ICV (version 1.0)

MISE_mixnorm: The MISE function in the case when the true density is the specified mixture of normal distributions and the selection kernel L_ICV is used in the cross-validation stage.

Description

Computing $MISE(h)$ for given $h$ in the case when the true density is the specified mixture of normal distributions and the kernel is L_ICV defined by expression (4) of Savchuk, Hart, and Sheather (2010).

Usage

MISE_mixnorm(h, n, alpha, sigma, w, mu, sdev)

Arguments

h

numerical vector of bandwidth values,

n

sample size,

alpha

first parameter of the selection kernel,

sigma

second parameter of the selection kernel,

w

vector of weighs (positive numbers between 0 and 1 that add up to one),

mu

vector of means,

sdev

vector of standard deviations.

Value

The vector of MISE values corresponding to the specified values of $h$.

Details

Calculation of $MISE(h)$ in the case when the true density is the mixture of normal distributions defined by the vector of weights $w$, the vector of means $\mu$, and the vector of standard deviations $\sigma$. See expression (2.3) of Marron and Wand (1992). It is assumed that the normals are defined as parsimonious as possible. The normal distributions in the mixture should be ordered such that the means in $\mu$ are arranged in a nondecreasing order. The MISE function is based on the selection kernel L_ICV defined by expression (4) of Savchuk, Hart, and Sheather (2010). Notice that the Gaussian kernel $\phi$ is the special case of L_ICV given that (Case 1) $\alpha=0$, $\sigma>0$ or (Case 2) $\sigma=1$, $-\infty

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.
• Marron, J.S., Wand, M.P. (1992). Exact Mean Integrated Squared Error. The Annals of Statistics, 20(2), 712-736.

See Also

mixnorm, ISE_mixnorm, h_isemixnorm, L_ICV, ICV, h_ICV, C_ICV.

Examples

## Not run: 
# # Example 1. MISE for the separated bimodal density of Marron and Wand (1992).
# # in the case (alpha,sigma)=(2.42,5.06), n=100.
# harray=seq(0.05,1,len=1000)
# w=c(1/2,1/2)
# m=c(-3/2,3/2)
# s=c(1/2,1/2)
# MISEarray=MISE_mixnorm(harray,100,2.42,5.06,w,m,s)
# hopt=round(optimize(MISE_mixnorm,c(0.01,1),n=100,alpha=2.42,sigma=5.06,w=w,mu=m,sdev=s)$minimum,
# digits=4)
# dev.new()
# plot(harray,MISEarray,'l',lwd=3,xlab="h",ylab="MISE",cex.lab=1.7,cex.axis=1.7,main="")
# title(main="MISE(h) for the separated bimodal density",cex.main=1.5)
# legend(0.45,0.45,legend=c(paste("h_MISE=",hopt),"n=100"),bty="n",cex=1.7)
# 
# # Example 2. MISE for the N(0,1) density in the case of the Gaussian kernel and n=500.
# harray=seq(0.03,1,len=1000)
# MISEarray=MISE_mixnorm(harray,500,1,1,1,0,1)
# hopt=round(optimize(MISE_mixnorm,c(0.01,1),n=500,alpha=1,sigma=1,w=1,mu=0,sdev=1)$minimum,digits=4)
# dev.new()
# plot(harray,MISEarray,'l',lwd=3,xlab="h",ylab="MISE",cex.lab=1.7,cex.axis=1.7,main="")
# title(main="MISE(h) for the standard normal density",cex.main=1.7)
# legend(0.2,0.02,legend=c(paste("h_MISE=",hopt),"n=500"),bty="n",cex=1.7)
# ## End(Not run)