
Using the moments up to a given order about one location, this function either returns the moments up to that given order about a new location as a vector or it returns a moment of a specific order defined by users (order <= maximum order of the given moments) about a new location as a single number. A generalization of using raw moments to obtain a central moment or using central moments to obtain a raw moment.
momChangeAbout(order = "all", oldMom, oldAbout, newAbout)
The moment of order order
about the location newAbout
when
order
is specified.
The vector of moments about the location newAbout
from first
order up to the maximum order of the oldMom
when order
takes the value "all"
or is not specified.
One of:
the character string "all", the default;
a positive integer less than the maximum order of oldMom
.
Numeric. Moments of orders 1, 2, ..., about the point
oldAbout
.
Numeric. The point about which the moments oldMom
have
been calculated.
Numeric. The point about which the desired moment or moments are to be obtained.
David Scott d.scott@auckland.ac.nz, Christine Yang Dong c.dong@auckland.ac.nz
Suppose momChangeAbout
. It is a generalization of the well-known
formulae used to change raw moments to central moments or to change
central moments to raw moments. See for example Kendall and Stuart
(1989), Chapter 3.
Kendall, M. G. and Stuart, A. (1969). The Advanced Theory of Statistics, Volume 1, 3rd Edition. London: Charles Griffin & Company.
### Gamma distribution
k <- 4
shape <- 2
old <- 0
new <- 1
sampSize <- 1000000
### Calculate 1st to 4th raw moments
m <- numeric(k)
for (i in 1:k){
m[i] <- gamma(shape + i)/gamma(shape)
}
m
### Calculate 4th moment about new
momChangeAbout(k, m, old, new)
### Calculate 3rd about new
momChangeAbout(3, m, old, new)
### Calculate 1st to 4th moments about new
momChangeAbout(oldMom = m, oldAbout = old, newAbout = new)
momChangeAbout(order = "all", m, old, new)
### Approximate kth moment about new using sampling
x <- rgamma(sampSize, shape)
mean((x - new)^k)
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