Contents

Cumulant

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 nth-order cumulant of their sum is equal to the sum of their nth-order cumulants. As well, the third and higher-order cumulants of a normal distribution 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[edit]

The cumulants of a random variable X are defined using the cumulant-generating function K(t), which is the natural logarithm of the moment-generating function: $${\displaystyle K($$t)=\log \operatorname {E} \left[e^{tX}\right].}

The cumulants κ_n are obtained from a power series expansion of the cumulant generating function: $${\displaystyle K($$t)=\sum _{n=1}^{\infty }\kappa _{n}{\frac {t^{n}}{n!}}=\kappa _{1}{\frac {t}{1!}}+\kappa _{2}{\frac {t^{2}}{2!}}+\kappa _{3}{\frac {t^{3}}{3!}}+\cdots =\mu t+\sigma ^{2}{\frac {t^{2}}{2}}+\cdots .}

This expansion is a Maclaurin series, so the nth cumulant can be obtained by differentiating the above expansion n times and evaluating the result at zero:^[1] $${\displaystyle \kappa _{n}=K^{($$n)}(0).}

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[edit]

Some writers^[2][3] prefer to define the cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function,^[4][5] $${\displaystyle H($$t)=\log \operatorname {E} \left[e^{itX}\right]=\sum _{n=1}^{\infty }\kappa _{n}{\frac {(it)^{n}}{n!}}=\mu it-\sigma ^{2}{\frac {t^{2}}{2}}+\cdots }

An advantage of H(t)—in some sense the function K(t) evaluated for purely imaginary arguments—is that E[e^itX] is 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 the function H(t) will be well defined, it will nonetheless mimic K(t) in terms of the length of its Maclaurin series, which may not extend beyond (or, rarely, even to) linear order in the argument t, and in particular the number of cumulants that are well defined will not change. 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 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[edit]

The ${\textstyle n}$th cumulant ${\textstyle \kappa _{n}($X)} of (the distribution of) a random variable ${\textstyle X}$ enjoys the following properties:

• If${\textstyle n>1}$ and ${\textstyle c}$ is constant (i.e. not random) then ${\textstyle \kappa _{n}(X+c)=\kappa _{n}($X),} i.e. the cumulant is translation invariant. (If${\textstyle n=1}$ then we have ${\textstyle \kappa _{1}(X+c)=\kappa _{1}($X)+c.)}
• If${\textstyle c}$ is constant (i.e. not random) then ${\textstyle \kappa _{n}($cX)=c^{n}\kappa _{n}(X),} i.e. the ${\textstyle n}$th cumulant is homogeneous of degree ${\textstyle n}$.
• If random variables ${\textstyle X_{1},\ldots ,X_{m}}$ are independent then $${\displaystyle \kappa _{n}(X_{1}+\cdots +X_{m})=\kappa _{n}(X_{1})+\cdots +\kappa _{n}(X_{m})\,.}$$ That is, the cumulant is cumulative — hence the name.

The cumulative property follows quickly by considering the cumulant-generating function: $${\displaystyle {\begin{aligned}K_{X_{1}+\cdots +X_{m}}($$t)&=\log \operatorname {E} \left[e^{t(X_{1}+\cdots +X_{m})}\right]\\[5pt]&=\log \left(\operatorname {E} \left[e^{tX_{1}}\right]\cdots \operatorname {E} \left[e^{tX_{m}}\right]\right)\\[5pt]&=\log \operatorname {E} \left[e^{tX_{1}}\right]+\cdots +\log \operatorname {E} \left[e^{tX_{m}}\right]\\[5pt]&=K_{X_{1}}(t)+\cdots +K_{X_{m}}(t),\end{aligned}}} 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 κ_n can be approximated through an Edgeworth series.

First several cumulants as functions of the moments[edit]

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.

