En teoría de probabilidad y estadística , los acumulados κ n de una distribución de probabilidad son un conjunto de cantidades que proporcionan una alternativa a los momentos de la distribución. Los momentos determinan los acumulados en el sentido de que cualesquiera dos distribuciones de probabilidad cuyos momentos sean idénticos también tendrán acumulados idénticos y, de manera similar, los acumulados determinan los momentos.
El primer acumulado es la media , el segundo acumulativo es la varianza y el tercer acumulado es el mismo que el tercer momento central . Pero los acumulados de cuarto orden y de orden superior no son iguales a los momentos centrales. En algunos casos, los tratamientos teóricos de los problemas en términos de acumulaciones son más simples que los que utilizan momentos. En particular, cuando dos o más variables aleatorias son estadísticamente independientes , la n º -order cumulante de su suma es igual a la suma de su n º cumulantes -order. Además, los acumulados de tercer orden y de orden superior de una distribución normal son cero y es la única distribución con esta propiedad.
Al igual que para los momentos, donde los momentos conjuntos se utilizan para colecciones de variables aleatorias, es posible definir acumuladores conjuntos .
Definición
Los acumulados de una variable aleatoria X se definen utilizando la función generadora de acumuladores K ( t ) , que es el logaritmo natural de la función generadora de momentos :
Los acumulados κ n se obtienen a partir de una expansión en serie de potencias de la función generadora de acumulados:
Esta expansión es una serie de Maclaurin , por lo que el n -ésimo acumulativo se puede obtener diferenciando la expansión anterior n veces y evaluando el resultado en cero: [1]
Si la función generadora de momentos no existe, los acumulados se pueden definir en términos de la relación entre los acumulados y los momentos que se comentan más adelante.
Definición alternativa de la función de generación acumulada
Algunos escritores [2] [3] prefieren definir la función generadora acumulativa como el logaritmo natural de la función característica , que a veces también se denomina segunda función característica, [4] [5]
Una ventaja de H ( t ) —en cierto sentido la función K ( t ) evaluada para argumentos puramente imaginarios — es que E [ e itX ] está bien definida para todos los valores reales de t incluso cuando E [ e tX ] no está bien definida para todos los valores reales de t , como puede ocurrir cuando hay "demasiada" probabilidad de que X tenga una gran magnitud. Aunque la función H ( t ) estará bien definida, no obstante imitará a K ( t ) en términos de la longitud de su serie de Maclaurin , que puede no extenderse más allá (o, raramente, incluso) del orden lineal en el argumento t , y, en particular, el número de acumulados que están bien definidos no cambiará. Sin embargo, incluso cuando H ( t ) no tiene una serie de Maclaurin larga, se puede utilizar directamente para analizar y, en particular, agregar variables aleatorias. Tanto la distribución de Cauchy (también llamada Lorentzian) como, de manera más general, las distribuciones estables (relacionadas con la distribución de Lévy) son ejemplos de distribuciones para las cuales las expansiones en series de potencia de las funciones generadoras tienen solo un número finito de términos bien definidos.
Usos en estadística
Trabajar con acumulados puede tener una ventaja sobre el uso de momentos porque para las variables aleatorias X e Y estadísticamente independientes ,
de modo que cada acumulado de una suma de variables aleatorias independientes es la suma de los correspondientes acumulados de los sumandos . Es decir, cuando los sumandos son estadísticamente independientes, la media de la suma es la suma de las medias, la varianza de la suma es la suma de las varianzas, el tercer acumulado (que resulta ser el tercer momento central) de la suma es la suma de los terceros acumulados, y así sucesivamente para cada orden de acumulados.
Una distribución con acumulados dados κ n se puede aproximar a través de una serie de Edgeworth .
Acumulantes de algunas distribuciones de probabilidad discretas
- Las variables aleatorias constantes X = μ . La función de generación acumulada es K ( t ) = μt . El primer acumulante es κ 1 = K '(0) = μ y los otros acumulados son cero, κ 2 = κ 3 = κ 4 = ... = 0 .
- Las distribuciones de Bernoulli , (número de éxitos en un ensayo con probabilidad p de éxito). La función de generación acumulada es K ( t ) = log (1 - p + p e t ) . Los primeros cumulantes son κ 1 = K '(0) = p y κ 2 = K' ' (0) = p · (1 - p ) . Los acumulados satisfacen una fórmula de recursividad
- Las distribuciones geométricas , (número de fracasos antes de un éxito con probabilidad p de éxito en cada ensayo). La función de generación acumulada es K ( t ) = log ( p / (1 + ( p - 1) e t )) . Los primeros cumulantes son κ 1 = K ' (0) = p -1 - 1 , y κ 2 = K' ' (0) = κ 1 p -1 . Sustituyendo p = ( μ + 1) −1 da K ( t ) = −log (1 + μ (1 − e t )) y κ 1 = μ .
