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== Cumulants of probability distributions ==
In
probability theory and
statistics, the '''cumulants''' κ
''n'' of a
probability distribution are given by
:<math>E\left(e^{tX}\right)=\exp\left(\sum_{n=1}^\infty\kappa_n t^n/n!\right)</math>
where ''X'' is any
random variable whose probability distribution is the one whose cumulants are taken. In other words, κ
''n''/''n''! is the ''n''th coefficient in the
power series representation of the
logarithm of the
moment-generating function. The logarithm of the moment-generating function is therefore called the '''cumulant-generating function'''.
The "problem of cumulants" attempts to recover a probability distribution from its sequence of cumulants. In some cases no solution exists; in some cases a unique solution exists; in some cases more than one solution exists.
==Some properties of cumulants==
===Invariance and equivariance===
The first cumulant is shift-equivariant; all of the others are shift-invariant. To state this less tersely, denote by κ
''n''(''X'') the ''n''th cumulant of the probability distribution of the random variable ''X''. The statement is that if ''c'' is constant then κ
1(''X'' + ''c'') = κ
1(''X'') + ''c'' and κ
''n''(''X'' + ''c'') = κ
''n''(''X'') for ''n''≥ 2, i.e., ''c'' is added to the first cumulant, but all higher cumulants are unchanged.
===Homogeneity===
The ''n''th cumulant is homogeneous of degree ''n'', i.e. if ''c'' is any constant, then
:<math>\kappa_n(cX)=c^n\kappa_n(X).</math>
===Additivity===
If ''X'' and ''Y'' are
independent random variables then κ
''n''(''X'' + ''Y'') = κ
''n''(''X'') + κ
''n''(''Y'').
===Cumulants and moments===
The cumulants are related to the
moments by the following recursion formula:
:<math>\kappa_n=\mu'_n-\sum_{k=1}^{n-1}{n-1 \choose k-1}\kappa_k \mu_{n-k}'.</math>
The ''n''th
moment μ′
''n'' is an ''n''th-degree polynomial in the first ''n'' cumulants, thus:
:<math>\mu'_1=\kappa_1</math>
:<math>\mu'_2=\kappa_2+\kappa_1^2</math>
:<math>\mu'_3=\kappa_3+3\kappa_2\kappa_1+\kappa_1^3</math>
:<math>\mu'_4
=\kappa_4+4\kappa_3\kappa_1+3\kappa_2^2+6\kappa_2\kappa_1^2+\kappa_1^4</math>
:<math>\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</math>
:<math>\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</math>
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.
The coefficients are precisely those that occur in
Faà di Bruno's formula.
===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
:<math>\mu'_n=\sum_{\pi}\prod_{B\in\pi}\kappa_{\left|B\right|}</math>
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
22 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.
==Cumulants of particular probability distributions==
The cumulants of the
normal distribution with
expected value μ and
variance σ
2 are κ
1 = μ, κ
2 = σ
2, and κ
''n'' = 0 for ''n'' > 2.
All of the cumulants of the
Poisson distribution are equal to the expected value.
A distribution with arbitrary given cumulants κ
''n'' can be approximated through the
Gram-Charlier or
Edgeworth series.
==Joint cumulants==
The '''joint cumulant''' of several random variables ''X''
1, ..., ''X''
''n'' is
:<math>\kappa(X_1,\dots,X_n)
=\sum_\pi\prod_{B\in\pi}(|B|-1)!(-1)^{|B|-1}E\left(\prod_{i\in B}X_i\right)</math>
where π runs through the list of all partitions of { 1, ..., ''n'' }, and ''B'' runs through the list of all block of the partition π. For example,
:<math>\kappa(X,Y,Z)=E(XYZ)-E(XY)E(Z)-E(XZ)E(Y)-E(YZ)E(X)+2E(X)E(Y)E(Z).\,</math>
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 idependent 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 ''n''th 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:
:<math>E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_B)</math>
where κ(''X''
''B'') is the joint cumulant of those among the random variables ''X''
1, ..., ''X''
''n'' whose indices are included in the block ''B''. For example:
:<math>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).</math>
===Conditional cumulants===
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
:<math>\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)).</math>
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
:<math>\kappa(X_1,\dots,X_n)=\sum_\pi \kappa(\kappa(X_{\pi_1}\mid Y),\dots,\kappa(X_{\pi_b}\mid Y))</math>
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''
π''k'') indicates that the joint cumulant of the random variables whose indices are in that block of the partition.
==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 1931 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''.
== "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
:<math>1+\sum_{n=1}^\infty m_n t^n/n!=\exp\left(\sum_{n=1}^\infty\kappa_n t^n/n!\right)</math>
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.
== One well-known example ==
In
combinatorics, the ''n''th
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.
== 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
:<math>\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}</math>
make a new polynomial in these plus one additional variable ''x'':
:<math>\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}</math>
... 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.
==Free cumulants==
In the identity
:<math>E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_B)</math>
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 Cartesion 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.
