PDQutils (version 0.1.6)

moment2cumulant: Convert moments to raw cumulants.

Description

Conversion of a vector of moments to raw cumulants.

Usage

moment2cumulant(moms)

Arguments

moms

a vector of the moments. The first element is the first moment (the mean). If centered moments are given, the first moment must be zero. If raw moments are given, the first moment must be the mean.

Value

a vector of the cumulants. The first element of the input shall be the same as the first element of the output.

Details

The 'raw' cumulants \(\kappa_i\) are connected to the 'raw' (uncentered) moments, \(\mu_i'\) via the equation $$\kappa_n = \mu_n' - \sum_{m=1}^{n-1} {n-1 \choose m-1} \kappa_m \mu_{n-m}'$$ Note that this formula also works for central moments, assuming the distribution has been normalized to zero mean.

See Also

cumulant2moment

Examples

# normal distribution, mean 0, variance 1
n.cum <- moment2cumulant(c(0,1,0,3,0,15))
# normal distribution, mean 1, variance 1
n.cum <- moment2cumulant(c(1,2,4,10,26))
# exponential distribution
lambda <- 0.7
n <- 1:6
e.cum <- moment2cumulant(factorial(n) / (lambda^n))