6.31 One way to choose the smoothing parameter, h, in kernel density estimation is to minimise the AMISE criterion, given by :

AMISE =

R(K)

nh 

+

1 4



η4h4R(f).

Here f(x) denotes the pdf being estimated, and K denotes the kernel being used (itself a pdf with mean zero and variance η2) and R(g) is the smoothness measure :

R(g) = 

g2(x)dx.

Derive the value of h which minimises the AMISE , as a function of R(f).

Base the procedure on the case of a normal form for f(x), with variance

σ2, for which R(f)=3/(8√πσ5) and assume also a N(0, 1) form for K, so that η = 1. Show that in this case the AMISE is minimised by setting h = * 4 3

+1/5

σn−1/5.