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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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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306 views

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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257 views

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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381 views

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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384 views

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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382 views

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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110 views

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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133 views

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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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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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} \...