ks (version 1.5.10)

Hamise.mixt, Hmise.mixt, amise.mixt, ise.mixt, mise.mixt: MISE- and AMISE-optimal bandwidth matrix selectors for normal mixture densities

Description

The global errors ISE (Integrated Squared Error), MISE (Mean Integrated Squared Error) and the AMISE (Asymptotic Mean Integrated Squared Error) for 1- to 6-dimensional data.

Normal mixture densities have closed form expressions for the MISE and AMISE. So in these cases, we can numerically minimise these criteria to find MISE- and AMISE-optimal matrices.

Usage

Hamise.mixt(mus, Sigmas, props, samp, Hstart, deriv.order=0)
Hmise.mixt(mus, Sigmas, props, samp, Hstart, deriv.order=0)
hamise.mixt(mus, sigmas, props, samp, hstart, deriv.order=0)
hmise.mixt(mus, sigmas, props, samp, hstart, deriv.order=0)
ise.mixt(x, H, mus, Sigmas, props, h, sigmas, deriv.order=0)  
mise.mixt(H, mus, Sigmas, props, samp, h, sigmas, deriv.order=0)
amise.mixt(H, mus, Sigmas, props, samp, h, sigmas, deriv.order=0)

Value

• -- Full MISE- or AMISE-optimal bandwidth matrix. Diagonal forms of these matrices are not available.

  -- ISE, MISE or AMISE value. ise is not yet available for deriv.order>0.

Details

For normal mixture densities, ISE, MISE and AMISE have exact formulas for all dimensions. See Chac'on, Duong & Wand (2008).

If Hstart is not given then it defaults to k*var(x) where k = $\left[\frac{4}{n(d+2r+2)}\right]^{2/(d+2r+4)}$, n = sample size, d = dimension of data, r= derivative order. The default for hstart is the square root of this expression.

References

Chac'on J.E., Duong, T. & Wand, M.P. (2008) Asymptotics for general multivariate kernel density derivative estimators. In preparation.

Examples

## 1-d
mus <- c(0, 2)
sigmas <- c(1, sqrt(0.7))
props <- c(1/2, 1/2)
samp <- 1000
h <- hmise.mixt(mus, sigmas, props, samp, deriv.order=0)
x <- rnorm.mixt(n=samp, mus=mus, sigmas=sigmas, props=props)
ise.mixt(x=x, h=h, mus=mus, sigmas=sigmas, props=props)
mise.mixt(h=h, mus=mus, sigmas=sigmas, props=props, samp=samp)

## 2-d 
mus <- rbind(c(0,0), c(2,2))
Sigma <- matrix(c(1, 0.7, 0.7, 1), nr=2, nc=2) 
Sigmas <- rbind(Sigma, Sigma)
props <- c(1/2, 1/2)
samp <- 100
H <- Hamise.mixt(mus, Sigmas, props, samp, deriv.order=2)
x <- rmvnorm.mixt(n=samp, mus=mus, Sigmas=Sigmas, props=props)
amise.mixt(H=H, mus=mus, Sigmas=Sigmas, props=props, samp=samp, deriv.order=2)