Grassberger's expectation of n^q

Source: R/e_n_q.R

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

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] 107.366
# and (naive estimation)
n^q
#> [1] 100