ICV (version 1.0)

LocICV: The local ICV function.

Description

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

Usage

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

Arguments

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.

Value

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)).

Details

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 $\alpha$ and $\sigma$. The minimizer of the local ICV function is to be used in computing the ultimate density estimate without additional rescaling. Parameter $\eta$ is a smoothing parameter that determines the degree to which the cross-validation is local. A suggested value of $\eta$ is $\eta=R/20$, where $R$ is the range of data.

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.
• 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.

See Also

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

Examples

## 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)