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 ...
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36<br />
Digression to density estimation<br />
•<br />
{ }<br />
AMISE ˆf (x) ; h = 1<br />
∫<br />
f ′′<br />
nh c (x) 2 dx<br />
K +<br />
h 4 d 2 K<br />
4<br />
• To find the optimal h, solve the following equation<br />
d<br />
{ }<br />
dh AMISE ˆf (x) ; h = − 1 ∫<br />
nh 2 c K + f ′′ (x) 2 dxh 3 d 2 K = 0<br />
• <strong>The</strong> optimal b<strong>and</strong>width is<br />
{<br />
h opt =<br />
c K<br />
d 2 K<br />
} 1/5<br />
1<br />
∫<br />
f<br />
′′<br />
(x) 2 n −1/5<br />
dx<br />
• <strong>The</strong> optimal b<strong>and</strong>width depends on curvature <strong>of</strong> the unknown<br />
density f, kernel K used <strong>and</strong> sample size n available. XploRe<br />
quantlets are denrot <strong>and</strong> denbwsel, we now explore their use<br />
ST5207 Nonparametric Regression, 27th January 2005