Obtain Moments About a New Location
Description
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.
Usage
1  momChangeAbout(order = "all", oldMom, oldAbout, newAbout)

Arguments
order 
One of:

oldMom 
Numeric. Moments of orders 1, 2, ..., about the point

oldAbout 
Numeric. The point about which the moments 
newAbout 
Numeric. The point about which the desired moment or moments are to be obtained. 
Details
Suppose m_k denotes the kth moment of a random variable X about a point a, and m_k^* denotes the kth moment about b. Then m_k^* may be determined from the moments m_1,m_2,...,m_k according to the formula
m_k^*=sum_{i=0}^k (ab)^i m^{ki}
This is the formula implemented by the function
momChangeAbout
. It is a generalization of the wellknown
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.
Value
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.
Author(s)
David Scott d.scott@auckland.ac.nz, Christine Yang Dong c.dong@auckland.ac.nz
References
Kendall, M. G. and Stuart, A. (1969). The Advanced Theory of Statistics, Volume 1, 3rd Edition. London: Charles Griffin & Company.
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  ### 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)
