ICV (version 1.0)

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

Usage

ICV(h, x)

Arguments

h

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

x

numerical vecror of data.

Value

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

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="" ultimate="" density="" estimate.="" parameters="" of="" selection="" are="" $(\alpha,\sigma)="(2.42," 5.06)$.="" icv="" bandwidth="" h_ICV is the minimizer of the ICV function.

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)