Cumulants of probability distributions

In probability theory and statistics, the cumulants κ_n of the probability distribution of a random variable X are given by Probability theory is the mathematical study of probability. ... Statistics is a type of data analysis which practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

In other words, κ_n/n! is the nth 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. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1. ... In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...

In case some of the moments of the probability distribution of the random variable X are infinite, one must take t to be a pure imaginary number. In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number or zero. ...

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 nth cumulant of the probability distribution of the random variable X. The statement is that if c is constant then κ₁(X + c) = κ₁(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. In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ... An invariant in mathematics is something that does not change under a set of transformations. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...

Homogeneity

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

κ_n(cX) = cⁿκ_n(X).

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: See also moment (physics). ... In mathematics and computer science, recursion is a particular way of specifying (or constructing) a class of objects (or an object from a certain class) with the help of a reference to other objects of the class: a recursive definition defines objects in terms of the already defined objects of...

The nth moment μ′_n is an nth-degree polynomial in the first n cumulants, thus: See also moment (physics). ...

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. In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k], where E is the expectation operator. ...

The coefficients are precisely those that occur in Faà di Bruno's formula. // The formula Faà di Brunos formula is an identity in mathematics generalizing the chain rule to higher derivatives, named in honor of Francesco Faà di Bruno (1825–1888), who was (in chronological order) a military officer, a mathematician, and a priest, and was beatified by the Pope a century...

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 Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria. ... A partition of U into 6 blocks: a Venn diagram representation. ...

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. In mathematics, a monomial is a particular kind of polynomial, having just one term. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

Cumulants of particular probability distributions

• The cumulants of the normal distribution with expected value μ and variance σ² are κ₁ = μ, κ₂ = σ², and κ_n = 0 for n > 2.

• All of the cumulants of the Poisson distribution are equal to the expected value.

• 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 Bernoulli distribution with expectation p are given by a recursion formula:

A distribution with arbitrary given cumulants κ_n can be approximated through the Gram-Charlier or Edgeworth series. The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution (discovered by Siméon-Denis Poisson (1781–1840) and published, together with his probability theory, in 1838 in his work Recherches sur la probabilité des jugements en matières criminelles et matière civile [Research on the... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In mathematics, the uniform distributions are simple probability distributions. ... In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... The Edgeworth series or Gram-Charlier A series, named in honor of Francis Ysidro Edgeworth, are series that approximate a probability distribution in terms of its cumulants. ...

Joint cumulants

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

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,

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. In probability theory and statistics, the covariance between two real_valued random variables X and Y, with expected values and is defined as: where E is the expected value. ...

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 that:

κ(X + Y,Z₁,Z₂,...) = κ(X,Z₁,Z₂,...) + κ(Y,Z₁,Z₂,...)

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 Main article: cumulant In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total variance. ... The proposition in probability theory known as the law of total expectation, or the law of iterated expectations, or perhaps by any of a variety of other names, states that if X is an integrable random variable (i. ... In probability theory, the law of total variance states that if X and Y are random variables on the same probability space, and the variance of X is finite, then In language perhaps better known to statisticians than to probabilists, the first term is the unexplained component of the variance... See also moment (physics). ...

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

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

A partition of U into 6 blocks: a Venn diagram representation. ...

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. Actuaries are professionals who analyze the financial impact of risk, particularly looking ahead far into the future. ... A mathematician is a person whose area of study and research is mathematics. ... Thorvald Nicolai Thiele (December 24, 1838 – September 26, 1910) was a Danish astronomer, actuary, and mathematician, most notable for his work in statistics, interpolation, and the three-body problem. ... The London Mathematical Society (LMS) is one of the leading mathematical societies in the United Kingdom. ... Sir Ronald Fisher Sir Ronald Aylmer Fisher, FRS (February 17, 1890 – July 29, 1962) was an evolutionary biologist, geneticist and statistician. ... John Wishart (28 November 1898 – 14 July 1956) was a Scottish agricultural statistician. ... In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ... Stephen Mack Stigler is Ernest DeWitt Burton Distinguished Service Professor at the Department of Statistics of the University of Chicago. ... Harold Hotelling (Fulda, Minnesota, September 29, 1895 - December 26, 1973) was a mathematical statistician. ...

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

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 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. Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria. ... The Bell numbers, named in honor of Eric Temple Bell, are a sequence of integers arising in combinatorics that begins thus (sequence A000110 in OEIS): In general, Bn is the number of partitions of a set of size n. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical...

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 Scalar is a concept that has meaning in mathematics, physics, and computing. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ...

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

... 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. In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are given by the sum extending over all sequences j1, j2, j3, ..., jn−k+1 of positive integers such that Combinatorial meaning If the integer n is partitioned into a sum in which 1 appears j1 times... Eric Temple Bell (1883 - 1960) was a mathematician born in Scotland who lived in the USA from 1903 until his death. ...

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. Definition In mathematics, a polynomial sequence, i. ...

Free cumulants

In the identity

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. In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory free probability. ... Free probability is a mathematical theory which studies non-commutative random variables. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In abstract algebra, the free product of groups constructs a group from two or more given ones. ...

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. The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse...

External references

• Cumulant on Wolfram Research mathworld.