss Nadaraya-Watson and local linear methods
• Conventions on multivariate kernel and bandwidth vector
d∏
( )
1 uα
K h (u) = K , u = (u 1 , ..., u d ) , h = (h 1 , ..., h d )
h α h α
α=1
• Wand & Jones (1995). Kernel Smoothing, Chapman and Hall,
London, and Wand & Ruppert (1994) (see reference list in syllabus)
use bandwidth matrix, instead of vector.
ST5207 Nonparametric Regression, 10th March 2005