divent (version 0.5-3)

e_n_q: 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)

Value

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

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 Grassberger1988;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 \(q > n\).

References

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