Obtain Moments About a New Location

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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:

  • the character string "all", the default;

  • a positive integer less than the maximum order of oldMom.

oldMom

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

oldAbout

Numeric. The point about which the moments oldMom have been calculated.

newAbout

Numeric. The point about which the desired moment or moments are to be obtained.

Details

Suppose m_k denotes the k-th moment of a random variable X about a point a, and m_k^* denotes the k-th 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 (a-b)^i m^{k-i}

This is the formula implemented by the function 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.

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

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### 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)