• ${\textstyle \kappa _{1}($X)=\operatorname {E} (X)={}} mean
• ${\textstyle \kappa _{2}($X)=\operatorname {var} (X)=\operatorname {E} {\big (}(X-\operatorname {E} (X))^{2}{\big )}={}} the variance, or second central moment.
• ${\textstyle \kappa _{3}($X)=\operatorname {E} {\big (}(X-\operatorname {E} (X))^{3}{\big )}={}} the third central moment.
• ${\textstyle \kappa _{4}($X)=\operatorname {E} {\big (}(X-\operatorname {E} (X))^{4}{\big )}-3\left(\operatorname {E} {\big (}(X-\operatorname {E} (X))^{2}{\big )}\right)^{2}={}} 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.
• ${\textstyle \kappa _{5}($X)=\operatorname {E} {\big (}(X-\operatorname {E} (X))^{5}{\big )}-10\operatorname {E} {\big (}(X-\operatorname {E} (X))^{3}{\big )}\operatorname {E} {\big (}(X-\operatorname {E} (X))^{2}{\big )}.}

Cumulants of some discrete probability distributions[edit]

• The constant random variables X = μ. The cumulant generating function is K(t) = μt. The first cumulant is κ₁ = K′(0) = μ and the other cumulants are zero, κ₂ = κ₃ = κ₄ = ⋅⋅⋅ = 0.
• The Bernoulli distributions, (number of successes in one trial with probability p of success). The cumulant generating function is K(t) = log(1 − p + pe^t). The first cumulants are κ₁ = K '(0) = p and κ₂ = K′′(0) = p·(1 − p). The cumulants satisfy a recursion formula $${\displaystyle \kappa _{n+1}=p(1-p){\frac {d\kappa _{n}}{dp}}.}$$
• The geometric distributions, (number of failures before one success with probability p of success on each trial). The cumulant generating function is K(t) = log(p / (1 + (p − 1)e^t)). The first cumulants are κ₁ = K′(0) = p⁻¹ − 1, and κ₂ = K′′(0) = κ₁p⁻¹. Substituting p = (μ + 1)⁻¹ gives K(t) = −log(1 + μ(1−e^t)) and κ₁ = μ.
• The Poisson distributions. The cumulant generating function is K(t) = μ(e^t − 1). 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 cumulant generating function is K(t) = n log(1 − p + pe^t). The first cumulants are κ₁ = K′(0) = np and κ₂ = K′′(0) = κ₁(1 − p). Substituting p = μ·n⁻¹ gives K '(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 r successes with probability p of success on each trial). The special case r = 1 is a geometric distribution. Every cumulant is just r times the corresponding cumulant of the corresponding geometric distribution. The derivative of the cumulant generating function is K′(t) = r·((1 − p)⁻¹·e^−t−1)⁻¹. The first cumulants are κ₁ = K′(0) = r·(p⁻¹−1), and κ₂ = K′′(0) = κ₁·p⁻¹. Substituting p = (μ·r⁻¹+1)⁻¹ gives K′(t) = ((μ⁻¹ + r⁻¹)e^−t − r⁻¹)⁻¹ and κ₁ = μ. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'. The limiting case r⁻¹ = 0 is a Poisson distribution.

Introducing the variance-to-mean ratio $${\displaystyle \varepsilon =\mu ^{-1}\sigma ^{2}=\kappa _{1}^{-1}\kappa _{2},}$$ the above probability distributions get a unified formula for the derivative of the cumulant generating function:^{[citation needed]} $${\displaystyle K'($$t)=(1+(e^{-t}-1)\varepsilon )^{-1}\mu }

The second derivative is $${\displaystyle K''($$t)=(\varepsilon -(\varepsilon -1)e^{t})^{-2}\mu \varepsilon e^{t}} confirming that the first cumulant is κ₁ = K′(0) = μ and the second cumulant is κ₂ = K′′(0) = με.

