ICV {ICV}

The ICV function.

Description

Computing ICV(h), the value of the ICV function, at a given bandwidth h (vector) for a data set x of size n<12,058. See Savchuk, Hart, and Sheather (2010).

Usage

ICV(h, x)

Arguments

h

numerical vector of bandwidth values (in the final scale),

x

numerical vecror of data.

Details

Computation of ICV(h) for given h (bandwidth vector) and x (data vector). The sample size n<12,058. The Gaussian kernel is to be used for computing the ultimate kernel density estimate. The parameters of the selection kernel are (\alpha,\sigma)=(2.42, 5.06). The ICV bandwidth h_ICV is the minimizer of the ICV function.

Value

The value of ICV(h) for given h and data (x).

References

Savchuk, O.Y., Hart, J.D., Sheather, S.J. (2010). Indirect cross-validation for density estimation. Journal of the American Statistical Association, 105(489), 415-423.

See Also

h_ICV, C_ICV, L_ICV, MISE_mixnorm, KDE_ICV, LocICV.

Examples

#Example 1. Computation of ICV(h) at h=0.4 for a random sample of size n=100 from a N(0,1)
#distribution.
ICV(0.4,rnorm(100))

## Not run: 
#Example 2. (Calculations for a random sample of size n=250 from the separated bimodal density).
w=c(1/2,1/2)
mu=c(-3/2,3/2)
sdev=c(1/2,1/2)
# Generating a sample of size n=250 from the separated bimodal density of Marron and Wand (1992).
dat=mixnorm(250,w,mu,sdev)
h_OS=(243/(35*length(dat)*2*sqrt(pi)))^0.2*sd(dat)    # Computing the oversmoothed bandwidth.
h_opt=round(h_ICV(dat),digits=4)
harg=seq(0.1,3,len=100)
dev.new()
plot(harg,ICV(harg,x=dat),'l',lwd=3,xlab="h",ylab="ICV",cex.lab=1.7,cex.axis=1.7)
title(main="ICV(h)",cex.main=1.7)
lines(c(h_OS,h_OS),c(-0.5,0.5),lty="dashed",lwd=3)
legend(0.75,-0.05,legend="Vertical line shows the oversmothed bandwidth")
legend(1.35,0.1,legend=paste("h_ICV=",h_opt),cex=2,bty="n")
# Notice that the scale of the ICV function is such that its minimizer is the ICV bandwidth h_ICV.
# Thus, no additional rescaling of the ICV function's minimizer is needed to obtain the ICV
# bandwidth.
dev.new()
dens=density(dat,bw=h_opt)
plot(dens,main="",cex.lab=1.7,cex.axis=1.7,lwd=3,xlab=paste("n=250,","h=h_ICV=",h_opt),
ylim=c(0,0.45))
title(main="KDE based on h_ICV",cex.main=1.7)
arg=seq(-3.5,3.5,len=1000)
lines(arg,w[1]*dnorm(arg,mu[1],sd=sdev[1])+w[2]*dnorm(arg,mu[2],sd=sdev[2]),lwd=3,lty="dashed")
legend(-1,0.45,lty=c("solid","dashed"),lwd=c(3,3),legend=c("ICV estimate","True density"),bty="n")

## End(Not run)