Questions tagged [cumulants]

The $n$th cumulant of a random variable $X$ is the $n$th derivative of the Taylor series expansion of $\log[E(e^{tX})]$ evaluated at zero.

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Does zero cumulants imply independence?

Question: Suppose we have two random variable $X$, $Y$ that follow non-Gaussian distribution, and we are given that: $$\operatorname { cum }(X, Y)=\operatorname { cum }(X, X, Y)=\operatorname { cum }(...
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Antiderivative of log-partition function of exponential family

Let $Y$ be a random variable with distribution belonging to a minimal regular exponential family. Let $\eta $ denote the scalar-valued canonical parameter of the exponential family and let $A$ be the ...
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Skewness and kurtosis for $\mathrm{MA}\left(\infty\right)$ model with non-gaussian noise

If an ARMA model formulation is written in infinite moving-average form: \begin{equation} X_t = C\left(B\right)\epsilon_t \quad \mbox{with} \quad C\left(B\right)=C_0+C_1B+C_2B^2 + \ldots \end{...
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Show that this probability distribution is an exponential family

We define a probability distribution on the non-negative integers 0,1,2,...,as having point probabilities $$P_\eta(k)=G(k,\rho)\left(\frac{1}{\rho+\eta}\right)^{\rho}\left(\frac{\eta}{\rho+ \eta}\...
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Skewness of the sum of dependent variables [closed]

I have two random variables $X$ and $Y$ which are both time series of observations. The two series are not independent. I want to find the skewness, kurtosis, hyper-skewness and hyper-kurtosis of $X+...
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asymptotic distribution of 3rd and 4th sample cumulants?

Suppose $X$ is distributed as standard normal, I take a sample of size $n$, and compute 3rd and 4th sample cumulants $\kappa_3$ and $\kappa_4$. I'm interested in the asymptotic distribution of $\kappa$...
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n'th cumulant (of a CGF) for exponential family / exponential dispersion model

The n'th cumulant is defined to be the n'th derivative of the CGF (cumulant generating function). $$\kappa_n = \frac{d^n K(t)}{dt^n} |_{t=0} $$ But I'm reading in a book (p.215, chapter5, eq. 5.8) ...
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Cumulant of sum of correlated random variables?

Let $X,Y$ be two random variables. We denote by $[X^k]$ and $[Y^k]$ the $k$'th order cumulants of $X$ and $Y$, respectively. I'm interested in computing the $k$'th order cumulant of $Z = X+Y$. If $X,Y$...
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Formulas for higher order cumulants

I want to calculate higher-order joint cumulants for 2 variables. I calculated the higher order single-variable and bivariate moments numerically. Now I need to combine them into cumulants (upto the ...
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What are the explicit formulas for the cumulants in terms of z-scores?

I'm trying to calculate the first few cumulants of a random variable using $Z$-scores. The situation Suppose we have a random variable $X$ with mean $\mu$ and standard deviation $\sigma$, and define ...
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Generating a 1-d sample with desired statistics

I'm interested in obtaining a sample of numbers $x_1,\ldots,x_n$ such that their cumulants approximately match user-provided set of cumulants. For instance, I can get a sample with first two ...
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Name of third cumulant?

The first cumulant is called the mean. The second is the variance. Does the third cumulant have a name? The fourth?
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Moments of an AR(1) Process

Definition of an AR(1) process In an Autoregressive Process, a time series can be generated based on a stochastic difference equation. \begin{align} X_t = c + \phi \, X_{t-1} + \epsilon \end{align} ...
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What is the cumulants of a whole data in terms of the cumulants of its parts?

I have around 8 billion data points, and I need to calculate the distribution and the cumulants of this distribution. However, due to technical restrictions, and time constraints, I can only ...
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Expectation Value of a Product of Many IID variables

First of all, I apologize for not being rigorous, but I am not a statistitian by background. Imagine you have $N$ i.i.d. positive random variables $X_1...X_N$ and you are trying to compute a ...
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Cumulants of Poisson random variable conditioned on a Bernoulli random variable

Consider a Bernoulli distributed random variable $Y$, which is 1 with probability $p$ and 0 with probability $1-p$. Further there is a random variable $X$ where the conditional probability ...
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Higher order moments of a multivariate Gaussian rv

Let $X~N_d(\mu,\Sigma)$ be a multivariate Gaussian random vector. Is there a convenient formula for each of $$ \mu_p\triangleq \mathbb{E}\left[\sum_{i=1}^d |X_i|^p\right], $$ in terms of $\mu$ and ...
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What is it called cumulative of every X previous days?