The constant random variables X = μ have ε = 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 sectionsbyeccentricity: circles ε = 0, ellipses 0 < ε <1, parabolas ε = 1, hyperbolas ε >1.

Cumulants of some continuous probability distributions[edit]

• For the normal distribution with expected value μ and variance σ², the cumulant generating function is K(t) = μt + σ²t²/2. The first and second derivatives of the cumulant generating function are K′(t) = μ + σ²·t and K′′(t) = σ². 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 nthBernoulli number.
• The cumulants of the exponential distribution with rate parameter λ are κ_n = λ⁻ⁿ (n − 1)!.

Some properties of the cumulant generating function[edit]

The cumulant generating function K(t), 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 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, that is, (see Big O notation) $${\displaystyle {\begin{aligned}&\exists c>0,\,\,F($$x)=O(e^{cx}),x\to -\infty ;{\text{ and}}\\[4pt]&\exists d>0,\,\,1-F(x)=O(e^{-dx}),x\to +\infty ;\end{aligned}}} where ${\textstyle F}$ is the cumulative distribution function. The cumulant-generating function will have vertical asymptote(s) at the negative supremum of such c, if such a supremum exists, and at the supremum of such d, if such a supremum exists, otherwise it will be defined for all real numbers.

If the support of a random variable X has finite upper or lower bounds, then its cumulant-generating function y = K(t), if it exists, approaches asymptote(s) whose slope is equal to the supremum or infimum of the support, $${\displaystyle {\begin{aligned}y&=(t+1)\inf \operatorname {supp} X-\mu ($$X),{\text{ and}}\\[5pt]y&=(t-1)\sup \operatorname {supp} X+\mu (X),\end{aligned}}} respectively, lying above both these lines everywhere. (The integrals $${\displaystyle \int _{-\infty }^{0}\left[t\inf \operatorname {supp} X-K'($$t)\right]\,dt,\qquad \int _{\infty }^{0}\left[t\inf \operatorname {supp} X-K'(t)\right]\,dt} yield the y-intercepts of these asymptotes, since K(0) = 0.)

For a shift of the distribution by c, ${\textstyle K_{X+c}($t)=K_{X}(t)+ct.} For a degenerate point mass at c, the cumulant generating function is the straight line ${\textstyle K_{c}($t)=ct} , and more generally, ${\textstyle K_{X+Y}=K_{X}+K_{Y}}$ if and only if X and Y are independent and their cumulant generating functions exist; (subindependence and the existence of second moments sufficing to imply independence.^[6])

The natural exponential family of a distribution may be realized by shifting or translating K(t), and adjusting it vertically so that it always passes through the origin: if f is the pdf with cumulant generating function ${\textstyle K($t)=\log M(t),} and ${\textstyle f|\theta }$ is its natural exponential family, then ${\textstyle f(x\mid \theta )={\frac {1}{M(\theta )}}e^{\theta x}f($x),} and ${\textstyle K(t\mid \theta )=K(t+\theta )-K(\theta ).}$

IfK(t) is finite for a range t₁ < Re(t) < t₂ then if t₁ < 0 < t₂ then K(t) is analytic and infinitely differentiable for t₁ < Re(t) < t₂. Moreover for t real and t₁ < t < t₂ K(t) is strictly convex, and K′(t) is strictly increasing. ^{[citation needed]}

Further properties of cumulants[edit]

A negative result[edit]

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.^[7] The underlying result here is that the cumulant generating function cannot be a finite-order polynomial of degree greater than 2.

