 Cumulant

In probability theory and statistics, the cumulants κ_{n} of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have identical cumulants as well, and similarly the cumulants determine the moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments.
Just as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.
Introduction
The cumulants κ_{n} of a random variable X are defined via the cumulantgenerating function
using the (noncentral) moments μ′_{n} of X and the momentgenerating function,
with a formal power series logarithm:
The cumulants of a distribution are closely related to the distribution's moments. For example, if a random variable X admits an expected value μ = E(X) and a variance σ^{2} = E((X − μ)^{2}), then these are the first two cumulants: μ = κ_{1} and σ^{2} = κ_{2}.
Generally, the cumulants can be extracted from the cumulantgenerating function via differentiation (at zero) of g(t). That is, the cumulants appear as the coefficients in the Maclaurin series of g(t):
Note that expectation values are sometimes denoted by angle brackets, e.g.,
and cumulants can be denoted by angle brackets with the subscript c,^{[citation needed]} e.g.,
Some writers^{[1]}^{[2]} prefer to define the cumulant generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function,^{[3]}^{[4]}
The advantage of h(t)—in some sense the function g(t) evaluated for (purely) imaginary arguments—is that E(e^{itX}) will be well defined for all real values of t even when E(e^{tX}) is not well defined for all real values of t, such as can occur when there is "too much" probability that X has a large magnitude. Although h(t) will be well defined, it nonetheless may mimic g(t) by not having a Maclaurin series beyond (or, rarely, even to) linear order in the argument t. Thus, many cumulants may still not be well defined. Nevertheless, even when h(t) does not have a long Maclaurin series it can be used directly in analyzing and, particularly, adding random variables. Both the Cauchy distribution (also called the Lorentzian) and stable distribution (related to the Lévy distribution) are examples of distributions for which the generating functions do not have powerseries expansions.
Uses in mathematical statistics
Working with cumulants can have an advantage over using moments because for independent variables X and Y,
so that each cumulant of a sum is the sum of the corresponding cumulants of the addends.
A distribution with given cumulants κ_{n} can be approximated through an Edgeworth series.
Cumulants of some discrete probability distributions
 The constant random variable X = 1. The derivative of the cumulant generating function is g '(t) = 1. The first cumulant is κ_{1} = g '(0) = 1 and the other cumulants are zero, κ_{2} = κ_{3} = κ_{4} = ... = 0.
 The constant random variables X = μ. Every cumulant is just μ times the corresponding cumulant of the constant random variable X = 1. The derivative of the cumulant generating function is g '(t) = μ. The first cumulant is κ_{1} = g '(0) = μ and the other cumulants are zero, κ_{2} = κ_{3} = κ_{4} = ... = 0. So the derivative of cumulant generating functions is a generalization of the real constants.
 The Bernoulli distributions, (number of successes in one trial with probability p of success). The special case p = 1 is the constant random variable X = 1. The derivative of the cumulant generating function is g '(t) = ((p^{ −1}−1)·e^{−t} + 1)^{−1}. The first cumulants are κ_{1} = g '(0) = p and κ_{2} = g ' '(0) = p·(1 − p) . The cumulants satisfy a recursion formula
 The geometric distributions, (number of failures before one success with probability p of success on each trial). The derivative of the cumulant generating function is g '(t) = ((1 − p)^{−1}·e^{−t} − 1)^{−1}. The first cumulants are κ_{1} = g '(0) = p^{−1} − 1, and κ_{2} = g ' '(0) = κ_{1}·p^{ − 1}. Substituting p = (μ+1)^{−1} gives g '(t) = ((μ^{−1} + 1)·e^{−t} − 1)^{−1} and κ_{1} = μ.
 The Poisson distributions. The derivative of the cumulant generating function is g '(t) = μ·e^{t}. All cumulants are equal to the parameter: κ_{1} = κ_{2} = κ_{3} = ...=μ.
 The binomial distributions, (number of successes in n independent trials with probability p of success on each trial). The special case n = 1 is a Bernoulli distribution. Every cumulant is just n times the corresponding cumulant of the corresponding Bernoulli distribution. The derivative of the cumulant generating function is g '(t) = n·((p^{−1}−1)·e^{−t} + 1)^{−1}. The first cumulants are κ_{1} = g '(0) = n·p and κ_{2} = g ' '(0) = κ_{1}·(1−p). Substituting p = μ·n^{−1} gives g '(t) = ((μ^{−1} − n^{−1})·e^{−t} + n^{−1})^{−1} and κ_{1} = μ. The limiting case n^{−1} = 0 is a Poisson distribution.
