ks (version 1.5.5)

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

Description

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. The global errors ISE (Integrated Squared Error), MISE (Mean Integrated Squared Error) of kernel density estimates for normal densities, for 2- to 6-dimensional data, and AMISE (Asymptotic Mean Integrated Squared Error) for 2-dimensional data.

Usage

Hmise.mixt(mus, Sigmas, props, samp, Hstart)
Hamise.mixt(mus, Sigmas, props, samp, Hstart)
ise.mixt(x, H, mus, Sigmas, props)  
mise.mixt(H, mus, Sigmas, props, samp)
amise.mixt(H, mus, Sigmas, props, samp)

Arguments

mus

(stacked) matrix of mean vectors

Sigmas

(stacked) matrix of variance matrices

props

vector of mixing proportions

samp

sample size

Hstart

initial bandwidth matrix, used in numerical optimisation

x

matrix of data values

H

bandwidth matrix

Value

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

  -- ISE, MISE or AMISE value.

Details

For normal mixture densities, ISE and MISE have exact formulas for all dimensions, and AMISE has an exact form for 2 dimensions. See Wand & Jones (1995).

If Hstart is not given then it defaults to k*var(x) where k = $\left[\frac{4}{n(d+2)}\right]^{2/(d+4)}$, n = sample size, d = dimension of data.

References

Wand, M.P. & Jones, M.C. (1995) Kernel Smoothing. Chapman & Hall. London.

Examples

mus <- rbind(c(0,0,0), c(2,2,2))
Sigma <- matrix(c(1, 0.7, 0.7, 0.7, 1, 0.7, 0.7, 0.7, 1), nr=3, nc=3) 
Sigmas <- rbind(Sigma, Sigma)
props <- c(1/2, 1/2)
samp <- 1000
x <- rmvnorm.mixt(n=samp, mus=mus, Sigmas=Sigmas, props=props)

H1 <- Hmise.mixt(mus, Sigmas, props, samp)
H2 <- Hamise.mixt(mus, Sigmas, props, samp)

ise.mixt(x, H2, mus, Sigmas, props)
mise.mixt(H2, mus, Sigmas, props, samp)