Package jline.api.moment

Class Moment_cumulant_from_raw

java.lang.Object
jline.api.moment.Moment_cumulant_from_raw
public final class Moment_cumulant_from_raw extends Object
Cumulants from power (raw) moments.

Converts the power moments m_n = E[X^n] of a random variable X into its cumulants kappa_n, the coefficients of the cumulant generating function log E[exp(sX)] = sum_{n>=1} kappa_n s^n / n!, by inverting the exponential-formula recursion 

   m_n = sum_{k=1}^{n} nchoosek(n-1,k-1) * kappa_k * m_(n-k)

Equivalently kappa_n is the Leonov-Shiryaev partition sum over the set partitions of {1,...,n}. The first cumulants are kappa_1 = m_1, kappa_2 = m_2 - m_1^2 (the variance) and kappa_3 = m_3 - 3 m_1 m_2 + 2 m_1^3 (the third central moment). The conversion is not restricted to discrete random variables.

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_cumulant_from_raw

      public static Matrix moment_cumulant_from_raw(Matrix m)
      Converts power (raw) moments into cumulants.
      Parameters:
      m - column vector of length n+1 holding m_0,...,m_n, with m_0 = 1
      Returns:
      column vector of length n+1 holding kappa_0,...,kappa_n, element 0 being kappa_0 = 0 and not m_0 = 1