# 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...