Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

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

33 questions
16
votes
1 answer

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 $$ \kappa(t) = \int_0 ^ 1 \frac{e^{tx} - 1}{x} \ dx. $$ I've run across it as the limiting…
guy
  • 7,737
  • 1
  • 26
  • 50
11
votes
1 answer

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 distribution of a random variable, i.e. an equivalent way to…
bolbteppa
  • 210
  • 2
  • 5
6
votes
1 answer

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}[X]&=\phi \mu^p \end{align} What is the formula for…
amoeba
  • 93,463
  • 28
  • 275
  • 317
5
votes
1 answer

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), apart from the former gives moments about the…
mchen
  • 710
  • 1
  • 6
  • 15
5
votes
1 answer

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?
vitsuk
  • 67
  • 5
5
votes
2 answers

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: $$\kappa_n = \theta_n - \sum_{m=1}^{n-1} \binom{n-1}{m-1} \kappa_m \theta_{n-m}$$ which allows one to compute the…
becko
  • 3,298
  • 1
  • 19
  • 36
4
votes
1 answer

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{equation} where $\epsilon_t$ is zero-mean white noise…
hydrologist
  • 722
  • 1
  • 5
  • 20
4
votes
1 answer

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 6th order cumulant, eg k_{3,3}) However,…
DankMasterDan
  • 1,188
  • 1
  • 10
  • 23
3
votes
0 answers

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 cumulants 0, 1 by sampling uniformly, then normalizing…
Yaroslav Bulatov
  • 5,167
  • 2
  • 24
  • 38
3
votes
1 answer

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} Typically, $\epsilon$ is chosen to be normally…
LBogaardt
  • 494
  • 1
  • 4
  • 18
3
votes
1 answer

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 distribution of $X$ on $Y$ is given by a Poisson random…
Trilarion
  • 260
  • 2
  • 13
3
votes
1 answer

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] \\ &= k_1t+k_2\frac{t^2}{2!}+\cdots+k_r\frac{t^r}{r!}+\cdots \\ &=…
emonhossain
  • 155
  • 7
3
votes
0 answers

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 to moments, as shown in first and second…
emberbillow
  • 417
  • 3
  • 10
2
votes
0 answers

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 $\kappa$'s as $n\to \infty$ I'm assuming…
Yaroslav Bulatov
  • 5,167
  • 2
  • 24
  • 38
2
votes
1 answer

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 $X,Y$ were independent, then a well-known property…
becko
  • 3,298
  • 1
  • 19
  • 36
1
2 3