# 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
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 have been translated from Steve Marron's Matlab code,
now at http://www.unc.edu/depts/stat-or/miscellaneous/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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | ```
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.)
``` |