- Las distribuciones de Poisson . La función de generación acumulada es K ( t ) = μ (e t - 1) . Todos los acumulados son iguales al parámetro: κ 1 = κ 2 = κ 3 = ... = μ .
- Las distribuciones binomiales , (número de éxitos en n ensayos independientes con probabilidad p de éxito en cada ensayo). El caso especial n = 1 es una distribución de Bernoulli. Cada acumulado es solo n veces el acumulado correspondiente de la distribución de Bernoulli correspondiente. La función de generación acumulada es K ( t ) = n log (1 - p + p e t ) . Los primeros acumulados son κ 1 = K ′ (0) = np y κ 2 = K ′ ′ (0) = κ 1 (1 - p ) . Sustituyendo p = μ · n −1 da K '( t ) = ((μ −1 - n −1 ) · e - t + n −1 ) −1 y κ 1 = μ . El caso límite n −1 = 0 es una distribución de Poisson.
- Las distribuciones binomiales negativas , (número de fracasos antes de r éxitos con probabilidad p de éxito en cada ensayo). El caso especial r = 1 es una distribución geométrica. Cada acumulado es solo r veces el acumulado correspondiente de la distribución geométrica correspondiente. La derivada de la función generadora acumulativa es K '( t ) = r · ((1 - p ) −1 · e - t −1) −1 . Los primeros acumulados son κ 1 = K '(0) = r · ( p −1 −1), y κ 2 = K ' '(0) = κ 1 · p −1 . Sustituyendo p = (μ · r −1 +1) −1 da K ′ ( t ) = (( μ −1 + r −1 ) e - t - r −1 ) −1 y κ 1 = μ . La comparación de estas fórmulas con las de las distribuciones binomiales explica el nombre "distribución binomial negativa". El caso límite r −1 = 0 es una distribución de Poisson.
Presentamos la relación entre varianza y media
las distribuciones de probabilidad anteriores obtienen una fórmula unificada para la derivada de la función generadora acumulativa: [ cita requerida ]
La segunda derivada es
confirmando que el primer acumulante es κ 1 = K ′ (0) = μ y el segundo acumulante es κ 2 = K ′ ′ (0) = με . Las variables aleatorias constantes X = μ tienen ε = 0 . Las distribuciones binomiales tienen ε = 1 - p de modo que 0 < ε <1 . Las distribuciones de Poisson tienen ε = 1 . Las distribuciones binomiales negativas tienen ε = p −1 de modo que ε > 1 . Nótese la analogía con la clasificación de las secciones cónicas por excentricidad : círculos ε = 0 , elipses 0 < ε <1 , parábolas ε = 1 , hipérbolas ε > 1 .
Acumulantes de algunas distribuciones de probabilidad continuas
- Para la distribución normal con valor esperado μ y varianza σ 2 , la función de generación de cumulante es K ( t ) = μ t + σ 2 t 2 /2. Las derivadas primera y segunda de la función generadora de acumuladores son K '( t ) = μ + σ 2 · t y K "( t ) = σ 2. Los acumulados son κ 1 = μ, κ 2 = σ 2 y κ 3 = κ 4 = ... = 0. El caso especial σ 2 = 0 es una variable aleatoria constante X = μ.
- Los acumulados de la distribución uniforme en el intervalo [−1, 0] son κ n = B n / n , donde B n es el n -ésimo número de Bernoulli .
- Los acumulados de la distribución exponencial con parámetro λ son κ n = λ - n ( n - 1) !.
Algunas propiedades de la función generadora de acumuladores
La función generadora acumulativa K ( t ), si existe, es infinitamente diferenciable y convexa , y pasa por el origen. Su primera derivada varía monótonamente en el intervalo abierto desde el mínimo hasta el superior del soporte de la distribución de probabilidad, y su segunda derivada es estrictamente positiva en todos los lugares donde se define, excepto por la distribución degenerada de una masa puntual única. La función generadora de acumuladores existe si y solo si las colas de la distribución están mayorizadas por una disminución exponencial , es decir, ( ver la notación Big O )
dónde es la función de distribución acumulativa . La función generadora de acumuladores tendrá una asíntota (s) vertical (s) en el mínimo de dicha c , si existe dicho mínimo, y en el superior de dicha d , si existe dicho superior; de lo contrario, se definirá para todos los números reales.
