Cumulant

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In probability theory and statistics, if a random variable X admits an expected value μ = E(X) and a variance σ² = E((X − μ)²), then these are the first two cumulants: μ = κ₁ and σ² = κ₂.

The cumulants κ_n are defined by the cumulant-generating function, the logarithm of the moment-generating function, if it exists:

$g(t)=\log(E (e^{t\cdot X}))=\sum_{n=1}^\infty\kappa_n \frac{t^n}{n!}=\mu t + \sigma^2 \frac{ t^2}{2} + \cdots.$

The cumulants are then given by derivatives (at zero) of g(t):

κ 1 = μ = g'(0),

κ 2 = σ 2 = g''(0),

κ n = g (n) (0).

A distribution with given cumulants κ_n can be approximated through the Edgeworth series.

The cumulants of a distribution are closely related to distribution's moments as described later. Working with cumulants can have an advantage over using moments because for independent variables X and Y,

$\begin{align} g_{X+Y}(t) & =\log(E(e^{t\cdot (X+Y)})) = \log(E(e^{tX}) \cdot E(e^{tY})) \\ & = \log(E(e^{tX})) + \log(E(e^{tY})) = g_{X}(t)+g_{Y}(t). \end{align}$

so that each cumulant of a sum is the sum of the corresponding cumulants of the addends.

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]

$h(t)=\log(E (e^{i t X}))=\sum_{n=1}^\infty\kappa_n \cdot\frac{(it)^n}{n!}=\mu it - \sigma^2 \frac{ t^2}{2} + \cdots.\,$

This characterization of cumulants is valid even for distributions whose higher moments do not exist, while the characterization in terms of the moment-generating function is not valid if the higher moments do not exist.

[edit] 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 κ₁ = g '(0) = 1 and the other cumulants are zero, κ₂ = κ₃ = κ₄ = ... = 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 κ₁ = g '(0) = μ and the other cumulants are zero, κ₂ = κ₃ = κ₄ = ... = 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 κ₁ = g '(0) = p and κ₂ = g ' '(0) = p·(1 − p) . The cumulants satisfy a recursion formula $\scriptstyle\kappa_{n+1}=p\cdot(1-p)\cdot\tfrac{d\kappa_n}{dp}.\,$

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 κ₁ = g '(0) = p⁻¹−1, and κ₂ = g ' '(0) = κ₁·p⁻¹. Substituting p = (μ+1)⁻¹ gives g '(t) = ((μ⁻¹ + 1)·e^−t − 1)⁻¹ and κ₁ = μ.

The Poisson distributions. The derivative of the cumulant generating function is g '(t) = μ·e^t. All cumulants are equal to the parameter: κ₁ = κ₂ = κ₃ = ...=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 κ₁ = g '(0) = n·p and κ₂ = g ' '(0) = κ₁·(1−p). Substituting p = μ·n⁻¹ gives g '(t) = ((μ⁻¹ − n⁻¹)·e^−t + n⁻¹)⁻¹ and κ₁ = 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 κ₁ = g '(0) = n·(p⁻¹−1), and κ₂ = g ' '(0) = κ₁·p⁻¹. Substituting p = (μ·n⁻¹+1)⁻¹ gives g '(t) = ((μ⁻¹+n⁻¹)·e^−t−n⁻¹)⁻¹ and κ₁ = 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 variance-to-mean ratio

$\varepsilon=\mu^{-1}\sigma^2=k_1^{-1}k_2, \,$

the above probability distributions get a unified formula for the derivative of the cumulant generating function:

$g'(t)=\mu\cdot(1+\varepsilon\cdot (e^{-t}-1))^{-1}. \,$

The second derivative is

$g''(t)=g'(t)\cdot(1+e^t\cdot (\varepsilon^{-1}-1))^{-1} \,$

confirming that the first cumulant is κ₁ = g '(0) = μ and the second cumulant is κ₂ = 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.

[edit] Cumulants of some continuous probability distributions

For the normal distribution with expected value μ and variance σ², the derivative of the cumulant generating function is g '(t) = μ + σ²·t. The cumulants are κ₁ = μ, κ₂ = σ², and κ₃ = κ₄ = ... = 0. The special case σ² = 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.

[edit] Some properties of cumulant generating function

The cumulant generating function g(t) is convex, 0 belong to the set {t: g(t)<infinite}, if the interior of the set is not empty, then g(t) is analytic there, and infinitely differentialbe there, on the set, g(t) is strctly convex, and g'(t) is strictly increasing.