 The negative binomial distributions, (number of failures before n successes with probability p of success on each trial). The special case n = 1 is a geometric distribution. Every cumulant is just n times the corresponding cumulant of the corresponding geometric distribution. The derivative of the cumulant generating function is g '(t) = n·((1−p)^{−1}·e^{−t}−1)^{−1}. The first cumulants are κ_{1} = g '(0) = n·(p^{−1}−1), and κ_{2} = g ' '(0) = κ_{1}·p^{−1}. Substituting p = (μ·n^{−1}+1)^{−1} gives g '(t) = ((μ^{−1}+n^{−1})·e^{−t}−n^{−1})^{−1} and κ_{1} = μ. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'. The limiting case n^{−1} = 0 is a Poisson distribution.
Introducing the variancetomean ratio
the above probability distributions get a unified formula for the derivative of the cumulant generating function:^{[citation needed]}
The second derivative is
confirming that the first cumulant is κ_{1} = g '(0) = μ and the second cumulant is κ_{2} = g ' '(0) = μ·ε. The constant random variables X = μ have є = 0. The binomial distributions have ε = 1 − p so that 0 < ε < 1. The Poisson distributions have ε = 1. The negative binomial distributions have ε = p^{−1} so that ε > 1. Note the analogy to the classification of conic sections by eccentricity: circles ε = 0, ellipses 0 < ε < 1, parabolas ε = 1, hyperbolas ε > 1.
Cumulants of some continuous probability distributions
 For the normal distribution with expected value μ and variance σ^{2}, the cumulant generating function is g(t) = μt + σ^{2}t^{2}/2. The first and second derivatives of the cumulant generating function are g '(t) = μ + σ^{2}·t and g"(t) = σ^{2}. The cumulants are κ_{1} = μ, κ_{2} = σ^{2}, and κ_{3} = κ_{4} = ... = 0. The special case σ^{2} = 0 is a constant random variable X = μ.
 The cumulants of the uniform distribution on the interval [−1, 0] are κ_{n} = B_{n}/n, where B_{n} is the nth Bernoulli number.
 The cumulants of the exponential distribution with parameter λ are κ_{n} = λ^{−n} (n − 1)!.
Some properties of the cumulant generating function
The cumulant generating function g(t) is convex. If g(t) is finite for a range t_{1} < Re(t) < t_{2} then if t_{1} < 0 < t_{2} then g(t) is analytic and infinitely differentiable for t_{1} < Re(t) < t_{2}. Moreover for t real and t_{1} < t < t_{2} g(t) is strictly convex, and g'(t) is strictly increasing.^{[citation needed]}
Some properties of cumulants
Invariance and equivariance
The first cumulant is shiftequivariant; all of the others are shiftinvariant. This means that, if we denote by κ_{n}(X) the nth cumulant of the probability distribution of the random variable X, then for any constant c:
In other words, shifting a random variable (adding c) shifts the first cumulant (the mean) and doesn't affect any of the others.
Homogeneity
The nth cumulant is homogeneous of degree n, i.e. if c is any constant, then
Additivity
If X and Y are independent random variables then κ_{n}(X + Y) = κ_{n}(X) + κ_{n}(Y).
A negative result
Given the results for the cumulants of the normal distribution, it might be hoped to find families of distributions for which κ_{m} = κ_{m+1} = ... = 0 for some m > 3, with the lowerorder cumulants (orders 3 to m − 1) being nonzero. There are no such distributions.^{[5]} The underlying result here is that the cumulant generating function cannot be a finiteorder polynomial of degree greater than 2.
Cumulants and moments
The moment generating function is:
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 (the second central moment is the variance); but the higher cumulants are neither moments nor central moments, but rather more complicated polynomial functions of the moments.
The cumulants are related to the moments by the following recursion formula:
The nth moment μ′_{n} is an nthdegree polynomial in the first n cumulants:
The coefficients are precisely those that occur in Faà di Bruno's formula.
The "prime" distinguishes the moments μ′_{n} from the central moments μ_{n}. To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which κ_{1} appears as a factor:
Likewise, the nth cumulant κ_{n} is an nthdegree polynomial in the first n noncentral moments:
To express the cumulants κ_{n} for n > 1 as functions of the central moments, drop from these polynomials all terms in which μ'_{1} appears as a factor:
Cumulants and setpartitions
These polynomials have a remarkable combinatorial interpretation: the coefficients count certain partitions of sets. A general form of these polynomials is
where
 π runs through the list of all partitions of a set of size n;
 "B π" means B is one of the "blocks" into which the set is partitioned; and
 B is the size of the set B.
