ks (version 1.2.1)

ise, mise, amise: ISE, MISE and AMISE of kernel density estimates for normal and t mixture densities

Description

The global errors ISE (Integrated Squared Error), MISE (Mean Integrated Squared Error) and AMISE (Asymptotic Mean Integrated Squared Error) of kernel density estimates for normal and t mixture densities.

Usage

ise.mixt(x, H, mus, Sigmas, props, lower, upper, gridsize=c(250,250),
         stepsize)
iset.mixt(x, H, mus, Sigmas, dfs, props, lower, upper, gridsize=c(250,250),
          stepsize)  
mise.mixt(H, mus, Sigmas, props, samp)
amise.mixt(H, mus, Sigmas, props, samp)

Arguments

x

matrix of data values

H

bandwidth matrix

mus

(stacked) matrix of mean vectors

Sigmas

(stacked) matrix of variance matrices

dfs

vector of degrees of freedom

props

vector of mixing proportions

samp

sample size

lower, upper

vectors of lower, upper bounds for numerical integration

gridsize

vector of number of points in each dimension

stepsize

vector of step sizes in each dimension

Value

• ISE, MISE or AMISE value.

Details

For normal mixture densities, the ISE, MISE and AMISE all have exact formulas. See Wand & Jones (1995). For the t mixture densities, we resort to using numerical integration, using a simple Riemann sum. A grid is set up and the function values are computed and then multiplied by the area of the grid element to give an approximation of the volume under the curve. The resolution of the grid is given either by gridsize or stepsize.

References

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

Examples

samp <- 100
mus <- rbind(c(-3/2,0), c(3/2,0))
Sigmas <- rbind(diag(c(1/16, 1)), rbind(c(1/16, 1/18), c(1/18, 1/16)))
props <- c(2/3, 1/3)
x <- rmvnorm.mixt(samp, mus, Sigmas, props)
H <- Hpi(x)
ise.mixt(x, H, mus, Sigmas, props, stepsize=0.01)
mise.mixt(H, mus, Sigmas, props, samp)
amise.mixt(H, mus, Sigmas, props, samp)

dfs <- c(7,5)
x <- rmvt.mixt(samp, mus, Sigmas, dfs, props)
H <- Hpi(x)
iset.mixt(x, H, mus, Sigmas, dfs, props, lower=c(-5,-5), upper=c(5,5))