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.

Filter by
Sorted by
Tagged with
2 votes
1 answer
99 views

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 ...
Abhinav Tahlani's user avatar
0 votes
0 answers
49 views

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) =...
Nick Green's user avatar
3 votes
0 answers
115 views

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 ...
Hiro's user avatar
  • 323
2 votes
0 answers
50 views

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\...
Phil's user avatar
  • 616
0 votes
0 answers
79 views

How to call $\mathrm{Cov}(X,Y,Z)$? [duplicate]

Let $$\mathrm{Cov}(X,Y,Z) = \mathrm E[(X-\mathrm E(X))(Y-\mathrm E(Y))(Z-\mathrm E(Z))]$$ This can be regarded as a generalization of covariance to three random variables. I am writing a text where I ...
a06e's user avatar
  • 4,420
1 vote
0 answers
99 views

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 }(...
graphitump's user avatar
0 votes
0 answers
140 views

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 ...
Student's user avatar
  • 221
4 votes
1 answer
194 views

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{...
hydrologist's user avatar
1 vote
1 answer
599 views

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}\...
Lifeni's user avatar
  • 305
1 vote
0 answers
115 views

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+...
hydrologist's user avatar
2 votes
0 answers
81 views

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 $\kappa$...
Yaroslav Bulatov's user avatar
1 vote
1 answer
1k views

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) ...
Maverick Meerkat's user avatar
2 votes
1 answer
472 views

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 $X,Y$...
a06e's user avatar
  • 4,420
6 votes
2 answers
1k views

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 ...
DankMasterDan's user avatar
1 vote
0 answers
63 views

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 ...
Tanner Swett's user avatar
3 votes
0 answers
37 views

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 ...
Yaroslav Bulatov's user avatar
2 votes
2 answers
1k views

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?
a06e's user avatar
  • 4,420
4 votes
1 answer
2k views

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} ...
LBogaardt's user avatar
  • 582
0 votes
1 answer
90 views

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 ...
Our's user avatar
  • 237
2 votes
1 answer
400 views

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 ...
Godzilla's user avatar
  • 133
3 votes
1 answer
296 views

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 ...
NoDataDumpNoContribution's user avatar
2 votes
1 answer
2k views

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 ...
ABIM's user avatar
  • 544
1 vote
0 answers
15 views

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 ...
SNaRe's user avatar
  • 111
2 votes
0 answers
177 views

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 $...
Justin MacCallum's user avatar
3 votes
1 answer
3k 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] \\ &= ...
emonhossain's user avatar
3 votes
1 answer
9k views

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 ...
PreeJackie's user avatar
3 votes
1 answer
532 views

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 ...
tunar's user avatar
  • 511
0 votes
1 answer
1k 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 ...
vitsuk's user avatar
  • 77
6 votes
1 answer
1k 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?
vitsuk's user avatar
  • 77
1 vote
0 answers
81 views

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 ...
Boris's user avatar
  • 993
6 votes
1 answer
981 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}[...
amoeba's user avatar
  • 105k
1 vote
1 answer
1k 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 ...
Daschin's user avatar
  • 215
3 votes
0 answers
104 views

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 ...
David Wright's user avatar
  • 2,261
5 votes
2 answers
2k views

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 \...
a06e's user avatar
  • 4,420
2 votes
3 answers
940 views

How to obtain the 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 $p(x)$ ...
a06e's user avatar
  • 4,420
1 vote
0 answers
33 views

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 \...
R S's user avatar
  • 537
11 votes
1 answer
5k views

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 ...
bolbteppa's user avatar
  • 210
1 vote
0 answers
252 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 ...
Michael Hardy's user avatar
5 votes
1 answer
4k views

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), ...
mchen's user avatar
  • 770
16 votes
1 answer
694 views

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} \...
guy's user avatar
  • 8,902