Let's say I have sales number of last 60 days. I can just draw a graph to see how sales has changed overtime. Numbers vary in different days, like on weekends sales numbers decrease. However, I ...
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Are the cumulants of sufficient statistics finite for the exponential family?

Under what conditions are the cumulants of the sufficient statistic finite for an exponential family? If we have $$ p(x \mid \theta) = \exp(\theta \cdot T(x) - A(\theta)) $$ then the derivatives of $...
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Relations between moments and cumulants

From the definition of KGF (cumulant generating function) we can write: \begin{align} K_x(t) &= \log_eM_x(t) \\ &= \log_e\left[1+\sum_{r=1}^{\infty}\frac{t^r}{r!}\mu_r^{'}\right] \\ &= ...
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Handling cumulative sum of variable in time series modelling

I would like to build a prediction model, based on the time series data, but all the features are a cumulative sum over the period. For example, the value of feature x is 50 at t = 1 (original value ...
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How can I get the cumulant expression from the recursive relation between cumulant and moment?

I am reading some paper about high-order statistics https://link.springer.com/article/10.1007%2Fs11004-009-9258-9?LI=true. The paper gives two recursive expressions relating the multivariate cumulants ...
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Moment generating function of the natural sufficient statistics of Gamma distribution

I have $X_1,X_2 ... X_n$ Gamma-distributed r.v with density: First of all I showed, that Gamma distribution belongs to exponential family and can be represented in form I found, that for Gamma ...
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Does exponential family of distributions have finite expected value?

I am curious about this question, because in definitions I have never seen this property. Is it true? If yes, why?
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Saddlepoint approximation with weibull distribution

I have some trouble with this computation, I have the moment generating function of a random variable $S$ by: $$M_S(t)=\frac{\beta\mu t}{1+(1+\beta)\mu t-M_X(t)}$$ According to the text that I am ...
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Skewness of Tweedie distribution

Tweedie distributions are a family of distributions from the exponential dispersion family that have power-law mean-variance relationship: \begin{align} \mathbb E[X] &= \mu \\ \operatorname{Var}[...
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Expansion of Cumulant Generating Function of Negbin

Let $X \thicksim Negbin(r,p)$ where $(0\lt p \lt 1) $ I want to derive skewness and kurtosis of $X$ by getting the Cgf of X. First, since Followance of Negative Binomial equals to the distribution ...
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Variance of k-statistic in large-n limit

k-statistics are unbiased estimators of cumulants. In the large-n limit, the formula for the $r$th k-statistic is just the formula for the $r$th cumulant in terms of central moments. But what, in the ...
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Proof of recurrence between cumulants and central-moments

According to Wikipedia, the $n$th cumulant $\kappa_n$ is related to the central-moments $\theta_n$ by the following recurrence: $$\kappa_n = \theta_n - \sum_{m=1}^{n-1} \binom{n-1}{m-1} \kappa_m \...
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2 answers
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Obtain probability density function from its cumulants?

Let $\sigma_1, \sigma_2, \dots, $ be the sequence of cumulants of a probability density function $p(x)$. How can we reconstruct $p(x)$ from its cumulants? P.S. If it helps, you can assume that $p(x)$ ...
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Joint cumulants of Zn2 characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \...
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Moment Generating Functions and Fourier Transforms?

Is a moment-generating function a Fourier transform of a probability density function? In other words, is a moment generating function just the spectral resolution of a probability density ...
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Relative influences of terms in the law of total cumulance

I posted this to mathoverflow and no one's saying anything. $\newcommand{\cum}{\operatorname{cum}}$ Denote the joint cumulant of several random variables by $\cum(A,B,C,\ldots)$ (more precisely, the ...
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5 votes
1 answer
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Difference between cumulants and moments

In particular, is the $n$th cumulant equivalent to the $n$th central moment (i.e. about the mean)? There's little difference I can see between MGFs (moment generating) and CGFs (cumulant generating), ...
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1 answer
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Distribution with $n$th cumulant given by $\frac 1 n$?

Is there any information out there about the distribution whose $n$th cumulant is given by $\frac 1 n$? The cumulant-generating function is of the form $$ \kappa(t) = \int_0 ^ 1 \frac{e^{tx} - 1}{x} \...
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