# 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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### By means of what distribution can I match the first n moments for arbitrary (i e. any) values of those moments?

Suppose I have the first n moments from some data set, either raw, centered or scaled, (or cumulants instead) whichever is more convenient for matching. Is there a continuous, continuously ...
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### How can I implement moment matching with kernel tricks if I do not have the complete distribution but only the higher-order moments or culuments?

As is said in Appendix B.3 of ref, "It is difficult to match high-order moments, because we have to deal with high order tensors directly. On the other hand, MMD can easily match high-order ...
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### Zero variance but non-zero skewness

I was thinking of a hypothetical distribution where the mean(first cumulant) is non-zero, second cumulant(variance) is zero, and the third cumulant(skewness) is non-zero. The higher order cumulants ...
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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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### 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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### 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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