Cumulants and moments[edit]

The moment generating function is given by: $${\displaystyle M($$t)=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(K(t)).}

So the cumulant generating function is the logarithm of the moment generating function $${\displaystyle K($$t)=\log M(t).}

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 moments can be recovered in terms of cumulants by evaluating the nth derivative of ${\textstyle \exp(K($t))} at${\displaystyle t=0}$, $${\displaystyle \mu '_{n}=M^{($$n)}(0)=\left.{\frac {\mathrm {d} ^{n}\exp(K(t))}{\mathrm {d} t^{n}}}\right|_{t=0}.}

Likewise, the cumulants can be recovered in terms of moments by evaluating the nth derivative of ${\textstyle \log M($t)} at${\displaystyle t=0}$, $${\displaystyle \kappa _{n}=K^{($$n)}(0)=\left.{\frac {\mathrm {d} ^{n}\log M(t)}{\mathrm {d} t^{n}}}\right|_{t=0}.}

The explicit expression for the nth moment in terms of the first n cumulants, and vice versa, can be obtained by using Faà di Bruno's formula for higher derivatives of composite functions. In general, we have $${\displaystyle \mu '_{n}=\sum _{k=1}^{n}B_{n,k}(\kappa _{1},\ldots ,\kappa _{n-k+1})}$$ $${\displaystyle \kappa _{n}=\sum _{k=1}^{n}(-1)^{k-1}(k-1)!B_{n,k}(\mu '_{1},\ldots ,\mu '_{n-k+1}),}$$ where ${\textstyle B_{n,k}}$ are incomplete (or partial) Bell polynomials.

In the like manner, if the mean is given by ${\textstyle \mu }$, the central moment generating function is given by $${\displaystyle C($$t)=\operatorname {E} [e^{t(x-\mu )}]=e^{-\mu t}M(t)=\exp(K(t)-\mu t),} and the nth central moment is obtained in terms of cumulants as $${\displaystyle \mu _{n}=C^{($$n)}(0)=\left.{\frac {\mathrm {d} ^{n}}{\mathrm {d} t^{n}}}\exp(K(t)-\mu t)\right|_{t=0}=\sum _{k=1}^{n}B_{n,k}(0,\kappa _{2},\ldots ,\kappa _{n-k+1}).}

Also, for n >1, the nth cumulant in terms of the central moments is $${\displaystyle {\begin{aligned}\kappa _{n}&=K^{($$n)}(0)=\left.{\frac {\mathrm {d} ^{n}}{\mathrm {d} t^{n}}}(\log C(t)+\mu t)\right|_{t=0}\\[4pt]&=\sum _{k=1}^{n}(-1)^{k-1}(k-1)!B_{n,k}(0,\mu _{2},\ldots ,\mu _{n-k+1}).\end{aligned}}}

The nthmoment μ′_n is an nth-degree polynomial in the first n cumulants. The first few expressions are:

$${\displaystyle {\begin{aligned}\mu '_{1}={}&\kappa _{1}\\[5pt]\mu '_{2}={}&\kappa _{2}+\kappa _{1}^{2}\\[5pt]\mu '_{3}={}&\kappa _{3}+3\kappa _{2}\kappa _{1}+\kappa _{1}^{3}\\[5pt]\mu '_{4}={}&\kappa _{4}+4\kappa _{3}\kappa _{1}+3\kappa _{2}^{2}+6\kappa _{2}\kappa _{1}^{2}+\kappa _{1}^{4}\\[5pt]\mu '_{5}={}&\kappa _{5}+5\kappa _{4}\kappa _{1}+10\kappa _{3}\kappa _{2}+10\kappa _{3}\kappa _{1}^{2}+15\kappa _{2}^{2}\kappa _{1}+10\kappa _{2}\kappa _{1}^{3}+\kappa _{1}^{5}\\[5pt]\mu '_{6}={}&\kappa _{6}+6\kappa _{5}\kappa _{1}+15\kappa _{4}\kappa _{2}+15\kappa _{4}\kappa _{1}^{2}+10\kappa _{3}^{2}+60\kappa _{3}\kappa _{2}\kappa _{1}+20\kappa _{3}\kappa _{1}^{3}\\&{}+15\kappa _{2}^{3}+45\kappa _{2}^{2}\kappa _{1}^{2}+15\kappa _{2}\kappa _{1}^{4}+\kappa _{1}^{6}.\end{aligned}}}$$