Si el apoyo de una variable aleatoria X tiene límites superiores o inferiores finitos, entonces su función generadora de acumuladores y = K ( t ), si existe, se aproxima a la (s) asíntota (s) cuya pendiente es igual al supremo y / o al mínimo del apoyo,
respectively, lying above both these lines everywhere. (The integrals
yield the y-intercepts of these asymptotes, since K(0) = 0.)
For a shift of the distribution by c, For a degenerate point mass at c, the cgf is the straight line , and more generally, if and only if X and Y are independent and their cgfs 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 cgf and is its natural exponential family, then and
If K(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]
Algunas propiedades de los acumulantes
Invariance and equivariance
The first cumulant is shift-equivariant; all of the others are shift-invariant. This means that, if we denote by κn(X) the n-th cumulant of the probability distribution of the random variable X, then for any constant c:
In other words, shifting a random variable (adding c) shifts the first cumulant (the mean) and doesn't affect any of the others.
Homogeneity
The n-th cumulant is homogeneous of degree n, i.e. if c is any constant, then
Additivity
If X and Y are independent random variables then κn(X + Y) = κn(X) + κn(Y).
A negative result
Given the results for the cumulants of the normal distribution, it might be hoped to find families of distributions for which κm = κm+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
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; 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 n-th derivative of at ,
Likewise, the cumulants can be recovered in terms of moments by evaluating the n-th derivative of at ,
The explicit expression for the n-th 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
where are incomplete (or partial) Bell polynomials.
In the like manner, if the mean is given by , the central moment generating function is given by
and the n-th central moment is obtained in terms of cumulants as
Also, for n > 1, the n-th cumulant in terms of the central moments is
The n-th moment μ′n is an nth-degree polynomial in the first n cumulants. The first few expressions are:
The "prime" distinguishes the moments μ′n from the central moments μn. To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which κ1 appears as a factor:
Similarly, the n-th cumulant κn is an n-th-degree polynomial in the first n non-central moments. The first few expressions are:
To express the cumulants κn for n > 1 as functions of the central moments, drop from these polynomials all terms in which μ'1 appears as a factor:
To express the cumulants κn for n > 2 as functions of the standardized central moments, also set μ'2=1 in the polynomials:
The cumulants can be related to the moments by differentiating the relationship log M(t) = K(t) with respect to t, giving M′(t) = K′(t) M(t), which conveniently contains no exponentiation or logarithms. Equating the coefficient of t n−1 on the left and right sides, using μ′0 = 1, and rearranging gives the following recursion formula for n ≥ 1:[8]
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
- 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.
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.[9]
Acumulantes conjuntos
The joint cumulant of several random variables X1, ..., Xn is defined by a similar cumulant generating function
A consequence is that
where π runs through the list of all partitions of { 1, ..., n }, B runs through the list of all blocks of the partition π, and |π| is the number of parts in the partition. For example,
If any of these random variables are identical, e.g. if X = Y, 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 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:
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 n = 3, expressed in the language of (central) moments rather than that of cumulants, says
In general,[10]
where
- the sum is over all partitions π of the set { 1, ..., n } of indices, and
- π1, ..., πb are all of the "blocks" of the partition π; the expression κ(Xπm) indicates that the joint cumulant of the random variables whose indices are in that block of the partition.
Relación con la física estadística
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 T have a fluctuating internal energy E, which can be considered a random variable drawn from a distribution . The partition function of the system is
where β = 1/(kT) and k is Boltzmann's constant and the notation has been used rather than for the expectation value to avoid confusion with the energy, E. Hence the first and second cumulant for the energy E give the average energy and heat capacity.
The Helmholtz free energy expressed in terms of
further connects thermodynamic quantities with cumulant generating function for the energy. Thermodynamics properties that are derivatives of the free energy, such as its internal energy, entropy, and specific heat capacity, all can be readily expressed in terms of these cumulants. Other free energy can be a function of other variables such as the magnetic field or chemical potential , e.g.
where N is the number of particles and is the grand potential. Again the close relationship between the definition of the free energy and the cumulant generating function implies that various derivatives of this free energy can be written in terms of joint cumulants of E and N.
