Grassberger's expectation of n^q

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.

Value

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

Details

The expectation of $n^{q}$ when $n$ follows a Poisson distribution was derived by Grassberger (1988) .

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

Grassberger P (1988). “Finite Sample Corrections to Entropy and Dimension Estimates.” Physics Letters A, 128(6-7), 369–373. doi:10.1016/0375-9601(88)90193-4 .

Examples

n <- 10
q <- 2
# Compare
e_n_q(n, q)
#> [1] 90
# with (empirical estimation)
mean(rpois(1000, lambda = n)^q)
#> [1] 107.82
# and (naive estimation)
n^q
#> [1] 100