dupp: Probability mass function (PMF) of discrete power law
Description
dupp returns the PMF at x for the discrete power law with exponent (1.0 / xi1 + 1.0), for values greater than or equal to u.
Usage
dupp(x, u, xi1, log = FALSE)
Value
A numeric vector of the same length as x
Arguments
x
Vector of positive integers
u
Scalar, non-negative integer threshold
xi1
Scalar, a positive real number representing the shape parameter
log
Boolean (default 'FALSE'), whether the PMF should be returned on the log scale.
Details
The PMF is proportional to x^(-alpha), where alpha = 1.0 / xi1 + 1.0. To be a proper PMF, it is normalised by 1/hzeta(alpha, u), where hzeta is the Hurwitz zeta function i.e. hzeta(y, z) = z^(-y) + (z+1)^(-y) + (z+2)^(-y) + ... Any values below u will have PMF equal to 0.0. That xi1 is used instead of alpha is for alignment with the parametrisation in dmix, Smix and mcmc_mix.
See Also
Supp for the corresponding survival function, dmix for the PMF of the discrete extreme value mixture distribution.