Thus each monomial is a constant times a product of cumulants in which the sum of the indices is n (e.g., in the term κ_{3} κ_{2}^{2} κ_{1}, the sum of the indices is 3 + 2 + 2 + 1 = 8; this appears in the polynomial that expresses the 8th moment as a function of the first eight cumulants). A partition of the integer n corresponds to each term. The coefficient in each term is the number of partitions of a set of n members that collapse to that partition of the integer n when the members of the set become indistinguishable.
Joint cumulants
The joint cumulant of several random variables X_{1}, ..., X_{n} is defined by a similar cumulant generating function
A consequence is that
where π runs through the list of all partitions of { 1, ..., n }, B runs through the list of all blocks of the partition π, and π is the number of parts in the partition. For example,
If any of these random variables are identical, e.g. if X = Y, then the same formulae apply, e.g.
although for such repeated variables there are more concise formulae. For zeromean random vectors,
The joint cumulant of just one random variable is its expected value, and that of two random variables is their covariance. If some of the random variables are independent of all of the others, then any cumulant involving two (or more) independent random variables is zero. If all n random variables are the same, then the joint cumulant is the nth ordinary cumulant.
The combinatorial meaning of the expression of moments in terms of cumulants is easier to understand than that of cumulants in terms of moments:
For example:
Another important property of joint cumulants is multilinearity:
Just as the second cumulant is the variance, the joint cumulant of just two random variables is the covariance. The familiar identity
generalizes to cumulants:
Conditional cumulants and the law of total cumulance
Main article: law of total cumulanceThe law of total expectation and the law of total variance generalize naturally to conditional cumulants. The case n = 3, expressed in the language of (central) moments rather than that of cumulants, says
The general result stated below first appeared in 1969 in The Calculation of Cumulants via Conditioning by David R. Brillinger in volume 21 of Annals of the Institute of Statistical Mathematics, pages 215–218.
In general, we have
where
 the sum is over all partitions π of the set { 1, ..., n } of indices, and
 π_{1}, ..., π_{b} are all of the "blocks" of the partition π; the expression κ(X_{πm}) indicates that the joint cumulant of the random variables whose indices are in that block of the partition.
Relation to statistical physics
In statistical physics many extensive quantities – that is quantities that are proportional to the volume or size of a given system – are related to cumulants of random variables. The deep connection is that in a large system an extensive quantity like the energy or number of particles can be thought of as the sum of (say) the energy associated with a number of nearly independent regions. The fact that the cumulants of these nearly independent random variables will (nearly) add make it reasonable that extensive quantities should be expected to be related to cumulants.
A system in equilibrium with a thermal bath at temperature T can occupy states of energy E. The energy E can be considered a random variable, having the probability density. The partition function of the system is
where β = 1/(kT) and k is Boltzmann's constant and the notation has been used rather than for the expectation value to avoid confusion with the energy, E. The Helmholtz free energy is then
and is clearly very closely related to the cumulant generating function for the energy. The free energy gives access to all of the thermodynamics properties of the system via its first second and higher order derivatives, such as its internal energy, entropy, and specific heat. Because of the relationship between the free energy and the cumulant generating function, all these quantities are related to cumulants e.g. the energy and specific heat are given by
and symbolizes the second cumulant of the energy. Other free energy is often also a function of other variables such as the magnetic field or chemical potential μ, e.g.
where N is the number of particles and Ω is the grand potential. Again the close relationship between the definition of the free energy and the cumulant generating function implies that various derivatives of this free energy can be written in terms of joint cumulants of E and N.
