Cumulant


In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.
The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the th-order cumulant of their sum is equal to the sum of their th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.
Just as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.

Definition

The cumulants of a random variable are defined using the cumulant-generating function, which is the natural logarithm of the moment-generating function:
The cumulants are obtained from a power series expansion of the cumulant generating function:
This expansion is a Maclaurin series, so the th cumulant can be obtained by differentiating the above expansion times and evaluating the result at zero:
If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.

Alternative definition of the cumulant generating function

Some writers prefer to define the cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function,
An advantage of — in some sense the function evaluated for purely imaginary arguments — is that is well defined for all real values of even when is not well defined for all real values of, such as can occur when there is "too much" probability that has a large magnitude. Although the function will be well defined, it will nonetheless mimic in terms of the length of its Maclaurin series, which may not extend beyond linear order in the argument , and in particular the number of cumulants that are well defined will not change. Nevertheless, even when does not have a long Maclaurin series, it can be used directly in analyzing and, particularly, adding random variables. Both the Cauchy distribution and more generally, stable distributions are examples of distributions for which the power-series expansions of the generating functions have only finitely many well-defined terms.

Some basic properties

The th cumulant of a random variable enjoys the following properties:
  • If and is constant then i.e. the cumulant is translation invariant. then i.e. the th cumulant is homogeneous of degree .
  • If random variables are independent then That is, the cumulant is cumulative — hence the name.
The cumulative property follows quickly by considering the cumulant-generating function:
so that each cumulant of a sum of independent random variables is the sum of the corresponding cumulants of the addends. That is, when the addends are statistically independent, the mean of the sum is the sum of the means, the variance of the sum is the sum of the variances, the third cumulant of the sum is the sum of the third cumulants, and so on for each order of cumulant.
A distribution with given cumulants can be approximated through an Edgeworth series.

First several cumulants as functions of the moments

All of the higher cumulants are polynomial functions of the central moments, with integer coefficients, but only in degrees 2 and 3 are the cumulants actually central moments.
Let be the cumulants, be the mean, and be the central moments. Then:
  • .
  • .
  • .
  • .
  • .

    Cumulants of some discrete probability distributions

  • The constant random variables. The cumulant generating function is. The first cumulant is and the other cumulants are zero,.
  • The Bernoulli distributions,. The cumulant generating function is. The first cumulants are and. The cumulants satisfy a recursion formula
  • The geometric distributions,. The cumulant generating function is. The first cumulants are, and. Substituting gives and.
  • The Poisson distributions. The cumulant generating function is. All cumulants are equal to the parameter:.
  • The binomial distributions,. The special case is a Bernoulli distribution. Every cumulant is just times the corresponding cumulant of the corresponding Bernoulli distribution. The cumulant generating function is. The first cumulants are and. Substituting gives and. The limiting case is a Poisson distribution.
  • The negative binomial distributions,. The special case is a geometric distribution. Every cumulant is just times the corresponding cumulant of the corresponding geometric distribution. The derivative of the cumulant generating function is. The first cumulants are, and. Substituting gives and. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'. The limiting case is a Poisson distribution.
Introducing the variance-to-mean ratio
the above probability distributions get a unified formula for the derivative of the cumulant generating function:
The second derivative is
confirming that the first cumulant is and the second cumulant is.
The constant random variables have.
The binomial distributions have so that.
The Poisson distributions have.
The negative binomial distributions have so that.
Note the analogy to the classification of conic sections by eccentricity: circles, ellipses, parabolas, hyperbolas.

Cumulants of some continuous probability distributions

The cumulant generating function, if it exists, is infinitely differentiable and convex, and passes through the origin. Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the degenerate distribution of a single point mass. The cumulant-generating function exists if and only if the tails of the distribution are majorized by an exponential decay, that is,
where is the cumulative distribution function. The cumulant-generating function will have vertical asymptote at the negative supremum of such, if such a supremum exists, and at the supremum of such, if such a supremum exists, otherwise it will be defined for all real numbers.
If the support of a random variable has finite upper or lower bounds, then its cumulant-generating function, if it exists, approaches asymptote whose slope is equal to the supremum or infimum of the support,
respectively, lying above both these lines everywhere.
For a shift of the distribution by, For a degenerate point mass at, the cumulant generating function is the straight line, and more generally, if and only if and are independent and their cumulant generating functions exist;
The natural exponential family of a distribution may be realized by shifting or translating, and adjusting it vertically so that it always passes through the origin: if is the pdf with cumulant generating function and is its natural exponential family, then and
If is finite for a range then if then is analytic and infinitely differentiable for. Moreover for real and is strictly convex, and is strictly increasing.

Further properties of cumulants

A negative result

Given the results for the cumulants of the normal distribution, it might be hoped to find families of distributions for which
for some, with the lower-order cumulants being non-zero. There are no such distributions. The underlying result here is that the cumulant generating function cannot be a finite-order polynomial of degree greater than 2.

Cumulants and moments

The moment generating function is given by:
So the cumulant generating function is the logarithm of the moment generating function
The first cumulant is the expected value; the second and third cumulants are respectively the second and third central moments ; but the higher cumulants are neither moments nor central moments, but rather more complicated polynomial functions of the moments.
The moments can be recovered in terms of cumulants by evaluating the th derivative of at,
Likewise, the cumulants can be recovered in terms of moments by evaluating the th derivative of at,
The explicit expression for the th moment in terms of the first cumulants, and vice versa, can be obtained by using Faà di Bruno's formula for higher derivatives of composite functions. In general, we have
where are incomplete Bell polynomials.
In the like manner, if the mean is given by, the central moment generating function is given by
and the th central moment is obtained in terms of cumulants as
Also, for, the th cumulant in terms of the central moments is
The th moment is an th-degree polynomial in the first cumulants. The first few expressions are:
The "prime" distinguishes the moments from the central moments. To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which appears as a factor:
Similarly, the th cumulant is an th-degree polynomial in the first non-central moments. The first few expressions are:
In general, the cumulant is the determinant of a matrix:
To express the cumulants for as functions of the central moments, drop from these polynomials all terms in which μ'1 appears as a factor:
The cumulants can be related to the moments by differentiating the relationship with respect to, giving, which conveniently contains no exponentials or logarithms. Equating the coefficient of on the left and right sides and using gives the following formulas for :
These allow either or to be computed from the other using knowledge of the lower-order cumulants and moments. The corresponding formulas for the central moments for are formed from these formulas by setting and replacing each with for :