R: Obtain Moments About a New Location

momChangeAbout {DistributionUtils} | R Documentation

## 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

      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

    ### 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)

[Package DistributionUtils version 0.6-0 Index]