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: $${\displaystyle {\begin{aligned}\mu _{1}&=0\\[4pt]\mu _{2}&=\kappa _{2}\\[4pt]\mu _{3}&=\kappa _{3}\\[4pt]\mu _{4}&=\kappa _{4}+3\kappa _{2}^{2}\\[4pt]\mu _{5}&=\kappa _{5}+10\kappa _{3}\kappa _{2}\\[4pt]\mu _{6}&=\kappa _{6}+15\kappa _{4}\kappa _{2}+10\kappa _{3}^{2}+15\kappa _{2}^{3}.\end{aligned}}}$$

Similarly, the nth cumulant κ_n is an nth-degree polynomial in the first n non-central moments. The first few expressions are: $${\displaystyle {\begin{aligned}\kappa _{1}={}&\mu '_{1}\\[4pt]\kappa _{2}={}&\mu '_{2}-{\mu '_{1}}^{2}\\[4pt]\kappa _{3}={}&\mu '_{3}-3\mu '_{2}\mu '_{1}+2{\mu '_{1}}^{3}\\[4pt]\kappa _{4}={}&\mu '_{4}-4\mu '_{3}\mu '_{1}-3{\mu '_{2}}^{2}+12\mu '_{2}{\mu '_{1}}^{2}-6{\mu '_{1}}^{4}\\[4pt]\kappa _{5}={}&\mu '_{5}-5\mu '_{4}\mu '_{1}-10\mu '_{3}\mu '_{2}+20\mu '_{3}{\mu '_{1}}^{2}+30{\mu '_{2}}^{2}\mu '_{1}-60\mu '_{2}{\mu '_{1}}^{3}+24{\mu '_{1}}^{5}\\[4pt]\kappa _{6}={}&\mu '_{6}-6\mu '_{5}\mu '_{1}-15\mu '_{4}\mu '_{2}+30\mu '_{4}{\mu '_{1}}^{2}-10{\mu '_{3}}^{2}+120\mu '_{3}\mu '_{2}\mu '_{1}\\&{}-120\mu '_{3}{\mu '_{1}}^{3}+30{\mu '_{2}}^{3}-270{\mu '_{2}}^{2}{\mu '_{1}}^{2}+360\mu '_{2}{\mu '_{1}}^{4}-120{\mu '_{1}}^{6}\,.\end{aligned}}}$$

In general,^[8] the cumulant is the determinant of a matrix: $${\displaystyle \kappa _{l}=(-1)^{l+1}\left|{\begin{array}{cccccccc}\mu '_{1}&1&0&0&0&0&\ldots &0\\\mu '_{2}&\mu '_{1}&1&0&0&0&\ldots &0\\\mu '_{3}&\mu '_{2}&\left({\begin{array}{l}2\\1\end{array}}\right)\mu '_{1}&1&0&0&\ldots &0\\\mu '_{4}&\mu '_{3}&\left({\begin{array}{l}3\\1\end{array}}\right)\mu '_{2}&\left({\begin{array}{l}3\\2\end{array}}\right)\mu '_{1}&1&0&\ldots &0\\\mu '_{5}&\mu '_{4}&\left({\begin{array}{l}4\\1\end{array}}\right)\mu '_{3}&\left({\begin{array}{l}4\\2\end{array}}\right)\mu '_{2}&\left({\begin{array}{c}4\\3\end{array}}\right)\mu '_{1}&1&\ldots &0\\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots &\ddots &\vdots \\\mu '_{l-1}&\mu '_{l-2}&\ldots &\ldots &\ldots &\ldots &\ddots &1\\\mu '_{l}&\mu '_{l-1}&\ldots &\ldots &\ldots &\ldots &\ldots &\left({\begin{array}{l}l-1\\l-2\end{array}}\right)\mu '_{1}\end{array}}\right|}$$