[edit] Some properties of cumulants

[edit] Invariance and equivariance

The first cumulant is shift-equivariant; all of the others are shift-invariant. 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:

$\kappa_1(X + c) = \kappa_1(X) + c ~ \text{ and}$
$\kappa_n(X + c) = \kappa_n(X) ~ \text{ for } ~ n \ge 2$

in other words, shifting a random variable (adding c) shifts the first cumulant (the mean) and doesn't affect any of the others.

[edit] Homogeneity

The nth cumulant is homogeneous of degree n, i.e. if c is any constant, then

$\kappa_n(cX)=c^n\kappa_n(X). \,$

[edit] Additivity

If X and Y are independent random variables then κ_n(X + Y) = κ_n(X) + κ_n(Y).

[edit] 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 lower order cumulants (orders 3 to m -1) being non-zero. There are no such distributions ^[5]. The underlying result here is that the cumulant generating function cannot be a finite order polynomial of order greater than 2.

[edit] Cumulants and moments

The moment generating function is:

$1+\sum_{n=1}^\infty \frac{\mu'_n t^n}{n!}=\exp\left(\sum_{n=1}^\infty \frac{\kappa_n t^n}{n!}\right) = \exp(g(t)).$

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:

$\kappa_n=\mu'_n-\sum_{k=1}^{n-1}{n-1 \choose k-1}\kappa_k \mu_{n-k}'.$

The nth moment μ′_n is an nth-degree polynomial in the first n cumulants, thus:

$\mu'_1=\kappa_1\,$

$\mu'_2=\kappa_2+\kappa_1^2\,$

$\mu'_3=\kappa_3+3\kappa_2\kappa_1+\kappa_1^3\,$

$\mu'_4=\kappa_4+4\kappa_3\kappa_1+3\kappa_2^2+6\kappa_2\kappa_1^2+\kappa_1^4\,$

$\mu'_5=\kappa_5+5\kappa_4\kappa_1+10\kappa_3\kappa_2 +10\kappa_3\kappa_1^2+15\kappa_2^2\kappa_1 +10\kappa_2\kappa_1^3+\kappa_1^5\,$

$\mu'_6=\kappa_6+6\kappa_5\kappa_1+15\kappa_4\kappa_2+15\kappa_4\kappa_1^2 +10\kappa_3^2+60\kappa_3\kappa_2\kappa_1+20\kappa_3\kappa_1^3+15\kappa_2^3 +45\kappa_2^2\kappa_1^2+15\kappa_2\kappa_1^4+\kappa_1^6.\,$

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 κ₁ appears as a factor:

$\mu_1=0\,$

$\mu_2=\kappa_2\,$

$\mu_3=\kappa_3\,$

$\mu_4=\kappa_4+3\kappa_2^2\,$

$\mu_5=\kappa_5+10\kappa_3\kappa_2\,$

$\mu_6=\kappa_6+15\kappa_4\kappa_2+10\kappa_3^2+15\kappa_2^3.\,$

[edit] Cumulants and set-partitions

These polynomials have a remarkable combinatorial interpretation: the coefficients count certain partitions of sets. A general form of these polynomials is

$\mu'_n=\sum_{\pi}\prod_{B\in\pi}\kappa_{\left|B\right|}$

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 κ₃ κ₂² κ₁, 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.

[edit] Joint cumulants

The joint cumulant of several random variables X₁, ..., X_n is

$\kappa(X_1,\dots,X_n) =\sum_\pi (|\pi|-1)!(-1)^{|\pi|-1}\prod_{B\in\pi}E\left(\prod_{i\in B}X_i\right)$

where π runs through the list of all partitions of { 1, ..., n }, and B runs through the list of all blocks of the partition π. For example,

$\kappa(X,Y,Z)=E(XYZ)-E(XY)E(Z)-E(XZ)E(Y)-E(YZ)E(X)+2E(X)E(Y)E(Z).\,$

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 the joint cumulant 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:

$E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_i : i \in B).$

For example:

$E(XYZ)=\kappa(X,Y,Z)+\kappa(X,Y)\kappa(Z)+\kappa(X,Z)\kappa(Y) +\kappa(Y,Z)\kappa(X)+\kappa(X)\kappa(Y)\kappa(Z).\,$

Another important property of joint cumulants is multilinearity:

$\kappa(X+Y,Z_1,Z_2,\dots)=\kappa(X,Z_1,Z_2,\dots)+\kappa(Y,Z_1,Z_2,\dots).\,$

Just as the second cumulant is the variance, the joint cumulant of just two random variables is the covariance. The familiar identity

