e_n_q: Grassberger's expectation of n^q

In divent: Entropy Partitioning to Measure Diversity

Grassberger's expectation of n^q

Description

Expected value of n^q when n follows a Poisson distribution of parameter n.

Usage

e_n_q(n, q)

Arguments

n	A positive integer vector.
q	A positive number.

Details

The expectation of n^q when n follows a Poisson distribution was derived by \insertCiteGrassberger1988;textualdivent.

It is computed using the beta function. Its value is 0 for n-q+1<0, and close to 0 when n=q, which is not a correct estimate: it should not be used when q > n.

Value

A vector of the same length as n containing the transformed values.

References

\insertAllCited

Examples

n <- 10
q <- 2
# Compare
e_n_q(n, q)
# with (empirical estimation)
mean(rpois(1000, lambda = n)^q)
# and (naive estimation)
n^q

divent documentation built on April 3, 2025, 7:40 p.m.