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Raw Moment

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A raw moment is a moment mu_n of a probability function P(x) taken about 0,

mu_n^'=<x^n>
(1)
=intx^nP(x)dx.
(2)

The raw moments mu_n^' (sometimes also called "crude moments") can be expressed as terms of the central moments mu_n (i.e., those taken about the mean mu) using the inverse binomial transform

mu_n^'=sum_(k=0)^n(n; k)mu_kmu_1^('n-k),
(3)

with mu_0=1 and mu_1=0 (Papoulis 1984, p. 146). The first few values are therefore

mu_2^'=mu_2+mu_1^('2)
(4)
mu_3^'=mu_3+3mu_2mu_1^'+mu_1^('3)
(5)
mu_4^'=mu_4+4mu_3mu_1^'+6mu_2mu_1^('2)+mu_1^('4)
(6)
mu_5^'=mu_5+5mu_4mu_1^'+10mu_3mu_1^('2)+10mu_2mu_1^('3)+mu_1^('5).
(7)

The raw moments mu_n^' can also be expressed in terms of the cumulants kappa_n by exponentiating both sides of the series

lnphi=ln(sum_(k=0)^infty((it)^k)/(k!)mu_k^')=sum_(n=0)^inftykappa_n((it)^n)/(n!),
(8)

where phi is the characteristic function, to obtain

sum_(k=0)^infty((it)^k)/(k!)mu_k^'=exp(sum_(n=0)^inftykappa_n((it)^n)/(n!)).
(9)

The first few terms are then given by

mu_1^'=kappa_1
(10)
mu_2^'=kappa_1^2+kappa_2
(11)
mu_3^'=kappa_1^3+3kappa_1kappa_2+kappa_3
(12)
mu_4^'=kappa_1^4+6kappa_1^2kappa_2+3kappa_2^2+4kappa_1kappa_3+kappa_4
(13)
mu_5^'=kappa_1^5+10kappa_1^3kappa_2+15kappa_1kappa_2^2+10kappa_1^2kappa_3+10kappa_2kappa_3+5kappa_1kappa_4+kappa_5.
(14)

These transformations can be obtained using RawToCumulant[n] in the Mathematica application package mathStatica.

The raw moment of a multivariate probability function P(x_1,x_2,...) can be similarly defined as

mu_(m,n,...)^'=<x_1^mx_2^n...>.
(15)

Therefore,

mu_(n,0,...,0)^'=mu_n^'.
(16)

The multivariate raw moments can be expressed in terms of the multivariate cumulants. For example,

mu_(1,1)^'=kappa_(0,1)kappa_(1,0)+kappa_(1,1)
(17)
mu_(2,1)^'=kappa_(0,1)kappa_(1,0)^2+2kappa_(1,0)kappa_(1,1)+kappa_(0,1)kappa_(2,0)+kappa_(2,1).
(18)

These transformations can be obtained using RawToCumulant[{m, n, ...}] in the Mathematica application package mathStatica.

See also

Absolute Moment, Central Moment, Complete Bell Polynomial, Cumulant, Mean, Moment, Partial Bell Polynomial, Sample Raw Moment

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References

Kendall, M. G. "The Derivation of Multivariate Sampling Formulae from Univariate Formulae by Symbolic Operation." Ann. Eugenics 10, 392-402, 1940.Kenney, J. F. and Keeping, E. S. "Moments about the Origin." §7.2 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 91-92, 1962.Kratky, J.; Reinfelds, J.; Hutcheson, K.; and Shenton, L. R. "Tables of Crude Moments Expressed in Terms of Cumulants." Technical Report. Athens, GA: University of Georgia, 1972.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984.Smith, P. J. "A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa." Amer. Stat. 49, 217-218, 1995.

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Raw Moment

Cite this as:

Weisstein, Eric W. "Raw Moment." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RawMoment.html

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