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Cross Validation for Kernel Density Estimation

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Cross Validation for Kernel Density Estimation

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In a post published in July, I mentioned the so-called Goldilocks principle, in the context of kernel density estimation, and bandwidth selection. The bandwidth should not be too small (the variance would be too large) and it should not be too large (the bias would be too large). Another standard method to select the bandwidth, as mentioned this afternoon in class is the cross-validation technique (described in Chiu (1991)). Here, we would like to minimize

http://latex.codecogs.com/gif.latex?\mathbb{E}\left[\int%20[\widehat{f}_h(x)-f(x)>^2dx\right]

The integral can be written

http://latex.codecogs.com/gif.latex?\int%20\widehat{f}_h(x)^2dx-2\int%20\widehat{f}_h(x)f(x)dx+\int%20f(x)^2dx

Since the third component is constant, we have to minimize the expected value of the sum of the first two.

The idea is to approximate it as

http://latex.codecogs.com/gif.latex?J(h)=\int%20\widehat{f}_h(x)^2dx-\frac{2}{n}\sum_{i=1}^n%20\widehat{f}_{(-i)}(X_i)

which can easily be computed. Consider here some sample, with 50 observations, from a Gaussian distribution,

> set.seed(1)
> X=rnorm(50)

From Silverman’s rule of thumb (which should be appropriate here since the sample has a Gaussian sample) the optimal bandwidth is

> 1.06*sd(X)*length(X)^(-1/5)
[1] 0.4030127

Using the cross-validation technique mentioned above, compute

> J=function(h){
+ fhat=Vectorize(function(x) density(X,from=x,to=x,n=1,bw=h)$y)
+ fhati=Vectorize(function(i) density(X[-i],from=X[i],to=X[i],n=1,bw=h)$y)
+ F=fhati(1:length(X))
+ return(integrate(function(x) fhat(x)^2,-Inf,Inf)$value-2*mean(F))
+ }
> vx=seq(.1,1,by=.01)
> vy=Vectorize(J)(vx)
> df=data.frame(vx,vy)
> library(ggplot2)
> qplot(vx,vy,geom="line",data=df)

The function has the following shape

and the optimal value is

> optimize(J,interval=c(.1,1))
$minimum
[1] 0.4687553

$objective
[1] -0.3355477

Note that, indeed, it is close to Siverman’s optimal bandwidth.

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