Marron-Wand Densities as 'norMix' Objects

Description

The fifteen density examples used in Marron and Wand (1992)'s simulation study have been used in quite a few subsequent studies, can all be written as normal mixtures and are provided here for convenience and didactical examples of normal mixtures. Number 16 has been added by Jansen et al.

Usage

 MW.nm1  # Gaussian
 MW.nm2  # Skewed
 MW.nm2.old # Skewed(old)
 MW.nm3  # Str Skew
 MW.nm4  # Kurtotic
 MW.nm5  # Outlier
 MW.nm6  # Bimodal
 MW.nm7  # Separated (bimodal)
 MW.nm8  # Asymmetric Bimodal
 MW.nm9  # Trimodal
 MW.nm10 # Claw
 MW.nm11 # Double Claw
 MW.nm12 # Asymmetric Claw
 MW.nm13 # Asymm. Double Claw
 MW.nm14 # Smooth   Comb
 MW.nm15 # Discrete Comb
 MW.nm16 # Distant Bimodal

Author(s)

Martin Maechler

Source

They are translated from Steve Marron's Matlab code at http://www.stat.unc.edu/postscript/papers/marron/parameters/nmpar.m, however for number 2, the Matlab code had MW.nm2.old; and I've defined MW.nm2 as from the Annals paper; see also the last example below.

References

Marron, S. and Wand, M. (1992) Exact Mean Integrated Squared Error; Annals of Statistcs 20, 712–736.

For number 16,
Janssen, Marron, Verb..., Sarle (1995) ....

Examples

MW.nm10
plot(MW.nm14)

## These are defined as norMix() calls in  ../R/zMarrWand-dens.R
nms <- ls(pat="^MW.nm", "package:nor1mix")
nms <- nms[order(as.numeric(substring(nms,6)))]
for(n in nms) {
   cat("\n",n,":\n"); print(get(n, "package:nor1mix"))
}

## Plot all of them:
op <- par(mfrow=c(4,4), mgp = c(1.2, 0.5, 0), tcl = -0.2,
          mar = .1 + c(2,2,2,1), oma = c(0,0,3,0))
for(n in nms[-17]) plot(get(n, "package:nor1mix"))
mtext("The Marron-Wand Densities", outer= TRUE, font= 2, cex= 1.6)

## and their Q-Q-plots (not really fast):
prob <- ppoints(N <- 100)
for(n in nms[-17])
   qqnorm(qnorMix(prob, get(n, "package:nor1mix")), main = n)
mtext("QQ-plots of Marron-Wand Densities", outer = TRUE,
      font = 2, cex = 1.6)
par(op)

## "object" overview:
cbind(sapply(nms, function(n) { o <- get(n)
      sprintf("%-18s: K =%2d; rng = [%3.1f, %2.1f]",
              attr(o, "name"), nrow(o),
              min(o[,"mu"] - 3*sqrt(o[,"sig2"])),
              max(o[,"mu"] + 3*sqrt(o[,"sig2"])) )
             }))

## Note that Marron-Wand (1992), p.720  give #2 as
MW.nm2
## the parameters of which at first look quite different from
MW.nm2.old
## which has been the definition in the above "Source" Matlab code.
## It's easy to see that mu_{nm2} = -.3 + 1.2   * mu_{paper},
## and correspondigly,   s2_{nm2} =       1.2^2 * s2_{paper}
## such that they are "identical" apart from scale and location:
op. <- par(mfrow=2:1, mgp= c(1.2,0.5,0), tcl= -0.2, mar=.1+c(2,2,2,1))
plot(MW.nm2)
plot(MW.nm2.old)
par(op.)