In
probability theory and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the cumulants of a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
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
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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 (or, rarely, even to) 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 (also called the Lorentzian) and more generally,
stable distributions (related to the Lévy distribution) 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 (the distribution of) a random variable
enjoys the following properties:
* If
and
is constant (i.e. not random) then
i.e. the cumulant is translation-invariant. (If
then we have
* If
is constant (i.e. not random) then
i.e. the
-th cumulant is homogeneous of degree
.
* If random variables
are independent then
::
: i.e. 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 (which happens to be the third central moment) 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 The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The se ...
.
The 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.
*
mean
*
the variance, or second central moment.
*
the third central moment.
*
the fourth central moment minus three times the square of the second central moment. Thus this is the first case in which cumulants are not simply moments or central moments. The central moments of degree more than 3 lack the cumulative property.
*
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, (number of successes in one trial with probability of success). The cumulant generating function is . The first cumulants are and . The cumulants satisfy a recursion formula
::
* The
geometric distributions, (number of failures before one success with probability of success on each trial). 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 distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
s, (number of successes in
independent trials with probability of success on each trial). 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 distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-r ...
s, (number of failures before successes with probability of success on each trial). 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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...
by
eccentricity: circles , ellipses , parabolas , hyperbolas .
Cumulants of some continuous probability distributions
* For the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
and
variance , the cumulant generating function is . The first and second derivatives of the cumulant generating function are and . The cumulants are , , and . The special case is a constant random variable .
* The cumulants of the
uniform distribution
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
on the interval are , where is the
th Bernoulli number.
* The cumulants of the
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
with parameter are .
Some properties of the cumulant generating function
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
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
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
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
, that is, (''see
Big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
'')
:
where
is the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
. The cumulant-generating function will have
vertical asymptote(s) at the negative
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of such , if such a supremum exists, and at the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of such , if such a supremum exists, otherwise it will be defined for all real numbers.
If the
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
of a random variable has finite upper or lower bounds, then its cumulant-generating function , if it exists, approaches
asymptote(s) whose slope is equal to the supremum and/or infimum of the support,
:
respectively, lying above both these lines everywhere. (The
integrals
:
yield the
-intercepts of these asymptotes, since .)
For a shift of the distribution by ,
For a degenerate point mass at , the cgf is the straight line
, and more generally,
if and only if and are independent and their cgfs exist; (
subindependence In probability theory and statistics, subindependence is a weak form of independence.
Two random variables ''X'' and ''Y'' are said to be subindependent if the characteristic function of their sum is equal to the product of their marginal character ...
and the existence of second moments sufficing to imply independence.)
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 cgf
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
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, it might be hoped to find families of distributions for which
for some , with the lower-order cumulants (orders 3 to ) 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
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
; 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 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 (or partial)
Bell polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's fo ...
.
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
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
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:
:
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:
:
:
:
:
:
To express the cumulants for as functions of the
standardized central moments , also set in the polynomials:
:
:
:
:
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
.
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
:
where
* runs through the list of all partitions of a set of size ;
*"" means is one of the "blocks" into which the set is partitioned; and
* is the size of the set .
Thus each
monomial is a constant times a product of cumulants in which the sum of the indices is (e.g., in the term , 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 corresponds to each term. The ''coefficient'' in each term is the number of partitions of a set of members that collapse to that partition of the integer when the members of the set become indistinguishable.
Cumulants and combinatorics
Further connection between cumulants and combinatorics can be found in the work of
Gian-Carlo Rota, where links to
invariant theory,
symmetric functions, and binomial sequences are studied via
umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...
.
Joint cumulants
The joint cumulant of several random variables is defined by a similar cumulant generating function
:
A consequence is that
:
where runs through the list of all partitions of , runs through the list of all blocks of the partition , and is the number of parts in the partition. For example,
:
is the
covariance, and
:
If any of these random variables are identical, e.g. if , then the same formulae apply, e.g.
:
although for such repeated variables there are more concise formulae. For zero-mean 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 random variables are the same, then the joint cumulant is the -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:
:
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
The
law of total expectation and the
law of total variance generalize naturally to conditional cumulants. The case , expressed in the language of (central)
moments rather than that of cumulants, says
:
In general,
:
where
* the sum is over all
partitions
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of the set of indices, and
*
1, ...,
b are all of the "blocks" of the partition ; the expression 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 have a fluctuating internal energy , which can be considered a random variable drawn from a distribution
. The
partition function of the system is
:
where
= and is
Boltzmann's constant and the notation
has been used rather than