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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Pooled Kurtosis Estimator Using Pooled Cumulant Estimators

I am trying to come up with a statistically sensible pooled kurtosis estimator that is based on pooled cumulant estimators. Specifically, I have unbiased estimators of the second and fourth cumulant ...
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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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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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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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Fenchel conjugate of the cumulant function (Exponential Family)

We have a minimal exponential family $f(x) = h(x) \exp(\langle\theta, t(x)\rangle - A(\theta)).$ The canonical parameter space is $\Theta = \{ \theta \in \mathbb{R}^d \colon \int h(x) \exp(\langle\...
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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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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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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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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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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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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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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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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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How do continuous partial derivatives depend on $n$ in maximum likelihood estimation?

I'm reading Tensor Methods in Statistics by McCullagh 1987, (P209 for this question) and I can't understand one step he uses. He begins with the usual log-likelihood \begin{equation*} l(\theta; Y) =...
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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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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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