Historia
The history of cumulants is discussed by Anders Hald.[11][12]
Cumulants were first introduced by Thorvald N. Thiele, in 1889, who called them semi-invariants.[13] They were first called cumulants in a 1932 paper[14] by Ronald Fisher and John Wishart. Fisher was publicly reminded of Thiele's work by Neyman, who also notes previous published citations of Thiele brought to Fisher's attention.[15] Stephen Stigler has said[citation needed] that the name cumulant was suggested to Fisher in a letter from Harold Hotelling. In a paper published in 1929,[16] Fisher had called them cumulative moment functions. The partition function in statistical physics was introduced by Josiah Willard Gibbs in 1901.[citation needed] 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.[citation needed]
Acumulantes en entornos generalizados
Formal cumulants
More generally, the cumulants of a sequence { mn : n = 1, 2, 3, ... }, not necessarily the moments of any probability distribution, are, by definition,
where the values of κn 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.
Bell numbers
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 { κn : 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.[clarification needed][citation needed] For those polynomials, construct a polynomial sequence in the following way. Out of the polynomial
make a new polynomial in these plus one additional variable x:
and then 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.[citation needed]
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.[citation needed]
Free cumulants
In the above moment-cumulant formula
for joint cumulants, one sums over all partitions of the set { 1, ..., n }. If instead, one sums only over the noncrossing partitions, then, by solving these formulae for the in terms of the moments, one gets free cumulants rather than conventional cumulants treated above. These free cumulants were introduced by Roland Speicher and play a central role in free probability theory.[17][18] In that theory, rather than considering independence of random variables, defined in terms of tensor products of algebras of random variables, one considers instead free independence of random variables, defined in terms of free products of algebras.[18]
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.[18] 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.
Ver también
- Entropic value at risk
- Cumulant generating function from a multiset
- Cornish–Fisher expansion
- Edgeworth expansion
- Polykay
- k-statistic, a minimum-variance unbiased estimator of a cumulant
- Ursell function
- Total position spread tensor as an application of cumulants to analyse the electronic wave function in quantum chemistry.
Referencias
- ^ Weisstein, Eric W. "Cumulant". From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/Cumulant.html
- ^ 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)
- ^ Hamedani, G. G.; Volkmer, Hans; Behboodian, J. (2012-03-01). "A note on sub-independent random variables and a class of bivariate mixtures". Studia Scientiarum Mathematicarum Hungarica. 49 (1): 19–25. doi:10.1556/SScMath.2011.1183.
- ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition), Griffin, London. (Theorem 7.3.5)
- ^ Smith, Peter J. (May 1995). "A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa". The American Statistician. 49 (2): 217–218. doi:10.2307/2684642.
- ^ Rota, G.-C.; Shen, J. (2000). "On the Combinatorics of Cumulants". Journal of Combinatorial Theory. Series A. 91 (1–2): 283–304. doi:10.1006/jcta.1999.3017.
- ^ Brillinger, D.R. (1969). "The Calculation of Cumulants via Conditioning". Annals of the Institute of Statistical Mathematics. 21: 215–218. doi:10.1007/bf02532246.
- ^ Hald, A. (2000) "The early history of the cumulants and the Gram–Charlier series" International Statistical Review, 68 (2): 137–153. (Reprinted in Steffen L. Lauritzen, ed. (2002). Thiele: Pioneer in Statistics. Oxford U. P. ISBN 978-0-19-850972-1.)
- ^ Hald, Anders (1998). A History of Mathematical Statistics from 1750 to 1930. New York: Wiley. ISBN 978-0-471-17912-2.
- ^ H. Cramér (1946) Mathematical Methods of Statistics, Princeton University Press, Section 15.10, p. 186.
- ^ Fisher, R.A. , John Wishart, J.. (1932) 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 doi: 10.1112/plms/s2-33.1.195
- ^ Neyman, J. (1956): ‘Note on an Article by Sir Ronald Fisher,’ Journal of the Royal Statistical Society, Series B (Methodological), 18, pp. 288–94.
- ^ Fisher, R. A. (1929). "Moments and Product Moments of Sampling Distributions" (PDF). Proceedings of the London Mathematical Society. 30: 199–238. doi:10.1112/plms/s2-30.1.199. hdl:2440/15200.
- ^ Speicher, Roland (1994). "Multiplicative functions on the lattice of non-crossing partitions and free convolution". Mathematische Annalen. 298 (4): 611–628. doi:10.1007/BF01459754.
- ^ a b c Novak, Jonathan; Śniady, Piotr (2011). "What Is a Free Cumulant?". Notices of the American Mathematical Society. 58 (2): 300–301. ISSN 0002-9920.
enlaces externos
- Weisstein, Eric W. "Cumulant". MathWorld.
- cumulant on the Earliest known uses of some of the words of mathematics