Grassberger's expectation of n^q

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

Enq(n, q)

Arguments

n

A positive integer vector.

q

A positive number.

Details

The expectation of \(n^q\) when \(n\) follows a Poisson ditribution has been derived by Grassberger (1988).

Value

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

References

Grassberger, P. (1988). Finite sample corrections to entropy and dimension estimates. Physics Letters A 128(6-7): 369-373.

Note

The function is computed using the beta.function.

Its value is 0 for \(n-q+1<0\).

Examples

# Compare
n <- c(2,3)
Enq(n, q=2)
#> [1] 2 6
# with
n^2
#> [1] 4 9

# Result is 1
Enq(n, q=0)
#> [1] 1 1
# Result is 0
Enq(n, q=5)
#> [1] -7.796344e-15  5.847258e-15