Package jline.api.moment

Class Moment_factcumulant_from_factorial

java.lang.Object
jline.api.moment.Moment_factcumulant_from_factorial
public final class Moment_factcumulant_from_factorial extends Object
Factorial cumulants from factorial moments.

The factorial cumulants of a discrete random variable N are the coefficients of the logarithm of the probability generating function expanded about z = 1, 

   log E[z^N] = sum_{n>=1} kappa_n (z-1)^n / n!

They stand to the factorial moments exactly as the cumulants stand to the power moments, so the same recursion applies. For a Poisson variable of rate lambda all factorial cumulants beyond the first vanish, which makes them the natural measure of departure from Poisson behaviour in the counting process of a MAP.

Reference: V. P. Leonov and A. N. Shiryaev. On a method of calculation of semi-invariants. Theory of Probability and its Applications, 4(3):319-329, 1959.

Since:
LINE 3.0

  • Method Details

    • moment_factcumulant_from_factorial

      public static Matrix moment_factcumulant_from_factorial(Matrix f)
      Converts factorial moments into factorial cumulants.
      Parameters:
      f - column vector of length n+1 holding f_0,...,f_n, with f_0 = 1
      Returns:
      column vector of length n+1 holding the factorial cumulants of order 0,...,n, element 0 being 0