LocICV: The local ICV function.
In ICV: Indirect Cross-Validation (ICV) for Kernel Density Estimation

Description Usage Arguments Details Value References See Also Examples

Computing the local ICV function at the given estimation point, as explained in Section 6 of Savchuk, Hart, and Sheather (2010).

1	LocICV(h, xest, x, eta, alpha, sigma)

`h`	bandwidth (scalar) in the final scale,
`xest`	estimation point (scalar),
`x`	numerical vector of data,
`eta`	smoothing parameter,
`alpha`	first parameter of the selection kernel,
`sigma`	second parameter of the selection kernel.

Calculation of the local ICV function at the given estimation point xest. The Gaussian kernel is used for local weighting. The ultimate kernel density estimate is computed based on the Gaussian kernel. The parameters of the selection kernel L_ICV are α and σ. The minimizer of the local ICV function is to be used in computing the ultimate density estimate without additional rescaling. Parameter η is a smoothing parameter that determines the degree to which the cross-validation is local. A suggested value of η is η=R/20, where R is the range of data.

The value of the local ICV function at the fixed estimation point and for the specified value of the bandwidth (see Section 6 of Savchuk, Hart, and Sheather (2010)).

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.
Savchuk, O.Y., Hart, J.D., Sheather, S.J. (2009). An empirical study of indirect cross-validation. Nonparametric Statistics and Mixture Models: A Festschrift in Honor of Thomas P. Hettmansperger. World Scientific Publishing, 288-308.
Hall, P., and Schukany, W. R. (1989). A local cross-validation algorithm. Statistics and Probability Letters, 8, 109-117.

h_ICV, C_ICV, L_ICV, MISE_mixnorm, ICV, KDE_ICV.

## Not run: 
# Local ICV function for a random sample of size n=150 from the kurtotic density of Marron and
# Wand (1992).
dat=mixnorm(150,c(2/3,1/3),c(0,0),c(1,1/10))
a=2.42   # alpha
s=5.06   # sigma
harg=seq(0.025,1,len=100)
Xest=0.1    # estimation point
LocICV_Xest=numeric(length(harg))
for(i in 1:length(harg))
  LocICV_Xest[i]=LocICV(harg[i],Xest,dat,0.2,a,s)
h_Xest=optimize(LocICV,c(0.001,0.2),tol=0.001,xest=Xest,eta=0.2,x=dat,alpha=a,sigma=s)$minimum
h_Xest=round(h_Xest,digits=4)
dev.new()
plot(harg,LocICV_Xest,'l',lwd=3,xlab="harg",ylab="LocICV_Xest",main="",cex.lab=1.7, cex.axis=1.7)
title(main=paste("Local ICV function at x=",Xest),cex.main=1.7)
legend(0.1,max(LocICV_Xest),legend=paste("h_x=",h_Xest),cex=1.7)
legend(0.2,max(LocICV_Xest)-0.15,legend="Note: first local minimizer is used", cex=1.5,bty="n")

## End(Not run)