# momChangeAbout: Obtain Moments About a New Location In DistributionUtils: Distribution Utilities

## 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

 ``` 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) ```

DistributionUtils documentation built on May 1, 2019, 9:12 p.m.