$\operatorname{var}(X+Y)=\operatorname{var}(X) +2\operatorname{cov}(X,Y)+\operatorname{var}(Y)\,$

generalizes to cumulants:

$\kappa_n(X+Y)=\sum_{j=0}^n {n \choose j} \kappa(\,\underbrace{X,\dots,X}_{j},\underbrace{Y,\dots,Y}_{n-j}).\,$

[edit] Conditional cumulants and the law of total cumulance

Main article: law of total cumulance

The 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

$\mu_3(X)=E(\mu_3(X\mid Y))+\mu_3(E(X\mid Y)) +3\,\operatorname{cov}(E(X\mid Y),\operatorname{var}(X\mid Y)).$

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

$\kappa(X_1,\dots,X_n)=\sum_\pi \kappa(\kappa(X_{\pi_1}\mid Y),\dots,\kappa(X_{\pi_b}\mid Y))$

where

the sum is over all partitions π of the set { 1, ..., n } of indices, and

π₁, ..., π_b are all of the "blocks" of the partition π; the expression κ(X_{π_k}) indicates that the joint cumulant of the random variables whose indices are in that block of the partition.

[edit] Relation to statistical physics

In statistical physics a given system at equilibrium with a thermal bath at temperature T = 1/β can occupy states of energy E. The energy E can be considered a random variable, having the probability density ƒ (also called the density of states of energy E). The partition function of the system is

$Z(\beta) = \langle\exp(-\beta E)\rangle.\,$

The free energy is defined as

$F(\beta) = \frac{-1}{\beta}\log(Z).$

The free energy gives access to all of the thermodynamics properties of the system, such as its internal energy, entropy, and specific heat.

[edit] History

Cumulants were first introduced by the Danish astronomer, actuary, mathematician, and statistician Thorvald N. Thiele (1838 – 1910) in 1889. Thiele called them half-invariants. They were first called cumulants in a 1932 paper, The derivation of the pattern formulae of two-way partitions from those of simpler patterns, Proceedings of the London Mathematical Society, Series 2, v. 33, pp. 195-208, by the great statistical geneticist Sir Ronald Fisher and the statistician John Wishart, eponym of the Wishart distribution. The historian Stephen Stigler has said that the name cumulant was suggested to Fisher in a letter from Harold Hotelling. In another paper published in 1929, Fisher had called them cumulative moment functions. The partition function in statistical physics was introduced by Josiah Willard Gibbs in 1901. 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.

[edit] 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

$1+\sum_{n=1}^\infty m_n t^n/n!=\exp\left(\sum_{n=1}^\infty\kappa_n t^n/n!\right)$

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.

[edit] 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. The Bell numbers are the moments of the Poisson distribution with expected value 1.

[edit] 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. For those polynomials, construct a polynomial sequence in the following way. Out the polynomial

$\begin{matrix}\mu'_6= & \kappa_6+6\kappa_5\kappa_1+15\kappa_4\kappa_2+15\kappa_4\kappa_1^2 +10\kappa_3^2+60\kappa_3\kappa_2\kappa_1 \\ \\ & {}+20\kappa_3\kappa_1^3+15\kappa_2^3 +45\kappa_2^2\kappa_1^2+15\kappa_2\kappa_1^4+\kappa_1^6\end{matrix}$

make a new polynomial in these plus one additional variable x:

$\begin{matrix}p_6(x)= & (\kappa_6)\,x+(6\kappa_5\kappa_1+15\kappa_4\kappa_2+10\kappa_3^2)\,x^2 +(15\kappa_4\kappa_1^2+60\kappa_3\kappa_2\kappa_1+15\kappa_2^3)\,x^3 \\ \\ & {}+(45\kappa_2^2\kappa_1^2)\,x^4+(15\kappa_2\kappa_1^4)\,x^5 +(\kappa_1^6)\,x^6\end{matrix}$

... and 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.

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.

[edit] Free cumulants

In the identity

$E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_i : i\in B)$

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. These play a central role in free probability theory. 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.

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. 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.

[edit] See also

[edit] 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)

[edit] External links

Eric W. Weisstein, Cumulant at MathWorld.
cumulant on the Earliest known uses of some of the words of mathematics

[0] Kendall, M.G., Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1 (3rd Edition). Griffin, London. (Section 3.12)

[1] Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Page 27)

[2] Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Section 2.4)

[3] Aapo Hyvarinen, Juha Karhunen, and Erkki Oja (2001) Independent Component Analysis, John Wiley & Sons. (Section 2.7.2)

[4] Lukacs, E. (1970) Characteristic Functions (2nd Edition), Griffin, London. (Theorem 7.3.5)

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