History
The history of cumulants is discussed by Hald.^{[6]}^{[7]}
Cumulants were first introduced by Thorvald N. Thiele, in 1889, who called them semiinvariants.^{[8]} They were first called cumulants in a 1932 paper^{[9]} by Ronald Fisher and John Wishart. Fisher was publicly reminded of Thiele's work by Neyman, who also notes previous published citations of Thiele brought to Fisher's attention.^{[10]} Stephen Stigler has said^{[citation needed]} that the name cumulant was suggested to Fisher in a letter from Harold Hotelling. In a paper published in 1929,^{[citation needed]} Fisher had called them cumulative moment functions. The partition function in statistical physics was introduced by Josiah Willard Gibbs in 1901.^{[citation needed]} The free energy is often called Gibbs free energy. In statistical mechanics, cumulants are also known as Ursell functions relating to a publication in 1927.^{[citation needed]}
Cumulants in generalized settings
Formal cumulants
More generally, the cumulants of a sequence { m_{n} : n = 1, 2, 3, ... }, not necessarily the moments of any probability distribution, are given by^{[citation needed]}
where the values of κ_{n} for n = 1, 2, 3, ... are found formally, i.e., by algebra alone, in disregard of questions of whether any series converges. All of the difficulties of the "problem of cumulants" are absent when one works formally. The simplest example is that the second cumulant of a probability distribution must always be nonnegative, and is zero only if all of the higher cumulants are zero. Formal cumulants are subject to no such constraints.
Bell numbers
In combinatorics, the nth Bell number is the number of partitions of a set of size n. All of the cumulants of the sequence of Bell numbers are equal to 1.^{[citation needed]} The Bell numbers are the moments of the Poisson distribution with expected value 1.^{[citation needed]}
Cumulants of a polynomial sequence of binomial type
For any sequence { κ_{n} : n = 1, 2, 3, ... } of scalars in a field of characteristic zero, being considered formal cumulants, there is a corresponding sequence { μ ′ : n = 1, 2, 3, ... } of formal moments, given by the polynomials above.^{[clarification needed]}^{[citation needed]} For those polynomials, construct a polynomial sequence in the following way. Out of the polynomial
make a new polynomial in these plus one additional variable x:
and then generalize the pattern. The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on x. Each coefficient is a polynomial in the cumulants; these are the Bell polynomials, named after Eric Temple Bell.^{[citation needed]}
This sequence of polynomials is of binomial type. In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants.^{[citation needed]}
Free cumulants
In the identity^{[clarification needed]}
one sums over all partitions of the set { 1, ..., n }. If instead, one sums only over the noncrossing partitions, then one gets "free cumulants" rather than conventional cumulants treated above.^{[clarification needed]} These play a central role in free probability theory.^{[11]} In that theory, rather than considering independence of random variables, defined in terms of Cartesian products of algebras of random variables, one considers instead "freeness" of random variables, defined in terms of free products of algebras rather than Cartesian products of algebras.^{[citation needed]}
The ordinary cumulants of degree higher than 2 of the normal distribution are zero. The free cumulants of degree higher than 2 of the Wigner semicircle distribution are zero.^{[11]} This is one respect in which the role of the Wigner distribution in free probability theory is analogous to that of the normal distribution in conventional probability theory.
See also
References
 ^ Kendall, M.G., Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1 (3rd Edition). Griffin, London. (Section 3.12)
 ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Page 27)
 ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Section 2.4)
 ^ Aapo Hyvarinen, Juha Karhunen, and Erkki Oja (2001) Independent Component Analysis, John Wiley & Sons. (Section 2.7.2)
 ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition), Griffin, London. (Theorem 7.3.5)
 ^ Hald, A. (2000) "The early history of the cumulants and the Gram–Charlier series" International Statistical Review, 68 (2): 137–153. (Reprinted in Steffen L. Lauritzen, ed (2002). Thiele: Pioneer in Statistics. Oxford U. P.. ISBN 9780198509721.)
 ^ Hald, Anders (1998). A History of Mathematical Statistics from 1750 to 1930. New York: Wiley. ISBN 0471179124.
 ^ H. Cramér (1946) Mathematical Methods of Statistics, Princeton University Press, Section 15.10, p. 186.
 ^ Fisher, R.A. , John Wishart, J.. (1932) The derivation of the pattern formulae of twoway partitions from those of simpler patterns, Proceedings of the London Mathematical Society, Series 2, v. 33, pp. 195–208 doi: 10.1112/plms/s233.1.195
 ^ Neyman, J. (1956): ‘Note on an Article by Sir Ronald Fisher,’ Journal of the Royal Statistical Society, Series B (Methodological), 18, pp. 288–94.
 ^ ^{a} ^{b} Novak, Jonathan; Śniady, Piotr (2011). "What Is a Free Cumulant?". Notices of the American Mathematical Society 58 (2): 300–301. ISSN 00029920.
External links
 Weisstein, Eric W., "Cumulant" from MathWorld.
 cumulant on the Earliest known uses of some of the words of mathematics
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