ks (version 1.6.13)

Hamise.mixt, Hmise.mixt, Hamise.mixt.diag, Hmise.mixt.diag, amise.mixt, ise.mixt, mise.mixt: Squared error 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.diag(mus, Sigmas, props, samp, Hstart, deriv.order=0)
Hmise.mixt.diag(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)

Arguments

mus

(stacked) matrix of mean vectors/vector of means

sigmas, Sigmas

vector of standard deviations/(stacked) matrix of variance matrices

props

vector of mixing proportions

samp

sample size

hstart, Hstart

initial bandwidth (matrix), used in numerical optimisation

deriv.order

derivative order

x

matrix of data values

h, 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, 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. (2009). Asymptotics for general multivariate kernel density derivative estimators. Statistica Sinica. Accepted.

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)
ise.mixt(x=x, H=H, mus=mus, Sigmas=Sigmas, props=props, deriv.order=2)
amise.mixt(H=H, mus=mus, Sigmas=Sigmas, props=props, samp=samp, deriv.order=2)